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Theorem conventions 21493
Description: Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they're identical):

  • Notation. Where possible, the notation attempts to conform to modern conventions, with variations due to our choice of the axiom system or to make proofs shorter. However, our notation is strictly sequential (left-to-right). For example, summation is written in the form  sum_ k  e.  A B (df-sum 12155) which denotes that index variable  k ranges over  A when evaluating  B. Thus,  sum_ k  e.  NN  ( 1  /  ( 2 ^ k ) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12334). Also, the method of definition, the axioms for predicate calculus, and the development of substitution are somewhat different from those found in standard texts. For example, the expressions for the axioms were designed for direct derivation of standard results without excessive use of metatheorems. (See Theorem 9.7 of [Megill] p. 448 for a rigorous justification.) The notation is usually explained in more detail when first introduced.
  • Axiomatic assertions ($a). All axiomatic assertions ($a statements) starting with "  |-" have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats $a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound, except for 4 (df-bi, df-cleq, df-clel, df-clab) that require a more complex metalogical justification by hand.
  • Proven axioms. In some cases we wish to treat an expression as an axiom in later theorems, even though it can be proved. For example, we derive the postulates or axioms of complex arithmetic as theorems of ZFC set theory. For convenience, after deriving the postulates we re-introduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. For more, see http://us.metamath.org/mpegif/mmcomplex.html. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is complex arithmetic axiom ax-1cn 8791, proven by the preceding theorem ax1cn 8767. The metamath.exe program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
  • Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
  • Statements with hypotheses. Many theorems and some axioms, such as ax-mp 8, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When presented in summarized form such as in the Theorem List (click on "Nearby theorems" on the ax-mp 8 page), the hypotheses are connected with an ampersand and separated from the conclusion with a big arrow, such as in "  |-  ph &  |-  ( ph  ->  ps ) =>  |-  ps". These symbols are not part of the Metamath language but are just informal notation meaning "and" and "implies".
  • Discouraged use and modification. If something should only be used in limited ways, it is marked with "(New usage is discouraged.)". This is used, for example, when something can be constructed in more than one way, and we do not want later theorems to depend on that specific construction. This marking is also used if we want later proofs to use proven axioms. For example, we want later proofs to use ax-1cn 8791 (not ax1cn 8767) and ax-1ne0 8802 (not ax1ne0 8778), as these are proven axioms for complex arithmetic. Thus, both ax1cn 8767 and ax1ne0 8778 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".
  • New definitions infrequent. Typically we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use  < and  <_ for inequality expressions, and use  ( ( sin `  ( _i  x.  A ) )  /  _i ) instead of  (sinh `  A ) for the hyperbolic sine.
  • Axiom of choice. The axiom of choice (df-ac 7739) is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 6977) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8057) or axiom of dependent choice (ax-dc 8068) can be used instead.
  • Variables. Typically, Greek letters ( ph = phi,  ps = psi,  ch = chi, etc.),... are used for propositional (wff) variables;  x,  y,  z,... for individual (set) variables; and  A,  B,  C,... for class variables.
  • Turnstile. " |-", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff  -.  ph".
  • Biconditional ( <->). There are basically two ways to maximize the effectiveness of biconditionals ( <->): you can either have one-directional simplifications of all theorems that produce biconditionals, or you can have one-directional simplifications of theorems that consume biconditionals. Some tools (like Lean) follow the first approach, but set.mm follows the second approach. Practically, this means that in set.mm, for every theorem that uses an implication in the hypothesis, like ax-mp 8, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 199 or mpbir 200. We prefer this second approach because the number of duplications in the second approach is bounded by the size of the propositional calculus section, which is much smaller than the number of possible theorems in all later sections that produce biconditionals. So although theorems like biimpi 186 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 886, sylbir 204, or 3imtr4i 257.
  • Substitution. " [ y  /  x ] ph" should be read "the wff that results from the proper substitution of  y for  x in wff  ph." See df-sb 1631 and the related df-sbc 2993 and df-csb 3083.
  • Is-a set. " A  e.  _V" should be read "Class  A is a set (i.e. exists)." This is a convenient convention based on Definition 2.9 of [Quine] p. 19. See df-v 2791 and isset 2793.
  • Converse. " `' R" should be read "converse of (relation)  R" and is the same as the more standard notation R^{-1} (the standard notation is ambiguous). See df-cnv 4696. This can be used to define a subset, e.g., df-tan 12349 notates "the set of values whose cosine is a nonzero complex number" as  ( `' cos " ( CC  \  { 0 } ) ).
  • Function application. "( F `  x)" should be read "the value of function  F at  x" and has the same meaning as the more familiar but ambiguous notation F(x). For example,  ( cos `  0 )  =  1 (see cos0 12426). 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. See df-fv 5229. In the ASCII (input) representation there are spaces around the grave accent; there is a single accent when it is used directly, and it is doubled within comments.
  • Infix and parentheses. When a function that takes two classes and produces a class is applied as part of an infix expression, the expression is always surrounded by parentheses (see df-ov 5823). For example, the  + in  ( 2  +  2 ); see 2p2e4 9838. Function application is itself an example of this. Similarly, predicate expressions in infix form that take two or three wffs and produce a wff are also always surrounded by parentheses, such as  ( ph  ->  ps ),  ( ph  \/  ps ),  ( ph  /\  ps ), and  ( ph  <->  ps ) (see wi 4, df-or 359, df-an 360, and df-bi 177 respectively). In contrast, a binary relation (which compares two classes and produces a wff) applied in an infix expression is not surrounded by parentheses. This includes set membership  A  e.  B (see wel 1686), equality  A  =  B (see df-cleq 2277), subset  A  C_  B (see df-ss 3167), and less-than  A  <  B (see df-lt 8746). For the general definition of a binary relation in the form  A R B, see df-br 4025. For example,  0  <  1 ( see 0lt1 9292) does not use parentheses.
  • Unary minus. The symbol  -u is used to indicate a unary minus, e.g.,  -u 1. It is specially defined because it is so commonly used. See cneg 9034.
  • Function definition. Functions are typically defined by first defining the constant symbol (using $c) and declaring that its symbol is a class with the label cNAME (e.g., ccos 12342). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 12348). Typically there are other proofs such as its closure labelled NAMEcl (e.g., coscl 12403), its function application form labelled NAMEval (e.g., cosval 12399), and at least one simple value (e.g., cos0 12426).
  • Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g.,  ( ! `  4 )  = ; 2 4 (df-fac 11285 and fac4 11292).
  • Unambiguous symbols. A given symbol has a single unambiguous meaning in general. Thus, where the literature might use the same symbol with different meanings, here we use different (variant) symbols for different meanings. These variant symbols often have suffixes, subscripts, or underlines to distinguish them. For example, here " 0" always means the value zero (df-0 8740), while " 0g" is the group identity element (df-0g 13400), " 0." is the poset zero (df-p0 14141), " 0 p" is the zero polynomial (df-0p 19021), " 0vec" is the zero vector in a normed complex vector space (df-0v 21148), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 14143). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including " .1.", " .+", " .*", and " .||". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use  NN (df-nn 9743) for the integer numbers starting from 1, and  NN0 (df-n0 9962) for the set of nonnegative integers starting at zero.
  • Decimal numbers. Numbers larger than ten are often expressed in base 10 using the decimal constructor df-dec 10121, e.g., ;;; 4 0 0 1 (see 4001prm 13139 for a proof that 4001 is prime).
  • Theorem forms. We often call a theorem a "deduction" whenever the conclusion and all hypotheses are each prefixed with the same antecedent  ph  ->. Deductions are often the preferred form for theorems because they allow us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used. See, for example, a1d 22. A deduction hypothesis can have an indirect antecedent via definitions, e.g., see lhop 19359. Deductions have a label suffix of "d" if there are other forms of the same theorem. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 10. Finally, a "tautology" would be the form with no hypotheses, and its label would have no suffix. For example, see pm2.43 47, pm2.43i 43, and pm2.43d 44.
  • Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see http://us.metamath.org/mpegif/mmdeduction.html. The Deduction Theorem is a metalogical theorem that cannot be applied directly in metamath, and the explosion of steps would be a problem anyway, so alternatives are used. One alternative we use sometimes is the "weak deduction theorem" dedth 3607, which works in certain cases in set theory. We also sometimes use dedhb 2936. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in the deduction theorem form (aka "deduction style") described earlier; the prefixed  ph  -> mimics the context in a deduction proof system. In practice this mechanism works very well. This approach is described in the deduction form and natural deduction page; a list of translations for common natural deduction rules is given in natded 21494.
  • Recursion. We define recursive functions using various "recursion constructors". These allow us to define, with compact direct definitions, functions that are usually defined in textbooks with indirect self-referencing recursive definitions. This produces compact definition and much simpler proofs, and greatly reduces the risk of creating unsound definitions. Examples of recursion constructors include recs ( F ) in df-recs 6384,  rec ( F ,  I ) in df-rdg 6419, seq𝜔 ( F ,  I ) in df-seqom 6456, and  seq  M (  .+  ,  F ) in df-seq 11043. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 13401 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 6384, but for the "average user" the most useful one is probably df-seq 11043- provided that a countable sequence is sufficient for the recursion.
  • Extensible structures. Mathematics includes many structures such as ring, group, poset, etc. We define an "extensible structure" which is then used to define group, ring, poset, etc. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them. See df-struct 13146.
  • Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 6967 is the first lemma for sbth 6977. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
  • Hypertext links. We strongly encourage comments to have many links to related material, with accompanying text that explains the relationship. These can help readers understand the context. Links to other statements, or to HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a space-separated tilde. When metamath.exe is used to generate HTML it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertion.
  • Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g. mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we require page numbers in comments for individual $a or $p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the metamath.exe "write bibliography" command.)
  • Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
  • Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see http://us.metamath.org/mpegif/mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.

Naming conventions

Every statement has a unique identifying label. We use various label naming conventions to provide easy-to-remember hints about their contents. Labels are not a 1-to-1 mapping, because that would create long names that would be difficult to remember and tedious to type. Instead, label names are relatively short while suggesting their purpose. Names are occasionally changed to make them more consistent or as we find better ways to name them. Here are a few of the label naming conventions:

  • Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
  • Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id.
  • Common names. If a theorem has a well-known name, that name (or a short version of it) is sometimes used directly. Examples include barbara 2241 and stirling 27249.
  • Syntax label fragments. Most theorems are named using syntax label fragments. Almost every syntactic construct has a definition labelled "df-NAME", and NAME normally is the syntax label fragment. For example, the difference construct  ( A  \  B ) is defined in df-dif 3156, and thus its syntax label fragment is "dif". Similarly, the singleton construct  { A } has syntax label fragment "sn" (because it is defined in df-sn 3647), the subclass (subset) relation  A  C_  B has "ss" (because it is defined in df-ss 3167), and the pair construct  { A ,  B } has "pr" (df-pr 3648). Theorem names are often a concatenation of the syntax label fragments (omitting variables). For example, a theorem about  ( A  \  B )  C_  A involves a difference ("dif") of a subset ("ss"), and thus is named difss 3304. Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
  • Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct  A  e.  B does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with  ( A  e.  ( B  \  { C } ) uses is-element-of ("el") of a difference ("dif") of a singleton ("sn"), it is named eldifsn 3750. An "n" is often used for negation ( -.), e.g., nan 563.
  • Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The table below attempts to list all such cases and marks them in bold. For example, label fragment "cn" represents complex numbers  CC (even though its definition is in df-c 8739) and "re" represents real numbers  RR. The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3457. Syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 8744), but "p" is used as the fragment for constant theorems. Equality  ( A  =  B ) often uses "e" as the fragment. As a result, "two plus two equals four" is named 2p2e4 9838.
  • Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
  • Principia Mathematica. Proofs of theorems from Principia Mathematica often use a different naming convention. They are instead often named "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 35.
  • Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value named "NAMEval" and its closure named "NAMEcl". E.g., for cosine (df-cos 12348) we have value cosval 12399 and closure coscl 12403.
  • Special cases. Sometimes syntax and related markings are insufficient to distinguish different theorems. For example, there are over 100 different implication-exclusive theorems. These are then grouped in a more ad-hoc way that attempts to make their distinctions clearer. These often use abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and "id" for "identity". It's especially hard to give good names in the propositional calculus section because there are so few primitives. However, in most cases this is not a serious problem. There are a few very common theorems like ax-mp 8 and syl 15 that you will have no trouble remembering, a few theorem series like syl*anc and simp* that you can use parametrically, and a few other useful glue things for destructuring 'and's and 'or's (see natded 21494 for a list), and that's about all you need for most things. As for the rest, you can just assume that if it involves three or less connectives we probably already have a proof, and searching for it will give you the name.
  • Suffixes. We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1793 via the use of distinct variable conditions combined with nfv 1605. 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 2150 derived from df-eu 2148. The "f" stands for "not free in" which is less restrictive than "does not occur in." 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 15) -type inference in a proof. When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the theorem form with "g" (for "more general") as in uniex 4515 vs. uniexg 4516. A theorem name is suffixed with "ALT" if it's an alternative less-preferred proof of a theorem.
  • Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly-used abbreviations in labels, 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. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
addadd (see "p") df-add 8744  ( A  +  B ) Yes addcl 8815, addcom 8994, addass 8820
ALTalternative/less preferred (suffix) No
anand df-an 360  ( ph  /\  ps ) Yes anor 475, iman 413, imnan 411
assassociative No biass 348, orass 510, mulass 8821
bibiconditional df-bi 177  ( ph  <->  ps ) Yes impbid 183
cncomplex numbers df-c 8739  CC Yes nnsscn 9747, nncn 9750
comcommutative No orcom 376, bicomi 193, eqcomi 2288
ddeduction form No idd 21, impbid 183
di, distrdistributive No andi 837, imdi 352, ordi 834, difindi 3424, ndmovdistr 5971
difdifference df-dif 3156  ( A  \  B ) Yes difss 3304, difindi 3424
divdivision df-div 9420  ( A  /  B ) Yes divcl 9426, divval 9422, divmul 9423
e, eqequals df-cleq 2277  A  =  B Yes 2p2e4 9838, uneqri 3318
elelement of  A  e.  B Yes eldif 3163, eldifsn 3750, elssuni 3856
f"not free in" (suffix) No
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 4516
ididentity No
idmidempotent No anidm 625, tpidm13 3730
im, impimplication (label often omitted) df-im 11582  ( A  ->  B ) Yes iman 413, imnan 411, impbidd 181
inintersection df-in 3160  ( A  i^i  B ) Yes elin 3359, incom 3362
is...is (something a) ...? No isrng 15341
mpmodus ponens ax-mp 8 No mpd 14, mpi 16
mulmultiplication (see "t") df-mul 8745  ( A  x.  B ) Yes mulcl 8817, divmul 9423, mulcom 8819, mulass 8821
n, notnot  -.  ph Yes nan 563, notnot2 104
ne0not equal to zero (see n0)  =/=  0 No negne0d 9151, ine0 9211, gt0ne0 9235
nnnatural numbers df-nn 9743  NN Yes nnsscn 9747, nncn 9750
n0not the empty set (see ne0)  =/=  (/) No n0i 3461, vn0 3463, ssn0 3488
oror df-or 359  ( ph  \/  ps ) Yes orcom 376, anor 475
pplus (see "add"), for all-constant theorems df-add 8744  ( 3  +  2 )  =  5 Yes 3p2e5 9851
pmPrincipia Mathematica No pm2.27 35
prpair df-pr 3648  { A ,  B } Yes elpr 3659, prcom 3706, prid1g 3733, prnz 3746
q  QQ (quotients) df-q 10313  QQ Yes elq 10314
rereal numbers df-r 8743  RR Yes recn 8823, 0re 8834
rngring df-rng 15336  Ring Yes rngidval 15339, isrng 15341, rnggrp 15342
rotrotation No 3anrot 939, 3orrot 940
seliminates need for syllogism (suffix) No
snsingleton df-sn 3647  { A } Yes eldifsn 3750
sssubset df-ss 3167  A  C_  B Yes difss 3304
subsubtract df-sub 9035  ( A  -  B ) Yes subval 9039, subaddi 9129
sylsyllogism syl 15 No 3syl 18
t times (see "mul"), for all-constant theorems df-mul 8745  ( 3  x.  2 )  =  6 Yes 3t2e6 9868
tptriple df-tp 3649  { A ,  B ,  C } Yes eltpi 3678, tpeq1 3716
ununion df-un 3158  ( A  u.  B ) Yes uneqri 3318, uncom 3320
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 1933
xreXtended reals df-xr 8867  RR* Yes ressxr 8872, rexr 8873, 0xr 8874
z  ZZ (integers, from German Zahlen) df-z 10021  ZZ Yes elz 10022, zcn 10025
0, z slashed zero (empty set) (see n0) df-nul 3457  (/) Yes n0i 3461, vn0 3463; snnz 3745, prnz 3746

Distinctness or freeness

Here are some conventions that address distinctness or freeness of a variable:

  •  F/ x ph is read "  x is not free in (wff)  ph"; see df-nf 1532 (whose description has some important technical details). Similarly,  F/_ x A is read  x is not free in (class)  A, see df-nfc 2409.
  • "$d x y $." should be read "Assume x and y are distinct variables."
  • "$d x  ph $." should be read "Assume x does not occur in phi $." Sometimes a theorem is proved using  F/ x ph (df-nf 1532) in place of "$d  x ph $." when a more general result is desired; ax-17 1603 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2150 from df-eu 2148.
  • "$d x A $." should be read "Assume x is not a variable occurring in class A."
  • "$d x A $. $d x ps $. $e |-  ( x  =  A  ->  ( ph  <->  ps ) ) $." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
  • "  |-  ( -.  A. x x  =  y  ->  ..." occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the $d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the $d eventually becomes a requirement, and after that this device is rarely used.

Here is more information about our processes for checking and contributing to this work:

  • Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
  • Rewrapped line length. The input file routinely has its text wrapped using metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' (so please do the same).
  • Discouraged information. A separate file named "discouraged" lists all discouraged statements and uses of them, and this file is checked. If you change the use of discouraged things, you will need to change this file. This makes it obvious when there is a change to anything discouraged (triggering further review).
  • Proposing specific changes. Please propose specific changes as pull requests (PRs) against the "develop" branch of set.mm, at: https://github.com/metamath/set.mm/tree/develop
  • Community. We encourage anyone interested in Metamath to join our mailing list: https://groups.google.com/forum/#!forum/metamath.

(Contributed by DAW, 27-Dec-2016.)

Hypothesis
Ref Expression
conventions.1  |-  ph
Assertion
Ref Expression
conventions  |-  ph

Proof of Theorem conventions
StepHypRef Expression
1 conventions.1 1  |-  ph
Colors of variables: wff set class
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