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Theorem conventions 20789
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 12159) 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 12338). 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/mpeuni/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 8795, proven by the preceding theorem ax1cn 8771. 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 8795 (not ax1cn 8771) and ax-1ne0 8806 (not ax1ne0 8782), as these are proven axioms for complex arithmetic. Thus, both ax1cn 8771 and ax1ne0 8782 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 7743) 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 6981) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8061) or axiom of dependent choice (ax-dc 8072) 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 1630 and the related df-sbc 2992 and df-csb 3082.
  • 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 2790 and isset 2792.
  • 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 4697. This can be used to define a subset, e.g., df-tan 12353 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 12430). 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 5263. 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 5861). For example, the  + in  ( 2  +  2 ); see 2p2e4 9842. 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 1685), equality  A  =  B (see df-cleq 2276), subset  A  C_  B (see df-ss 3166), and less-than  A  <  B (see df-lt 8750). For the general definition of a binary relation in the form  A R B, see df-br 4024. For example,  0  <  1 ( see 0lt1 9296) 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 9038.
  • 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 12346). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 12352). Typically there are other proofs such as its closure labelled NAMEcl (e.g., coscl 12407), its function application form labelled NAMEval (e.g., cosval 12403), and at least one simple value (e.g., cos0 12430).
  • 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 11289 and fac4 11296).
  • 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 8744), while " 0g" is the group identity element (df-0g 13404), " 0." is the poset zero (df-p0 14145), " 0 p" is the zero polynomial (df-0p 19025), " 0vec" is the zero vector in a normed complex vector space (df-0v 21154), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 14147). 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 9747) for the integer numbers starting from 1, and  NN0 (df-n0 9966) 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 10125, e.g., ;;; 4 0 0 1 (see 4001prm 13143 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 19363. 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/mpeuni/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 3606, which works in certain cases in set theory. We also sometimes use dedhb 2935. 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 20790.
  • 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 6388,  rec ( F ,  I ) in df-rdg 6423, seq𝜔 ( F ,  I ) in df-seqom 6460, and  seq  M (  .+  ,  F ) in df-seq 11047. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 13405 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 6388, but for the "average user" the most useful one is probably df-seq 11047- 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 13150.
  • 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 6971 is the first lemma for sbth 6981. 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/mpeuni/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 2240 and stirling 27838.
  • 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 3155, 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 3646), the subclass (subset) relation  A  C_  B has "ss" (because it is defined in df-ss 3166), and the pair construct  { A ,  B } has "pr" (df-pr 3647). 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 3303. 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 3749. 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 8743) and "re" represents real numbers  RR. The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3456. Syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 8748), 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 9842.
  • 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 12352) we have value cosval 12403 and closure coscl 12407.
  • 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 20790 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 1791 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 2149 derived from df-eu 2147. 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 4516 vs. uniexg 4517. 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 8748  ( A  +  B ) Yes addcl 8819, addcom 8998, addass 8824
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 8825
bibiconditional df-bi 177  ( ph  <->  ps ) Yes impbid 183
cncomplex numbers df-c 8743  CC Yes nnsscn 9751, nncn 9754
comcommutative No orcom 376, bicomi 193, eqcomi 2287
ddeduction form No idd 21, impbid 183
di, distrdistributive No andi 837, imdi 352, ordi 834, difindi 3423, ndmovdistr 6009
difdifference df-dif 3155  ( A  \  B ) Yes difss 3303, difindi 3423
divdivision df-div 9424  ( A  /  B ) Yes divcl 9430, divval 9426, divmul 9427
e, eqequals df-cleq 2276  A  =  B Yes 2p2e4 9842, uneqri 3317
elelement of  A  e.  B Yes eldif 3162, eldifsn 3749, elssuni 3855
f"not free in" (suffix) No
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 4517
ididentity No
idmidempotent No anidm 625, tpidm13 3729
im, impimplication (label often omitted) df-im 11586  ( A  ->  B ) Yes iman 413, imnan 411, impbidd 181
inintersection df-in 3159  ( A  i^i  B ) Yes elin 3358, incom 3361
is...is (something a) ...? No isrng 15345
mpmodus ponens ax-mp 8 No mpd 14, mpi 16
mulmultiplication (see "t") df-mul 8749  ( A  x.  B ) Yes mulcl 8821, divmul 9427, mulcom 8823, mulass 8825
n, notnot  -.  ph Yes nan 563, notnot2 104
ne0not equal to zero (see n0)  =/=  0 No negne0d 9155, ine0 9215, gt0ne0 9239
nnnatural numbers df-nn 9747  NN Yes nnsscn 9751, nncn 9754
n0not the empty set (see ne0)  =/=  (/) No n0i 3460, vn0 3462, ssn0 3487
oror df-or 359  ( ph  \/  ps ) Yes orcom 376, anor 475
pplus (see "add"), for all-constant theorems df-add 8748  ( 3  +  2 )  =  5 Yes 3p2e5 9855
pmPrincipia Mathematica No pm2.27 35
prpair df-pr 3647  { A ,  B } Yes elpr 3658, prcom 3705, prid1g 3732, prnz 3745
q  QQ (quotients) df-q 10317  QQ Yes elq 10318
rereal numbers df-r 8747  RR Yes recn 8827, 0re 8838
rngring df-rng 15340  Ring Yes rngidval 15343, isrng 15345, rnggrp 15346
rotrotation No 3anrot 939, 3orrot 940
seliminates need for syllogism (suffix) No
snsingleton df-sn 3646  { A } Yes eldifsn 3749
sssubset df-ss 3166  A  C_  B Yes difss 3303
subsubtract df-sub 9039  ( A  -  B ) Yes subval 9043, subaddi 9133
sylsyllogism syl 15 No 3syl 18
t times (see "mul"), for all-constant theorems df-mul 8749  ( 3  x.  2 )  =  6 Yes 3t2e6 9872
tptriple df-tp 3648  { A ,  B ,  C } Yes eltpi 3677, tpeq1 3715
ununion df-un 3157  ( A  u.  B ) Yes uneqri 3317, uncom 3319
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 1930
xreXtended reals df-xr 8871  RR* Yes ressxr 8876, rexr 8877, 0xr 8878
z  ZZ (integers, from German Zahlen) df-z 10025  ZZ Yes elz 10026, zcn 10029
0, z slashed zero (empty set) (see n0) df-nul 3456  (/) Yes n0i 3460, vn0 3462; snnz 3744, prnz 3745

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 2408.
  • "$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 $e version, see the proof of euf 2149 from df-eu 2147.
  • "$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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