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Theorem List for Metamath Proof Explorer - 25901-26000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrgrareg 25901 If a finite friendship graph is k-regular, then k must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( ( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( K  =  0  \/  K  =  2 ) ) )
 
Theoremfrgraregord013 25902 If a finite friendship graph is k-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K ) 
 ->  ( ( # `  V )  =  0  \/  ( # `  V )  =  1  \/  ( # `
  V )  =  3 ) )
 
Theoremfrgraregord13 25903 If a nonempty finite friendship graph is k-regular, then it must have order 1 or 3. Special case of frgraregord013 25902. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( ( # `  V )  =  1  \/  ( # `  V )  =  3 )
 )
 
Theoremfrgraogt3nreg 25904* If a finite friendship graph has an order greater than 3, it cannot be k-regular for any k. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  ->  A. k  e.  NN0  -.  <. V ,  E >. RegUSGrph  k )
 
Theoremfriendshipgt3 25905* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E )
 
Theoremfriendship 25906* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  =/=  (/)  /\  V  e.  Fin )  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E )
 
PART 17  GUIDES AND MISCELLANEA
 
17.1  Guides (conventions, explanations, and examples)
 
17.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. For the general conventions, see conventions 25907, and for conventions related to labels, see conventions-label 25908. Logic and set theory provide a foundation for all of mathematics. To learn about them, you should study one or more of the references listed below. We indicate references using square brackets. 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:

  • Axioms of propositional calculus - [Margaris].
  • Axioms of predicate calculus - [Megill] (System S3' in the article referenced).
  • Theorems of propositional calculus - [WhiteheadRussell].
  • Theorems of pure predicate calculus - [Margaris].
  • Theorems of equality and substitution - [Monk2], [Tarski], [Megill].
  • Axioms of set theory - [BellMachover].
  • Development of set theory - [TakeutiZaring]. (The first part of [Quine] has a good explanation of the powerful device of "virtual" or class abstractions, which is essential to our development.)
  • Construction of real and complex numbers - [Gleason]
  • Theorems about real numbers - [Apostol]
 
Theoremconventions 25907

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 are identical). For conventions related to labels, see conventions-label 25908.

  • 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 13808) 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 13993). 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 as discussed below, except for 4 definitions (df-bi 190, df-cleq 2455, df-clel 2458, df-clab 2449) 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 reintroduce 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 9628, proven by the preceding theorem ax1cn 9604. 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 5, 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 5 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 9628 (not ax1cn 9604) and ax-1ne0 9639 (not ax1ne0 9615), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9604 and ax1ne0 9615 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.
  • Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, we prefer proofs that do not use the axiom of choice (df-ac 8578) where such proofs can be found. The axiom of choice 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 7723) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8896) or axiom of dependent choice (ax-dc 8907) can be used instead.
  • Alternative (ALT) proofs. If a different proof is significantly shorter or clearer but uses more or stronger axioms, we prefer to make that proof an "alternative" proof (marked with an ALT label suffix), even if this alternative proof was formalized first. We then make the proof that requires fewer axioms the main proof. This has the effect of reducing (over time) the number and strength of axioms used by any particular proof. There can be multiple alternatives if it makes sense to do so. Alternative (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternative and main statement comments should use hyperlinks to refer to each other (so that a reader of one will become easily aware of the other).
  • Alternative (ALTV) versions. If a theorem or definition is an alternative/variant of an already existing theorem resp. definition, its label should have the same name with suffix ALTV. Such alternatives should be temporary only, until it is decided which alternative should be used in the future. Alternative (*ALTV) theorems or definitions are usually contained in mathboxes. Their comments need not to contain "(Proof modification is discouraged.) (New usage is discouraged.)". Alternative statements should follow the main statement, so that people reading the text in order will see the main statement first.
  • Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first. This typically happens when a shorter proof to an existing theorem is found: the existing theorem is kept as an *OLD statement for one year. When a proof is shortened automatically (using Metamath's minimize_with command), then it is not necessary to keep the old proof, nor to add credit for the shortening.
  • Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are normally used in this order:  ph = phi,  ps = psi,  ch = chi,  th = theta,  ta = tau,  et = eta,  ze = zeta, and  si = sigma. Individual setvar variables are represented with lowercase Latin letters and are normally used in this order:  x,  y,  z,  w,  v,  u, and  t. Variables that represent classes are often represented by uppercase Latin letters:  A,  B,  C,  D,  E, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g.,  .0. for poset zero (see p0val 16342) and connective symbols such as  .+ for some group addition operation. (See prdsplusgval 15426 for an example of the use of  .+). Class variables are selected in alphabetical order starting from  A if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.
  • 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 5, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 213 or mpbir 214. 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 199 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 936, sylbir 218, or 3imtr4i 274.
  • 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 1809 and the related df-sbc 3280 and df-csb 3376.
  • Is-a-set. " A  e.  _V" should be read "Class  A is a set (i.e. exists)." This is a convention based on Definition 2.9 of [Quine] p. 19. See df-v 3059 and isset 3061. However, instead of using  I  e.  _V in the antecedent of a theorem for some variable  I, we now prefer to use  I  e.  V (or another variable if  V is not available) to make it more general. That way we can often avoid needing extra uses of elex 3066 and syl 17 in the common case where  I is already a member of something.
  • 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 4864. This can be used to define a subset, e.g., df-tan 14180 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 14259). 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 5613. 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 6323). For example, the  + in  ( 2  +  2 ); see 2p2e4 10761. 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 376, df-an 377, and df-bi 190 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 1899), equality  A  =  B (see df-cleq 2455), subset  A  C_  B (see df-ss 3430), and less-than  A  <  B (see df-lt 9583). For the general definition of a binary relation in the form  A R B, see df-br 4419. For example,  0  <  1 (see 0lt1 10169) 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 9892.
  • 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 14172). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 14179). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 14236), its function application form labeled NAMEval (e.g., cosval 14232), and at least one simple value (e.g., cos0 14259).
  • 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 12498 and fac4 12505).
  • 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 9577), while " 0g" is the group identity element (df-0g 15395), " 0." is the poset zero (df-p0 16340), " 0p" is the zero polynomial (df-0p 22684), " 0vec" is the zero vector in a normed complex vector space (df-0v 26273), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 16342). 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.
  • ASCII representation of symbols. We must have an ASCII representation for each symbol. We generally choose short sequences, ideally digraphs, and generally choose sequences that vaguely resemble the mathematical symbol. Here are some of the conventions we use when selecting an ASCII representation.
    We generally do not include parentheses inside a symbol because that confuses text editors (such as emacs). Greek letters for wff variables always use the first two letters of their English names, making them easy to type and easy to remember. Symbols that almost look like letters, such as  A., are often represented by that letter followed by a period. For example, "A." is used to represent  A., "e." is used to represent  e., and "E." is used to represent  E.. Single letters are now always variable names, so constants that are often shown as single letters are now typically preceded with "_" in their ASCII representation, for example, "_i" is the ASCII representation for the imaginary unit  _i. A script font constant is often the letter preceded by "~" meaning "curly", such as "~P" to represent the power class  ~P.
    Originally, all setvar and class variables used only single letters a-z and A-Z, respectively. A big change in recent years was to allow the use of certain symbols as variable names to make formulas more readable, such as a variable representing an additive group operation. The convention is to take the original constant token (in this case "+" which means complex number addition) and put a period in front of it to result in the ASCII representation of the variable ".+", shown as  .+, that can be used instead of say the letter "P" that had to be used before.
    Choosing tokens for more advanced concepts that have no standard symbols but are represented by words in books, is hard. A few are reasonably obvious, like "Grp" for group and "Top" for topology, but often they seem to end up being either too long or too cryptic. It would be nice if the math community came up with standardized short abbreviations for English math terminology, like they have more or less done with symbols, but that probably won't happen any time soon.
    Another informal convention that we've somewhat followed, that is also not uncommon in the literature, is to start tokens with a capital letter for collection-like objects and lower case for function-like objects. For example, we have the collections On (ordinal numbers), Fin, Prime, Grp, and we have the functions sin, tan, log, sup. Predicates like Ord and Lim also tend to start with upper case, but in a sense they are really collection-like, e.g. Lim indirectly represents the collection of limit ordinals, but it can't be an actual class since not all limit ordinals are sets. This initial capital vs. lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).
  • Typography conventions. Class symbols for functions (e.g.,  abs,  sin) should usually not have leading or trailing blanks in their HTML/Latex representation. This is in contrast to class symbols for operations (e.g.,  gcd, sadd, eval), which usually do include leading and trailing blanks in their representation. If a class symbol is used for a function as well as an operation (according to the definition df-ov 6323, each operation value can be written as function value of an ordered pair), the convention for its primary usage should be used, e.g.  (iEdg `  G ) versus  ( ViEdg E ) for the edges of a graph  G  =  <. V ,  E >..
  • Number construction independence. There are many ways to model complex numbers. After deriving the complex number postulates we reintroduce 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. This also lets us be independent of the specific construction, which we believe is valuable. See mmcomplex for details. Thus, for example, we don't allow the use of  (/)  e/  CC, as handy as that would be, because that would be construction-specific. We want proofs about  CC to be independent of whether or not  (/)  e.  CC.
  • Minimize hypotheses (except for construction independence and number theorem domains). In most cases we try to minimize hypotheses, that is, we eliminate or reduce what must be true to prove something, so that the proof is more general and easier to use. There are exceptions. For example, we intentionally add hypotheses if they help make proofs independent of a particular construction (e.g., the contruction of complex numbers  CC). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they aren't strictly needed. For example, we could show that  ( A  <  B  ->  B  =/=  A ) without any other hypotheses, but in practice we also require proving at least some domains (e.g., see ltnei 9789). Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ :
    1. Having the hypotheses immediately shows the intended domain of applicability (is it  RR,  RR*,  om, or something else?), without having to trace back to definitions.
    2. Having the hypotheses forces its use in the intended domain, which generally is desirable.
    3. The behavior is dependent on accidental behavior of definitions outside of their domains, so the theorems are non-portable and "brittle".
    4. Only a few theorems can have their hypotheses removed in this fashion due to happy coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who hasn't traced back the set-theoretical foundations of the definitions, it is seemingly random and isn't intuitive at all.
    5. The consensus of opinion of people on this group seemed to be against doing this.
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use  NN (df-nn 10643) for the set of positive integers starting from 1, and  NN0 (df-n0 10904) 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 11086, e.g., ;;; 4 0 0 1 (see 4001prm 15171 for a proof that 4001 is prime).
  • Theorem forms. We will use the following descriptive terms to categorize theorems:
    • A theorem is in "closed form" if it has no $e hypotheses (e.g., unss 3620). The term "tautology" is also used, especially in propositional calculus. This form was formerly called "theorem form" or "closed theorem form".
    • A theorem is in "deduction form" (or is a "deduction") if it has one or more $e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually  ph), and every hypothesis ($e statement) is either:
      1. an implication with the same antecedent as the conclusion, or
      2. a definition. A definition can be for a class variable (this is a class variable followed by  =, e.g. the definition of  D in lhop 23024) or a wff variable (this is a wff variable followed by  <->); class variable definitions are more common.
      In practice, a proof of a theorem in deduction form will also contain many steps that are implications where the antecedent is either that wff variable (usually  ph) or is a conjunction  ( ph  i^i  ... ) including that wff variable ( ph). E.g. a1d 26, unssd 3622.
    • A theorem is in "inference form" (or is an "inference") if it has one or more $e hypotheses, but is not in deduction form, i.e. there is no common antecedent (e.g., unssi 3621).
    Any theorem whose conclusion is an implication has an associated inference, whose hypotheses are the hypotheses of that theorem together with the antecedent of its conclusion, and whose conclusion is the consequent of that conclusion. When both theorems are in set.mm, then the associated inference is often labeled by adding the suffix "i" to the label of the original theorem (for instance, con3i 142 is the inference associated with con3 141). The inference associated with a theorem is easily derivable from that theorem by a simple use of ax-mp 5. The other direction is the subject of the Deduction Theorem discussed below. We may also use the term "associated inference" when the above process is iterated. For instance, syl 17 is an inference associated with imim1 79 because it is the inference associated with imim1i 60 which is itself the inference associated with imim1 79.
    "Deduction form" is the preferred form for theorems because this form allows us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that would end up requiring us to use the deduction theorem.
    Deductions have a label suffix of "d", especially if there are other forms of the same theorem (e.g., pm2.43d 50). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 49). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 53). When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the closed form with "g" (for "more general") as in uniex 6619 vs. uniexg 6620.
  • 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 (its associated inference). See for instance Theorem 3 in [Margaris] p. 56. 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 3944, which works in certain cases in set theory. We also sometimes use dedhb 3220. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in deduction form (aka "deduction style") as 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 25909.
  • 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 7121,  rec ( F ,  I ) in df-rdg 7159, seq𝜔 ( F ,  I ) in df-seqom 7196, and  seq M (  .+  ,  F ) in df-seq 12252. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 15396 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7121, but for the "average user" the most useful one is probably df-seq 12252- 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 15178.
  • Undefined results and "junk theorems". Some expressions are only expected to be meaningful in certain contexts. For example, consider Russell's definition description binder iota, where  ( iota x ph ) is meant to be "the  x such that  ph" (where  ph typically depends on x). What should that expression produce when there is no such  x? In set.mm we primarily use one of two approaches. One approach is to make the expression evaluate to the empty set whenever the expression is being used outside of its expected context. While not perfect, it makes it a bit more clear when something is undefined, and it has the advantage that it makes more things equal outside their domain which can remove hypotheses when you feel like exploiting these so-called junk theorems. Note that Quine does this with iota (his definition of iota evaluates to the empty set when there is no unique value of  x). Quine has no problem with that and we don't see why we should, so we define iota exactly the same way that Quine does. The main place where you see this being systematically exploited is in "reverse closure" theorems like  A  e.  ( F `  B )  ->  B  e.  dom  F, which is useful when  F is a family of sets. (by this we mean it's a set set even in a type theoretic interpretation.) The second approach uses "(New usage is discouraged.)" to prevent unintentional uses of certain properties. For example, you could define some construct df-NAME whose usage is discouraged, and prove only the specific properties you wish to use (and add those proofs to the list of permitted uses of "discouraged" information). From then on, you can only use those specific properties without a warning. Other approaches often have hidden problems. For example, you could try to "not define undefined terms" by creating definitions like ${ $d  y x $. $d  y ph $. df-iota $a  |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } ) $. $}. This will be rejected by the definition checker, but the bigger theoretical reason to reject this axiom is that it breaks equality - the metatheorem  ( x  =  y  -> P(x)  = P(y)  ) fails to hold if definitions don't unfold without some assumptions. (That is, iotabidv 5590 is no longer provable and must be added as an axiom.) It is important for every syntax constructor to satisfy equality theorems *unconditionally*, e.g., expressions like  ( 1  /  0 )  =  ( 1  /  0 ) should not be rejected. This is forced on us by the context free term language, and anything else requires a lot more infrastructure (e.g., a type checker) to support without making everything else more painful to use. Another approach would be to try to make nonsensical statements syntactically invalid, but that can create its own complexities; in some cases that would make parsing itself undecidable. In practice this does not seem to be a serious issue. No one does these things deliberately in "real" situations, and some knowledgeable people (such as Mario Carneiro) have never seen this happen accidentally. Norman Megill doesn't agree that these "junk" consequences are necessarily bad anyway, and they can significantly shorten proofs in some cases. This database would be much larger if, for example, we had to condition fvex 5902 on the argument being in the domain of the function. It is impossible to derive a contradiction from sound definitions (i.e. that pass the definition check), assuming ZFC is consistent, and he doesn't see the point of all the extra busy work and huge increase in set.mm size that would result from restricting *all* definitions. So instead of implementing a complex system to counter a problem that does not appear to occur in practice, we use a significantly simpler set of approaches.
  • 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 7713 is the first lemma for sbth 7723. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
  • Limit proof size. It is often preferable to break longer proofs into smaller parts, just as you would do with traditional proofs. One reason is that humans have trouble understanding long proofs. Another reason is that it's generally best to prove reusable results separately, so that others will be able to easily reuse them. Finally, the "minimize" routine can take much longer with very long proofs. We encourage proofs to be no more than 200 essential steps, and generally no more than 500 essential steps, though these are simply guidelines and not hard-and-fast rules. Much smaller proofs are fine! We also acknowledge that some proofs, especially autogenerated ones, should sometimes not be broken up (e.g., because breaking them up might be useless and inefficient due to many interconnections and reused terms within the proof). In Metamath, breaking up longer proofs is done by creating multiple separate proofs of 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 7713 is the first lemma for sbth 7723.
  • 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 (e.g., " ~ df-prm " results in " df-prm 14678"). 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.
  • Hypertext links to section headers. Some section headers have text under them that describes or explains the section. However, they are not part of the description of axioms or theorems, and there is no way to link to them directly. To provide for this, section headers with accompanying text (indicated with "*" prefixed to mmtheorems.html#mmdtoc entries) have an anchor in mmtheorems.html whose name is the first $a or $p statement that follows the header. For example there is a glossary under the section heading called GRAPH THEORY. The first $a or $p statement that follows is cuhg 25073, which you can see two lines down. To reference it we link to the anchor using a space-separated tilde followed by the space-separated link mmtheorems.html#cuhg, which will become the hyperlink mmtheorems.html#cuhg. Note that no theorem in set.mm is allowed to begin with "mm" (enforced by "verify markup" in the metamath program). Whenever the software sees a tilde reference beginning with "http:", "https:", or "mm", the reference is assumed to be a link to something other than a statement label, and the tilde reference is used as is. This can also be useful for relative links to other pages such as mmcomplex.html.
  • 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.)
  • Acceptable shorter proofs Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem's description. However, in some cases a proof may be "shorter" or not depending on how it is formatted. This section provides general guidelines.

    Usually we automatically accept shorter proofs that (1) shorten the set.mm file (with compressed proofs), (2) reduce the size of the HTML file generated with SHOW STATEMENT xx / HTML, (3) use only existing, unmodified theorems in the database (the order of theorems may be changed, though), and (4) use no additional axioms. Usually we will also automatically accept a _new_ theorem that is used to shorten multiple proofs, if the total size of set.mm (including the comment of the new theorem, not including the acknowledgment) decreases as a result.

    In borderline cases, we typically place more importance on the number of compressed proof steps and less on the length of the label section (since the names are in principle arbitrary). If two proofs have the same number of compressed proof steps, we will typically give preference to the one with the smaller number of different labels, or if these numbers are the same, the proof with the fewest number of characters that the proofs happen to have by chance when label lengths are included.

    A few theorems have a longer proof than necessary in order to avoid the use of certain axioms, for pedagogical purposes, and for other reasons. These theorems will (or should) have a "(Proof modification is discouraged.)" tag in their description. For example, idALT 23 shows a proof directly from axioms. Shorter proofs for such cases won't be accepted, of course, unless the criteria described continues to be satisfied.

  • 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.
  • Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.
  • Language and spelling. It is preferred to use American English for comments and symbols, e.g. we use "neighborhood" instead of the British English "neighbourhood". An exception is the word "analog", which can be either a noun or an adjective. Furthermore, "analog" has the confounding meaning "not digital", whereas "analogue" is often used in the sense something that bears analogy to something else also in American English. Therefore, "analogue" is used for the noun and "analogous" for the adjective in set.mm.


The challenge of varying mathematical conventions

We try to follow mathematical conventions, but in many cases different texts use different conventions. In those cases we pick some reasonably common convention and stick to it. We have already mentioned that the term "natural number" has varying definitions (some start from 0, others start from 1), but that is not the only such case. A useful example is the set of metavariables used to represent arbitrary well-formed formulas (wffs). We use an open phi, φ, to represent the first arbitrary wff in an assertion with one or more wffs; this is a common convention and this symbol is easily distinguished from the empty set symbol. That said, it is impossible to please everyone or simply "follow the literature" because there are many different conventions for a variable that represents any arbitrary wff. To demonstrate the point, here are some conventions for variables that represent an arbitrary wff and some texts that use each convention:
  • open phi φ (and so on): Tarski's papers, Rasiowa & Sikorski's The Mathematics of Metamathematics (1963), Monk's Introduction to Set Theory (1969), Enderton's Elements of Set Theory (1977), Bell & Machover's A Course in Mathematical Logic (1977), Jech's Set Theory (1978), Takeuti & Zaring's Introduction to Axiomatic Set Theory (1982).
  • closed phi ϕ (and so on): Levy's Basic Set Theory (1979), Kunen's Set Theory (1980), Paulson's Isabelle: A Generic Theorem Prover (1994), Huth and Ryan's Logic in Computer Science (2004/2006).
  • Greek α, β, γ: Duffy's Principles of Automated Theorem Proving (1991).
  • Roman A, B, C: Kleene's Introduction to Metamathematics (1974), Smullyan's First-Order Logic (1968/1995).
  • script A, B, C: Hamilton's Logic for Mathematicians (1988).
  • italic A, B, C: Mendelson's Introduction to Mathematical Logic (1997).
  • italic P, Q, R: Suppes's Axiomatic Set Theory (1972), Gries and Schneider's A Logical Approach to Discrete Math (1993/1994), Rosser's Logic for Mathematicians (2008).
  • italic p, q, r: Quine's Set Theory and Its Logic (1969), Kuratowski & Mostowski's Set Theory (1976).
  • italic X, Y, Z: Dijkstra and Scholten's Predicate Calculus and Program Semantics (1990).
  • Fraktur letters: Fraenkel et. al's Foundations of Set Theory (1973).


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 1679 (whose description has some important technical details). Similarly,  F/_ x A is read  x is not free in (class)  A, see df-nfc 2592.
  • "$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 $." Sometimes a theorem is proved using  F/ x ph (df-nf 1679) in place of "$d  x ph $." when a more general result is desired; ax-5 1769 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2318 from df-eu 2314.
  • "$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.

There is a general technique to replace a $d x A or $d x ph condition in a theorem with the corresponding  F/_ x A or  F/ x ph; here it is.  |- T[x, A] where , and you wish to prove  |-  F/_ x A =>  |- T[x, A]. You apply the theorem substituting  y for  x and  A for  A, where  y is a new dummy variable, so that $d y A is satisfied. You obtain  |- T[y, A], and apply chvar to obtain  |- T[x, A] (or just use mpbir 214 if T[x, A] binds  x). The side goal is  |-  ( x  =  y  ->  ( T[y, A]  <-> T[x, A]  ) ), where you can use equality theorems, except that when you get to a bound variable you use a non-dv bound variable renamer theorem like cbval 2125. The section mmtheorems32.html#mm3146s also describes the metatheorem that underlies this.

Standard Metamath verifiers do not distinguish between axioms and definitions (both are $a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions ($a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 190, df-clab 2449, df-cleq 2455, and df-clel 2458); see those definitions for more information. These additional rules for definitions are checked by at least mmj2's definition check (see mmj2 master file mmj2jar/macros/definitionCheck.js). This definition check relies on the database being very much like set.mm, down to the names of certain constants and types, so it cannot apply to all Metamath databases... but it is useful in set.mm. In this definition check, a $a-statement with a given label and typecode  |- passes the test if and only if it respects the following rules (these rules require that we have an unambiguous tree parse, which is checked separately):

  1. The expression must be a biconditional or an equality (i.e. its root-symbol must be  <-> or  =). If the proposed definition passes this first rule, we then define its definiendum as its left hand side (LHS) and its definiens as its right hand side (RHS). We define the *defined symbol* as the root-symbol of the LHS. We define a *dummy variable* as a variable occurring in the RHS but not in the LHS. Note that the "root-symbol" is the root of the considered tree; it need not correspond to a single token in the database (e.g., see w3o 990 or wsb 1808).
  2. The defined expression must not appear in any statement between its syntax axiom () and its definition, and the defined expression must be not be used in its definiens. See df-3an 993 for an example where the same symbol is used in different ways (this is allowed).
  3. No two variables occurring in the LHS may share a disjoint variable (DV) condition.
  4. All dummy variables are required to be disjoint from any other (dummy or not) variable occurring in this labeled expression.
  5. Either (a) there must be no non-setvar dummy variables, or (b) there must be a justification theorem. The justification theorem must be of form  |-  ( definiens root-symbol definiens'  ) where definiens' is definiens but the dummy variables are all replaced with other unused dummy variables of the same type. Note that root-symbol is  <-> or  =, and that setvar variables are simply variables with the  setvar typecode.
  6. One of the following must be true: (a) there must be no setvar dummy variables, (b) there must be a justification theorem as described in rule 5, or (c) if there are setvar dummy variables, every one must not be free. That is, it must be true that  ( ph  ->  A. x ph ) for each setvar dummy variable  x where  ph is the definiens. We use two different tests for nonfreeness; one must succeed for each setvar dummy variable  x. The first test requires that the setvar dummy variable  x be syntactically bound (this is sometimes called the "fast" test, and this implies that we must track binding operators). The second test requires a successful search for the directly-stated proof of  ( ph  ->  A. x ph ) Part c of this rule is how most setvar dummy variables are handled.

Rule 3 may seem unnecessary, but it is needed. Without this rule, you can define something like cbar $a wff Foo x y $. ${ $d x y $. df-foo $a |- ( Foo x y <-> x = y ) $. $} and now "Foo x x" is not eliminable; there is no way to prove that it means anything in particular, because the definitional theorem that is supposed to be responsible for connecting it to the original language wants nothing to do with this expression, even though it is well formed.

A justification theorem for a definition (if used this way) must be proven before the definition that depends on it. One example of a justification theorem is vjust 3058. The definition df-v 3059  |-  _V  =  { x  |  x  =  x } is justified by the justification theorem vjust 3058  |-  { x  |  x  =  x }  =  { y  |  y  =  y }. Another example of a justification theorem is trujust 1457; the definition df-tru 1458  |-  ( T.  <->  ( A. x x  =  x  ->  A. x x  =  x ) ) is justified by trujust 1457  |-  ( ( A. x x  =  x  ->  A. x x  =  x )  <->  ( A. y y  =  y  ->  A. y y  =  y ) ).

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.
  • Maximum text line length is 79 characters. You can fix comment line length by running the commands scripts/rewrap or metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' quit . As a general rule, a math string in a comment should be surrounded by backquotes on the same line, and if it is too long it should be broken into multiple adjacent mathstrings on multiple lines. Those commands don't modify the math content of statements. In statements we try to break before the outermost important connective (not including the typecode and perhaps not the antecedent). For examples, see sqrtmulii 13504 and absmax 13447.
  • 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).
  • LRParser check. Metamath verifiers ensure that $p statements follow from previous $a and $p statements. However, by itself the Metamath language permits certain kinds of syntactic ambiguity that we choose to avoid in this database. Thus, we require that this database unambiguously parse using the "LRParser" check (implemented by at least mmj2). (For details, see mmj2 master file src/mmj/verify/LRParser.java). This check counters, for example, a devious ambiguous construct developed by saueran at oregonstate dot edu posted on Mon, 11 Feb 2019 17:32:32 -0800 (PST) based on creating definitions with mismatched parentheses.
  • 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.)

 |-  ph   =>    |-  ph
 
Theoremconventions-label 25908

The following explains some of the label conventions in use in the Metamath Proof Explorer ("set.mm"). For the general conventions, see conventions 25907.

Every statement has a unique identifying label, which serves the same purpose as an equation number in a book. 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 (these ids are usually consecutive integers starting with 1, e.g. for rgen 2759"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 19686: "mdet0.d $e |- D = ( N maDet R ) $.").
  • 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 2403 and stirling 38052.
  • Principia Mathematica. Proofs of theorems from Principia Mathematica often use a special naming convention: "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 40.
  • 19.x series of theorems. Similar to the conventions for the theorems from Principia Mathematica, theorems from Section 19 of [Margaris] p. 90 often use a special naming convention: "19." resp. "r19." (for corresponding restricted quantifier versions) followed by its identifier. For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled 19.38 1723, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 2799.
  • Characters to be used for labels Although the specification of Metamath allows for dots/periods "." in any label, it is usually used only in labels for hypotheses (see above). Exceptions are the labels of theorems from Principia Mathematica and the 19.x series of theorems from Section 19 of [Margaris] p. 90 (see above) and 0.999... 13992. Furthermore, the underscore "_" should not be used.
  • Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct  ( A  \  B ) is defined in df-dif 3419, and thus its syntax label fragment is "dif". Similarly, the subclass relation  A  C_  B has syntax label fragment "ss" because it is defined in df-ss 3430. Most theorem names follow from these fragments, for example, the theorem proving  ( A  \  B )  C_  A involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 3572. There are many other syntax label fragments, e.g., singleton construct  { A } has syntax label fragment "sn" (because it is defined in df-sn 3981), and the pair construct  { A ,  B } has fragment "pr" ( from df-pr 3983). 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 class difference ("dif") of a singleton ("sn"), it is labeled eldifsn 4110. An "n" is often used for negation ( -.), e.g., nan 588.
  • Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The definition should note this exception as part of its definition. In addition, the table below attempts to list all such cases and marks them in bold. For example, the label fragment "cn" represents complex numbers  CC (even though its definition is in df-c 9576) and "re" represents real numbers  RR ( definition df-r 9580). The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3744. The syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 9581), 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 labeled 2p2e4 10761.
  • Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
  • Focus on the important part of the conclusion. Typically the conclusion is the part the user is most interested in. So, a rough guideline is that a label typically provides a hint about only the conclusion; a label rarely says anything about the hypotheses or antecedents. If there are multiple theorems with the same conclusion but different hypotheses/antecedents, then the labels will need to differ; those label differences should emphasize what is different. There is no need to always fully describe the conclusion; just identify the important part. For example, cos0 14259 is the theorem that provides the value for the cosine of 0; we would need to look at the theorem itself to see what that value is. The label "cos0" is concise and we use it instead of "cos0eq1". There is no need to add the "eq1", because there will never be a case where we have to disambiguate between different values produced by the cosine of zero, and we generally prefer shorter labels if they are unambiguous.
  • Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl". E.g., for cosine (df-cos 14179) we have value cosval 14232 and closure coscl 14236.
  • Special cases. Sometimes, syntax and related markings are insufficient to distinguish different theorems. For example, there are over a hundred different implication-only theorems. They are 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 is 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 5 and syl 17 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 25909 for a list), and that is about all you need for most things. As for the rest, you can just assume that if it involves at most three connectives, then it is probably already proved in set.mm, and searching for it will give you the label.
  • Suffixes. Suffixes are used to indicate the form of a theorem (see above). Additionally, we sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1998 via the use of disjoint variable conditions combined with nfv 1772. If two (or three) such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used, e.g. exlimivv 1789. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the disjoint variable condition; e.g. euf 2318 derived from df-eu 2314. The "f" stands for "not free in" which is less restrictive than "does not occur in." The suffix "b" often means "biconditional" ( <->, "iff" , "if and only if"), e.g. sspwb 4666. 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 17) -type inference in a proof. A theorem label is suffixed with "ALT" if it provides an alternate less-preferred proof of a theorem (e.g., the proof is clearer but uses more axioms than the preferred version). The "ALT" may be further suffixed with a number if there is more than one alternate theorem. Furthermore, a theorem label is suffixed with "OLD" if there is a new version of it and the OLD version is obsolete (and will be removed within one year). Finally, it should be mentioned that suffixes can be combined, for example in cbvaldva 2135 (cbval 2125 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). Here is a non-exhaustive list of common suffixes:
    • a : theorem having a conjunction as antecedent
    • b : theorem expressing a logical equivalence
    • c : contraction (e.g., sylc 62, syl2anc 671), commutes (e.g., biimpac 493)
    • d : theorem in deduction form
    • f : theorem with a hypothesis such as  F/ x ph
    • g : theorem in closed form having an "is a set" antecedent
    • i : theorem in inference form
    • l : theorem concerning something at the left
    • r : theorem concerning something at the right
    • r : theorem with something reversed (e.g., a biconditional)
    • s : inference that manipulates an antecedent ("s" refers to an application of syl 17 that is eliminated)
    • v : theorem with one (main) disjoint variable condition
    • vv : theorem with two (main) disjoint variable conditions
    • w : weak(er) form of a theorem
    • ALT : alternate proof of a theorem
    • ALTV : alternate version of a theorem or definition
    • OLD : old/obsolete version of a theorem/definition/proof
  • 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, the source theorem or the 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)
aand (suffix) No biimpa 491, rexlimiva 2887
ablAbelian group df-abl 17488  Abel Yes ablgrp 17490, zringabl 19098
absabsorption No ressabs 15243
absabsolute value (of a complex number) df-abs 13354  ( abs `  A ) Yes absval 13356, absneg 13395, abs1 13415
adadding No adantr 471, ad2antlr 738
addadd (see "p") df-add 9581  ( A  +  B ) Yes addcl 9652, addcom 9850, addass 9657
al"for all"  A. x ph No alim 1694, alex 1709
ALTalternative/less preferred (suffix) No aevALT 2166
anand df-an 377  ( ph  /\  ps ) Yes anor 496, iman 430, imnan 428
antantecedent No adantr 471
assassociative No biass 365, orass 531, mulass 9658
asymasymmetric, antisymmetric No intasym 5237, asymref 5238, posasymb 16253
axaxiom No ax6dgen 1913, ax1cn 9604
bas, base base (set of an extensible structure) df-base 15181  ( Base `  S ) Yes baseval 15223, ressbas 15234, cnfldbas 19029
b, bibiconditional ("iff", "if and only if") df-bi 190  ( ph  <->  ps ) Yes impbid 195, sspwb 4666
brbinary relation df-br 4419  A R B Yes brab1 4464, brun 4467
cbvchange bound variable No cbvalivw 1861, cbvrex 3028
clclosure No ifclda 3925, ovrcl 6353, zaddcl 11011
cncomplex numbers df-c 9576  CC Yes nnsscn 10647, nncn 10650
cnfldfield of complex numbers df-cnfld 19026 fld Yes cnfldbas 19029, cnfldinv 19054
cntzcentralizer df-cntz 17026  (Cntz `  M ) Yes cntzfval 17029, dprdfcntz 17703
cnvconverse df-cnv 4864  `' A Yes opelcnvg 5036, f1ocnv 5853
cocomposition df-co 4865  ( A  o.  B ) Yes cnvco 5042, fmptco 6085
comcommutative No orcom 393, bicomi 207, eqcomi 2471
concontradiction, contraposition No condan 808, con2d 120
csbclass substitution df-csb 3376  [_ A  /  x ]_ B Yes csbid 3383, csbie2g 3406
cygcyclic group df-cyg 17568 CycGrp Yes iscyg 17569, zringcyg 19115
ddeduction form (suffix) No idd 25, impbid 195
df(alternate) definition (prefix) No dfrel2 5308, dffn2 5757
di, distrdistributive No andi 883, imdi 369, ordi 880, difindi 3709, ndmovdistr 6490
difclass difference df-dif 3419  ( A  \  B ) Yes difss 3572, difindi 3709
divdivision df-div 10303  ( A  /  B ) Yes divcl 10309, divval 10305, divmul 10306
dmdomain df-dm 4866  dom  A Yes dmmpt 5353, iswrddm0 12732
e, eq, equequals df-cleq 2455  A  =  B Yes 2p2e4 10761, uneqri 3588, equtr 1876
elelement of  A  e.  B Yes eldif 3426, eldifsn 4110, elssuni 4241
eu"there exists exactly one" df-eu 2314  E! x ph Yes euex 2334, euabsn 4057
exexists (i.e. is a set) No brrelex 4895, 0ex 4551
ex"there exists (at least one)" df-ex 1675  E. x ph Yes exim 1717, alex 1709
expexport No expt 161, expcom 441
f"not free in" (suffix) No equs45f 2192, sbf 2220
ffunction df-f 5609  F : A --> B Yes fssxp 5768, opelf 5772
falfalse df-fal 1461 F. Yes bifal 1468, falantru 1479
fifinite intersection df-fi 7956  ( fi `  B ) Yes fival 7957, inelfi 7963
fi, finfinite df-fin 7604  Fin Yes isfi 7624, snfi 7681, onfin 7794
fldfield (Note: there is an alternative definition  Fld of a field, see df-fld 26198) df-field 18033 Field Yes isfld 18039, fldidom 18584
fnfunction with domain df-fn 5608  A  Fn  B Yes ffn 5755, fndm 5701
frgpfree group df-frgp 17415  (freeGrp `  I ) Yes frgpval 17463, frgpadd 17468
fsuppfinitely supported function df-fsupp 7915  R finSupp  Z Yes isfsupp 7918, fdmfisuppfi 7923, fsuppco 7946
funfunction df-fun 5607  Fun  F Yes funrel 5622, ffun 5758
fvfunction value df-fv 5613  ( F `  A ) Yes fvres 5906, swrdfv 12823
fzfinite set of sequential integers df-fz 11820  ( M ... N ) Yes fzval 11821, eluzfz 11830
fz0finite set of sequential nonnegative integers  ( 0 ... N ) Yes nn0fz0 11925, fz0tp 11928
fzohalf-open integer range df-fzo 11953  ( M..^ N ) Yes elfzo 11959, elfzofz 11972
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 6620
gragraph No uhgrav 25079, isumgra 25098, usgrares 25152
grpgroup df-grp 16728  Grp Yes isgrp 16732, tgpgrp 21148
gsumgroup sum df-gsum 15396  ( G  gsumg  F ) Yes gsumval 16569, gsumwrev 17072
hashsize (of a set) df-hash 12554  ( # `  A ) Yes hashgval 12556, hashfz1 12567, hashcl 12576
hbhypothesis builder (prefix) No hbxfrbi 1705, hbald 1937, hbequid 32526
hm(monoid, group, ring) homomorphism No ismhm 16639, isghm 16938, isrhm 18004
iinference (suffix) No eleq1i 2531, tcsni 8258
iimplication (suffix) No brwdomi 8114, infeq5i 8172
ididentity No biid 244
idmidempotent No anidm 654, tpidm13 4087
im, impimplication (label often omitted) df-im 13219  ( A  ->  B ) Yes iman 430, imnan 428, impbidd 193
imaimage df-ima 4869  ( A " B ) Yes resima 5159, imaundi 5270
impimport No biimpa 491, impcom 436
inintersection df-in 3423  ( A  i^i  B ) Yes elin 3629, incom 3637
infinfimum df-inf 7988 inf ( RR+ ,  RR* ,  <  ) Yes fiinfcl 8048, infiso 8054
is...is (something a) ...? No isring 17839
jjoining, disjoining No jc 152, jaoi 385
lleft No olcd 399, simpl 463
mapmapping operation or set exponentiation df-map 7505  ( A  ^m  B ) Yes mapvalg 7513, elmapex 7523
matmatrix df-mat 19488  ( N Mat  R ) Yes matval 19491, matring 19523
mdetdeterminant (of a square matrix) df-mdet 19665  ( N maDet  R ) Yes mdetleib 19667, mdetrlin 19682
mgmmagma df-mgm 16543  Magma Yes mgmidmo 16557, mgmlrid 16564, ismgm 16544
mgpmultiplicative group df-mgp 17779  (mulGrp `  R ) Yes mgpress 17789, ringmgp 17841
mndmonoid df-mnd 16592  Mnd Yes mndass 16601, mndodcong 17246
mo"there exists at most one" df-mo 2315  E* x ph Yes eumo 2339, moim 2359
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mptmodus ponendo tollens No mptnan 1662, mptxor 1663
mptmaps-to notation for a function df-mpt 4479  ( x  e.  A  |->  B ) Yes fconstmpt 4900, resmpt 5176
mpt2maps-to notation for an operation df-mpt2 6325  ( x  e.  A ,  y  e.  B  |->  C ) Yes mpt2mpt 6420, resmpt2 6426
mulmultiplication (see "t") df-mul 9582  ( A  x.  B ) Yes mulcl 9654, divmul 10306, mulcom 9656, mulass 9658
n, notnot  -.  ph Yes nan 588, notnot2 117
nenot equaldf-ne  A  =/=  B Yes exmidne 2645, neeqtrd 2705
nelnot element ofdf-nel  A  e/  B Yes neli 2738, nnel 2745
ne0not equal to zero (see n0)  =/=  0 No negne0d 10015, ine0 10087, gt0ne0 10112
nf "not free in" (prefix) No nfnd 1995
ngpnormed group df-ngp 21653 NrmGrp Yes isngp 21665, ngptps 21671
nmnorm (on a group or ring) df-nm 21652  ( norm `  W ) Yes nmval 21659, subgnm 21696
nnpositive integers df-nn 10643  NN Yes nnsscn 10647, nncn 10650
nn0nonnegative integers df-n0 10904  NN0 Yes nnnn0 10910, nn0cn 10913
n0not the empty set (see ne0)  =/=  (/) No n0i 3748, vn0 3751, ssn0 3779
OLDold, obsolete (to be removed soon) No 19.43OLD 1757
opordered pair df-op 3987  <. A ,  B >. Yes dfopif 4177, opth 4693
oror df-or 376  ( ph  \/  ps ) Yes orcom 393, anor 496
otordered triple df-ot 3989  <. A ,  B ,  C >. Yes euotd 4719, fnotovb 6361
ovoperation value df-ov 6323  ( A F B ) Yes fnotovb 6361, fnovrn 6476
pplus (see "add"), for all-constant theorems df-add 9581  ( 3  +  2 )  =  5 Yes 3p2e5 10776
pfxprefix df-pfx 39060  ( W prefix  L ) Yes pfxlen 39069, ccatpfx 39087
pmPrincipia Mathematica No pm2.27 40
pmpartial mapping (operation) df-pm 7506  ( A  ^pm  B ) Yes elpmi 7521, pmsspw 7537
prpair df-pr 3983  { A ,  B } Yes elpr 3998, prcom 4063, prid1g 4091, prnz 4104
prm, primeprime (number) df-prm 14678  Prime Yes 1nprm 14684, dvdsprime 14692
pssproper subset df-pss 3432  A  C.  B Yes pssss 3540, sspsstri 3547
q rational numbers ("quotients") df-q 11299  QQ Yes elq 11300
rright No orcd 398, simprl 769
rabrestricted class abstraction df-rab 2758  { x  e.  A  |  ph } Yes rabswap 2982, df-oprab 6324
ralrestricted universal quantification df-ral 2754  A. x  e.  A ph Yes ralnex 2846, ralrnmpt2 6443
rclreverse closure No ndmfvrcl 5917, nnarcl 7348
rereal numbers df-r 9580  RR Yes recn 9660, 0re 9674
relrelation df-rel 4863  Rel  A Yes brrelex 4895, relmpt2opab 6910
resrestriction df-res 4868  ( A  |`  B ) Yes opelres 5132, f1ores 5855
reurestricted existential uniqueness df-reu 2756  E! x  e.  A ph Yes nfreud 2975, reurex 3021
rexrestricted existential quantification df-rex 2755  E. x  e.  A ph Yes rexnal 2848, rexrnmpt2 6444
rmorestricted "at most one" df-rmo 2757  E* x  e.  A ph Yes nfrmod 2976, nrexrmo 3024
rnrange df-rn 4867  ran  A Yes elrng 5048, rncnvcnv 5080
rng(unital) ring df-ring 17837  Ring Yes ringidval 17792, isring 17839, ringgrp 17840
rotrotation No 3anrot 996, 3orrot 997
seliminates need for syllogism (suffix) No ancoms 459
sb(proper) substitution (of a set) df-sb 1809  [ y  /  x ] ph Yes spsbe 1812, sbimi 1814
sbc(proper) substitution of a class df-sbc 3280  [. A  /  x ]. ph Yes sbc2or 3288, sbcth 3294
scascalar df-sca 15261  (Scalar `  H ) Yes resssca 15330, mgpsca 17785
simpsimple, simplification No simpl 463, simp3r3 1124
snsingleton df-sn 3981  { A } Yes eldifsn 4110
spspecialization No spsbe 1812, spei 2116
sssubset df-ss 3430  A  C_  B Yes difss 3572
structstructure df-struct 15178 Struct Yes brstruct 15184, structfn 15189
subsubtract df-sub 9893  ( A  -  B ) Yes subval 9897, subaddi 9993
supsupremum df-sup 7987  sup ( A ,  B ,  <  ) Yes fisupcl 8016, supmo 7997
suppsupport (of a function) df-supp 6947  ( F supp  Z ) Yes ressuppfi 7940, mptsuppd 6970
swapswap (two parts within a theorem) No rabswap 2982, 2reuswap 3254
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 3675, cnvsym 5236
symgsymmetric group df-symg 17074  ( SymGrp `  A ) Yes symghash 17081, pgrpsubgsymg 17104
t times (see "mul"), for all-constant theorems df-mul 9582  ( 3  x.  2 )  =  6 Yes 3t2e6 10795
ththeorem No nfth 1687, sbcth 3294, weth 8956
tptriple df-tp 3985  { A ,  B ,  C } Yes eltpi 4028, tpeq1 4073
trtransitive No bitrd 261, biantr 947
trutrue df-tru 1458 T. Yes bitru 1467, truanfal 1478
ununion df-un 3421  ( A  u.  B ) Yes uneqri 3588, uncom 3590
unitunit (in a ring) df-unit 17925  (Unit `  R ) Yes isunit 17940, nzrunit 18546
vdisjoint variable conditions used when a not-free hypothesis (suffix) No spimv 2112
vv2 disjoint variables (in a not-free hypothesis) (suffix) No 19.23vv 1830
wweak (version of a theorem) (suffix) No ax11w 1915, spnfw 1855
wrdword df-word 12703 Word  S Yes iswrdb 12716, wrdfn 12724, ffz0iswrd 12736
xpcross product (Cartesian product) df-xp 4862  ( A  X.  B ) Yes elxp 4873, opelxpi 4888, xpundi 4909
xreXtended reals df-xr 9710  RR* Yes ressxr 9715, rexr 9717, 0xr 9718
z integers (from German "Zahlen") df-z 10972  ZZ Yes elz 10973, zcn 10976
zn ring of integers  mod  n df-zn 19133  (ℤ/n `  N ) Yes znval 19161, zncrng 19170, znhash 19184
zringring of integers df-zring 19095 ring Yes zringbas 19100, zringcrng 19096
0, z slashed zero (empty set) (see n0) df-nul 3744  (/) Yes n0i 3748, vn0 3751; snnz 4103, prnz 4104

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

 |-  ph   =>    |-  ph
 
17.1.2  Natural deduction
 
Theoremnatded 25909 Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

NameNatural Deduction RuleTranslation RecommendationComments
IT  _G |-  ps =>  _G |-  ps idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
 /\I  _G |-  ps &  _G |-  ch =>  _G |-  ps  /\  ch jca 539 jca 539, pm3.2i 461 Definition  /\I in [Pfenning] p. 18, definition I /\m,n in [Clemente] p. 10, and definition  /\I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\EL  _G |-  ps  /\  ch =>  _G |-  ps simpld 465 simpld 465, adantr 471 Definition  /\EL in [Pfenning] p. 18, definition E /\(1) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\ER  _G |-  ps  /\  ch =>  _G |-  ch simprd 469 simpr 467, adantl 472 Definition  /\ER in [Pfenning] p. 18, definition E /\(2) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 ->I  _G ,  ps |-  ch =>  _G |-  ps  ->  ch ex 440 ex 440 Definition  ->I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition  ->I in [Indrzejczak] p. 33.
 ->E  _G |-  ps  ->  ch &  _G |-  ps =>  _G |-  ch mpd 15 ax-mp 5, mpd 15, mpdan 679, imp 435 Definition  ->E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition  ->E in [Indrzejczak] p. 33.
 \/IL  _G |-  ps =>  _G |-  ps  \/  ch olcd 399 olc 390, olci 397, olcd 399 Definition  \/I in [Pfenning] p. 18, definition I \/n(1) in [Clemente] p. 12
 \/IR  _G |-  ch =>  _G |-  ps  \/  ch orcd 398 orc 391, orci 396, orcd 398 Definition  \/IR in [Pfenning] p. 18, definition I \/n(2) in [Clemente] p. 12.
 \/E  _G |-  ps  \/  ch &  _G ,  ps |-  th &  _G ,  ch |-  th =>  _G |-  th mpjaodan 800 mpjaodan 800, jaodan 799, jaod 386 Definition  \/E in [Pfenning] p. 18, definition E \/m,n,p in [Clemente] p. 12.
 -.I  _G ,  ps |- F. =>  _G |-  -.  ps inegd 1474 pm2.01d 174
 -.I  _G ,  ps |-  th &  _G |-  -.  th =>  _G |-  -.  ps mtand 669 mtand 669 definition I -.m,n,p in [Clemente] p. 13.
 -.I  _G ,  ps |-  ch &  _G ,  ps |-  -.  ch =>  _G |-  -.  ps pm2.65da 584 pm2.65da 584 Contradiction.
 -.I  _G ,  ps |-  -.  ps =>  _G |-  -.  ps pm2.01da 448 pm2.01d 174, pm2.65da 584, pm2.65d 180 For an alternative falsum-free natural deduction ruleset
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |- F. pm2.21fal 1476 pm2.21dd 179
 -.E  _G ,  -.  ps |- F. =>  _G |-  ps pm2.21dd 179 definition  ->E in [Indrzejczak] p. 33.
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  th pm2.21dd 179 pm2.21dd 179, pm2.21d 110, pm2.21 112 For an alternative falsum-free natural deduction ruleset. Definition  -.E in [Pfenning] p. 18.
T.I  _G |- T. a1tru 1471 tru 1459, a1tru 1471, trud 1464 Definition T.I in [Pfenning] p. 18.
F.E  _G , F.  |-  th falimd 1470 falim 1469 Definition F.E in [Pfenning] p. 18.
 A.I  _G |-  [ a  /  x ] ps =>  _G |-  A. x ps alrimiv 1784 alrimiv 1784, ralrimiva 2814 Definition  A.Ia in [Pfenning] p. 18, definition I A.n in [Clemente] p. 32.
 A.E  _G |-  A. x ps =>  _G |-  [ t  /  x ] ps spsbcd 3293 spcv 3152, rspcv 3158 Definition  A.E in [Pfenning] p. 18, definition E A.n,t in [Clemente] p. 32.
 E.I  _G |-  [ t  /  x ] ps =>  _G |-  E. x ps spesbcd 3362 spcev 3153, rspcev 3162 Definition  E.I in [Pfenning] p. 18, definition I E.n,t in [Clemente] p. 32.
 E.E  _G |-  E. x ps &  _G ,  [ a  /  x ] ps |-  th =>  _G |-  th exlimddv 1792 exlimddv 1792, exlimdd 2081, exlimdv 1790, rexlimdva 2891 Definition  E.Ea,u in [Pfenning] p. 18, definition E E.m,n,p,a in [Clemente] p. 32.
F.C  _G ,  -.  ps |- F. =>  _G |-  ps efald 1475 efald 1475 Proof by contradiction (classical logic), definition F.C in [Pfenning] p. 17.
F.C  _G ,  -.  ps |-  ps =>  _G |-  ps pm2.18da 449 pm2.18da 449, pm2.18d 116, pm2.18 114 For an alternative falsum-free natural deduction ruleset
 -.  -.C  _G |-  -.  -.  ps =>  _G |-  ps notnotrd 118 notnotrd 118, notnot2 117 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E -.n in [Clemente] p. 14.
EM  _G |-  ps  \/  -.  ps exmidd 422 exmid 421 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
 =I  _G |-  A  =  A eqidd 2463 eqid 2462, eqidd 2463 Introduce equality, definition =I in [Pfenning] p. 127.
 =E  _G |-  A  =  B &  _G [. A  /  x ]. ps =>  _G |-  [. B  /  x ]. ps sbceq1dd 3285 sbceq1d 3284, equality theorems Eliminate equality, definition =E in [Pfenning] p. 127. (Both E1 and E2.)

Note that MPE uses classical logic, not intuitionist logic. As is conventional, the "I" rules are introduction rules, "E" rules are elimination rules, the "C" rules are conversion rules, and  _G represents the set of (current) hypotheses. We use wff variable names beginning with  ps to provide a closer representation of the Metamath equivalents (which typically use the antedent  ph to represent the context  _G).

Most of this information was developed by Mario Carneiro and posted on 3-Feb-2017. For more information, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer.

For annotated examples where some traditional ND rules are directly applied in MPE, see ex-natded5.2 25910, ex-natded5.3 25913, ex-natded5.5 25916, ex-natded5.7 25917, ex-natded5.8 25919, ex-natded5.13 25921, ex-natded9.20 25923, and ex-natded9.26 25925.

(Contributed by DAW, 4-Feb-2017.) (New usage is discouraged.)

 |-  ph   =>    |-  ph
 
17.1.3  Natural deduction examples

These are examples of how natural deduction rules can be applied in metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 25909 and http://us.metamath.org/mpeuni/mmnatded.html. Since these examples should not be used within proofs of other theorems, especially in Mathboxes, they are marked with "(New usage is discouraged.)".

 
Theoremex-natded5.2 25910 Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15  ( ( ps  /\  ch )  ->  th )  ( ph  ->  ( ( ps  /\  ch )  ->  th ) ) Given $e.
22  ( ch  ->  ps )  ( ph  ->  ( ch  ->  ps ) ) Given $e.
31  ch  ( ph  ->  ch ) Given $e.
43  ps  ( ph  ->  ps )  ->E 2,3 mpd 15, the MPE equivalent of  ->E, 1,2
54  ( ps  /\  ch )  ( ph  ->  ( ps  /\  ch ) )  /\I 4,3 jca 539, the MPE equivalent of  /\I, 3,1
66  th  ( ph  ->  th )  ->E 1,5 mpd 15, the MPE equivalent of  ->E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 25911. A proof without context is shown in ex-natded5.2i 25912. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2-2 25911 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 25910 and ex-natded5.2i 25912. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2i 25912 The same as ex-natded5.2 25910 and ex-natded5.2-2 25911 but with no context. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ps  /\  ch )  ->  th )   &    |-  ( ch  ->  ps )   &    |-  ch   =>    |- 
 th
 
Theoremex-natded5.3 25913 Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 25914. A proof without context is shown in ex-natded5.3i 25915. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 471 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 471 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 467, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15, the MPE equivalent of  ->E, 1.3. adantr 471 was used to transform its dependency (we could also use imp 435 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15, the MPE equivalent of  ->E, 4,6. adantr 471 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 539, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 440, the MPE equivalent of  ->I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3-2 25914 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 25913 and ex-natded5.3i 25915. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3i 25915 The same as ex-natded5.3 25913 and ex-natded5.3-2 25914 but with no context. Identical to jccir 546, which should be used instead. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ch  /\  th ) )
 
Theoremex-natded5.5 25916 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 471 to move it into the ND hypothesis
25  -.  ch  ( ph  ->  -.  ch ) Given $e; we'll use adantr 471 to move it into the ND hypothesis
31 ...|  ps  ( ph  ->  ps ) ND hypothesis assumption simpr 467
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15 1,3
56 ...  -.  ch  ( ( ph  /\  ps )  ->  -.  ch ) IT 2 adantr 471 5
67  -.  ps  ( ph  ->  -.  ps )  /\I 3,4,5 pm2.65da 584 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 471; simpr 467 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 182; a proof without context is shown in mto 181.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.7 25917 Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 25918. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16  ( ps  \/  ( ch  /\  th ) )  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) ) Given $e. No need for adantr 471 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 467
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 398, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 467
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 465, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 399, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 800, the MPE equivalent of  \/E, 2,5,6

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.7-2 25918 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 25917. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.8 25919 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11  ( ( ps  /\  ch )  ->  -.  th )  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) ) Given $e; adantr 471 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 471 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 471 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 471 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 467. New ND hypothesis scope, each reference outside the scope must change antecedent  ph to  ( ph  /\  ps ).
69 ...  ( ps  /\  ch )  ( ( ph  /\  ps )  ->  ( ps  /\  ch ) )  /\I 5,3 jca 539 ( /\I), 6,8 (adantr 471 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 15 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 584 ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 471; simpr 467 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 25920.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.8-2 25920 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 25919. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.13 25921 Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 25922. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) ) Given $e.
2;32  ( ps  ->  th )  ( ph  ->  ( ps  ->  th ) ) Given $e. adantr 471 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 737 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 467
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 398 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 467
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 467
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 15 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 471 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 584 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 118 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 399 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 800 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 471; simpr 467 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded5.13-2 25922 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 25921. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded9.20 25923 Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11  ( ps  /\  ( ch  \/  th ) )  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) ) Given $e
22  ps  ( ph  ->  ps )  /\EL 1 simpld 465 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 469 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 467
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 539 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 398 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 467
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 539 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 399 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 800 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 471; simpr 467 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 25924. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.20-2 25924 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 25923. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.26 25925* Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that  x is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13  E. x A. y ps ( x ,  y )  ( ph  ->  E. x A. y ps ) Given $e.
26 ...|  A. y ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  A. y ps ) ND hypothesis assumption simpr 467. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3293 ( A.E), 5,6. To use it we need a1i 11 and vex 3060. This could be immediately done with 19.21bi 1958, but we want to show the general approach for substitution.
412;8,9,10,11 ...  E. x ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  E. x ps )  E.I 3,a spesbcd 3362 ( E.I), 11. To use it we need sylibr 217, which in turn requires sylib 201 and two uses of sbcid 3296. This could be more immediately done using 19.8a 1946, but we want to show the general approach for substitution.
513;1,2  E. x ps ( x ,  y )  ( ph  ->  E. x ps )  E.E 1,2,4,a exlimdd 2081 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1772 and nfe1 1929 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1784 ( A.I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof,  ps ( x ,  y ) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 25926.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
Theoremex-natded9.26-2 25926* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 25925. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
17.1.4  Definitional examples
 
Theoremex-or 25927 Example for df-or 376. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  3  \/  4  =  4 )
 
Theoremex-an 25928 Example for df-an 377. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  2 
 /\  3  =  3 )
 
Theoremex-dif 25929 Example for df-dif 3419. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  \  {
 1 ,  8 } )  =  { 3 }
 
Theoremex-un 25930 Example for df-un 3421. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  u.  {
 1 ,  8 } )  =  { 1 ,  3 ,  8 }
 
Theoremex-in 25931 Example for df-in 3423. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  i^i  {
 1 ,  8 } )  =  { 1 }
 
Theoremex-uni 25932 Example for df-uni 4213. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |- 
 U. { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
 1 ,  3 ,  8 }
 
Theoremex-ss 25933 Example for df-ss 3430. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C_  { 1 ,  2 ,  3 }
 
Theoremex-pss 25934 Example for df-pss 3432. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C.  { 1 ,  2 ,  3 }
 
Theoremex-pw 25935 Example for df-pw 3965. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  =  {
 3 ,  5 ,  7 }  ->  ~P A  =  ( ( { (/) }  u.  { { 3 } ,  { 5 } ,  { 7 } }
 )  u.  ( { { 3 ,  5 } ,  { 3 ,  7 } ,  { 5 ,  7 } }  u.  { { 3 ,  5 ,  7 } }
 ) ) )
 
Theoremex-pr 25936 Example for df-pr 3983. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( A  e.  {
 1 ,  -u 1 }  ->  ( A ^
 2 )  =  1 )
 
Theoremex-br 25937 Example for df-br 4419. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  3 R
 9 )
 
Theoremex-opab 25938* Example for df-opab 4478. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
 )  =  y ) }  ->  3 R
 4 )
 
Theoremex-eprel 25939 Example for df-eprel 4767. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  5  _E  { 1 ,  5 }
 
Theoremex-id 25940 Example for df-id 4771. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 5  _I  5  /\  -.  4  _I  5
 )
 
Theoremex-po 25941 Example for df-po 4777. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  (  <  Po  RR  /\ 
 -.  <_  Po  RR )
 
Theoremex-xp 25942 Example for df-xp 4862. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( { 1 ,  5 }  X.  {
 2 ,  7 } )  =  ( { <. 1 ,  2 >. ,  <. 1 ,  7
 >. }  u.  { <. 5 ,  2 >. ,  <. 5 ,  7 >. } )
 
Theoremex-cnv 25943 Example for df-cnv 4864. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  `' { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  =  { <. 6 ,  2 >. ,  <. 9 ,  3 >. }
 
Theoremex-co 25944 Example for df-co 4865. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( exp  o.  cos ) `  0 )  =  _e
 
Theoremex-dm 25945 Example for df-dm 4866. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  dom  F  =  { 2 ,  3 } )
 
Theoremex-rn 25946 Example for df-rn 4867. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ran  F  =  { 6 ,  9 } )
 
Theoremex-res 25947 Example for df-res 4868. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
 >. } )
 
Theoremex-ima 25948 Example for df-ima 4869. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } )
 
Theoremex-fv 25949 Example for df-fv 5613. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ( F `  3 )  =  9 )
 
Theoremex-1st 25950 Example for df-1st 6825. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 1st `  <. 3 ,  4 >. )  =  3
 
Theoremex-2nd 25951 Example for df-2nd 6826. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 2nd `  <. 3 ,  4 >. )  =  4
 
Theorem1kp2ke3k 25952 Example for df-dec 11086, 1000 + 2000 = 3000.

This proof disproves (by counter-example) 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 11086 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.)

 |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
 
Theoremex-fl 25953 Example for df-fl 12066. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ( |_ `  (
 3  /  2 )
 )  =  1  /\  ( |_ `  -u (
 3  /  2 )
 )  =  -u 2
 )
 
Theoremex-dvds 25954 3 divides into 6. A demonstration of df-dvds 14361. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  3  ||  6
 
17.1.5  Other examples
 
Theoremex-ind-dvds 25955 Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
 |-  ( N  e.  NN0  -> 
 3  ||  ( (
 4 ^ N )  +  2 ) )
 
17.2  Humor
 
17.2.1  April Fool's theorem
 
Theoremavril1 25956 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object  x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpeuni/mmnotes.txt, under the 1-Apr-2006 entry.

 |- 
 -.  ( A ~P RR ( _i `  1
 )  /\  F (/) ( 0  x.  1 ) )
 
Theorem2bornot2b 25957 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 2  x.  B  \/  -.  2  x.  B )
 
Theoremhelloworld 25958 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( h  e.  ( L L 0 )  /\  W (/) ( R. 1 d ) )
 
Theorem1p1e2apr1 25959 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  +  1 )  =  2
 
Theoremeqid1 25960 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  A  =  A
 
Theorem1div0apr 25961 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  /  0
 )  =  (/)
 
Theoremtopnfbey 25962 Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( B  e.  (
 0 ... +oo )  -> +oo  <  B )
 
17.3  (Future - to be reviewed and classified)
 
17.3.1  Planar incidence geometry
 
Syntaxcplig 25963 Extend class notation with the class of all planar incidence geometries.
 class  Plig
 
Definitiondf-plig 25964* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry.  e. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)
 |- 
 Plig  =  { x  |  ( A. a  e. 
 U. x A. b  e.  U. x ( a  =/=  b  ->  E! l  e.  x  (
 a  e.  l  /\  b  e.  l )
 )  /\  A. l  e.  x  E. a  e. 
 U. x E. b  e.  U. x ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  U. x E. b  e.  U. x E. c  e.  U. x A. l  e.  x  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l
 ) ) }
 
Theoremisplig 25965* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  A  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
 
Theoremtncp 25966* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) )
 
Theoremlpni 25967* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
 |-  P  =  U. G   =>    |-  (
 ( G  e.  Plig  /\  L  e.  G ) 
 ->  E. a  e.  P  a  e/  L )
 
17.3.2  Algebra preliminaries
 
Syntaxcrpm 25968 Ring primes.
 class RPrime
 
Definitiondf-rprm 25969* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 14720. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |- RPrime  =  ( w  e.  _V  |->  [_ ( Base `  w )  /  b ]_ { p  e.  ( b  \  (
 (Unit `  w )  u.  { ( 0g `  w ) } )
 )  |  A. x  e.  b  A. y  e.  b  [. ( ||r `  w )  /  d ]. ( p d ( x ( .r `  w ) y )  ->  ( p d x  \/  p d y ) ) } )
 
PART 18  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)

This part contains an earlier development of groups, rings, and fields that was defined before extensible structures were introduced.

Theorem grpo2grp 26018 shows the relationship between the older group definition and the extensible structure definition.

The intent is for this deprecated section to be deleted once its theorems have extensible structure versions (or are not useful). You can make a list of "terminal" theorems (i.e. theorems not referenced by anything else) and for each theorem see if there exists an extensible structure version (or decide it's not useful), and if so, delete it. Then repeat this recursively. One way to search for terminal theorems, for example in deprecated group theory, is to log the output ("open log x.txt") of "show usage cgr~circgrp" in metamath.exe and search for "(None)".

 
18.1  Additional material on group theory
 
18.1.1  Definitions and basic properties for groups
 
Syntaxcgr 25970 Extend class notation with the class of all group operations.
 class  GrpOp
 
Syntaxcgi 25971 Extend class notation with a function mapping a group operation to the group's identity element.
 class GId
 
Syntaxcgn 25972 Extend class notation with a function mapping a group operation to the inverse function for the group.
 class  inv
 
Syntaxcgs 25973 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
 class  /g
 
Syntaxcgx 25974 Extend class notation with a function mapping a group operation to the power operation for the group.
 class  ^g
 
Definitiondf-grpo 25975* Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |- 
 GrpOp  =  { g  |  E. t ( g : ( t  X.  t ) --> t  /\  A. x  e.  t  A. y  e.  t  A. z  e.  t  (
 ( x g y ) g z )  =  ( x g ( y g z ) )  /\  E. u  e.  t  A. x  e.  t  (
 ( u g x )  =  x  /\  E. y  e.  t  ( y g x )  =  u ) ) }
 
Definitiondf-gid 25976* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId 
 =  ( g  e. 
 _V  |->  ( iota_ u  e. 
 ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
 
Definitiondf-ginv 25977* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
 |- 
 inv  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g  |->  ( iota_ z  e.  ran  g (
 z g x )  =  (GId `  g
 ) ) ) )
 
Definitiondf-gdiv 25978* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |- 
 /g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
 ) ) ) )
 
Definitiondf-gx 25979* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |- 
 ^g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ZZ  |->  if ( y  =  0 ,  (GId `  g ) ,  if ( 0  <  y ,  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( ( inv `  g ) `  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) ) )
 
Theoremisgrpo 25980* The predicate "is a group operation." Note that  X is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) 
 /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) ) ) )
 
Theoremisgrpo2 25981* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y )  e.  X  /\  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. n  e.  X  ( n G x )  =  u ) ) ) )
 
Theoremisgrpoi 25982* Properties that determine a group operation. Read  N as  N ( x ). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  X  e.  _V   &    |-  G : ( X  X.  X ) --> X   &    |-  (
 ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  U  e.  X   &    |-  ( x  e.  X  ->  ( U G x )  =  x )   &    |-  ( x  e.  X  ->  N  e.  X )   &    |-  ( x  e.  X  ->  ( N G x )  =  U )   =>    |-  G  e.  GrpOp
 
Theoremgrpofo 25983 A group operation maps onto the group's underlying set. (Contributed by NM, 30-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  G : ( X  X.  X ) -onto-> X )
 
Theoremgrpocl 25984 Closure law for a group operation. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremgrpolidinv 25985* A group has a left identity element, and every member has a left inverse. (Contributed by NM, 2-Nov-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) )
 
Theoremgrpon0 25986 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  X  =/= 
 (/) )
 
Theoremgrpoass 25987 A group operation is associative. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( G  e.  GrpOp  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A G B ) G C )  =  ( A G ( B G C ) ) )
 
Theoremgrpoidinvlem1 25988 Lemma for grpoidinv 25992. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( Y G A )  =  U  /\  ( A G A )  =  A ) )  ->  ( U G A )  =  U )
 
Theoremgrpoidinvlem2 25989 Lemma for grpoidinv 25992. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  ( Y  e.  X  /\  A  e.  X ) )  /\  ( ( U G Y )  =  Y  /\  ( Y G A )  =  U ) )  ->  ( ( A G Y ) G ( A G Y ) )  =  ( A G Y ) )
 
Theoremgrpoidinvlem3 25990* Lemma for grpoidinv 25992. (Contributed by NM, 11-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  ( ph 
 <-> 
 A. x  e.  X  ( U G x )  =  x )   &    |-  ( ps 
 <-> 
 A. x  e.  X  E. z  e.  X  ( z G x )  =  U )   =>    |-  ( ( ( ( G  e.  GrpOp  /\  U  e.  X )  /\  ( ph  /\  ps ) ) 
 /\  A  e.  X )  ->  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )
 
Theoremgrpoidinvlem4 25991* Lemma for grpoidinv 25992. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  (
 ( ( G  e.  GrpOp  /\  A  e.  X ) 
 /\  E. y  e.  X  ( ( y G A )  =  U  /\  ( A G y )  =  U ) )  ->  ( A G U )  =  ( U G A ) )
 
Theoremgrpoidinv 25992* A group has a left and right identity element, and every member has a left and right inverse. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E. u  e.  X  A. x  e.  X  ( ( ( u G x )  =  x  /\  ( x G u )  =  x )  /\  E. y  e.  X  (
 ( y G x )  =  u  /\  ( x G y )  =  u ) ) )
 
Theoremgrpoideu 25993* The left identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 14-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  GrpOp  ->  E! u  e.  X  A. x  e.  X  ( u G x )  =  x )
 
Theoremgrporndm 25994 A group's range in terms of its domain. (Contributed by NM, 6-Apr-2008.) (New usage is discouraged.)
 |-  ( G  e.  GrpOp  ->  ran  G  =  dom  dom  G )
 
Theorem0ngrp 25995 The empty set is not a group. (Contributed by NM, 25-Apr-2007.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  GrpOp
 
Theoremgrporn 25996 The range of a group operation. Useful for satisfying group base set hypotheses of the form  X  =  ran  G. (Contributed by NM, 5-Nov-2006.) (New usage is discouraged.)
 |-  G  e.  GrpOp   &    |- 
 dom  G  =  ( X  X.  X )   =>    |-  X  =  ran  G
 
Theoremgidval 25997* The value of the identity element of a group. (Contributed by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  V  ->  (GId `  G )  =  (
 iota_ u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  ( x G u )  =  x ) ) )
 
Theoremfngid 25998 GId is a function. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId  Fn  _V
 
Theoremgrposn 25999 The group operation for the singleton group. (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  { <. <. A ,  A >. ,  A >. }  e.  GrpOp
 
Theoremgrpoidval 26000* Lemma for grpoidcl 26001 and others. (Contributed by NM, 5-Feb-2010.) (Proof shortened by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   =>    |-  ( G  e.  GrpOp  ->  U  =  ( iota_ u  e.  X  A. x  e.  X  ( u G x )  =  x ) )
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