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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrgrreg 28101 If a finite nonempty friendship graph is 𝐾-regular, then 𝐾 must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))
 
Theoremfrgrregord013 28102 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.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
 
Theoremfrgrregord13 28103 If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 28102. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))
 
Theoremfrgrogt3nreg 28104* 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.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)
 
Theoremfriendshipgt3 28105* 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.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
 
Theoremfriendship 28106* 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 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))
 
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. They are organized as follows:

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 28107

Here are some of the conventions we use in the Metamath Proof Explorer (MPE, set.mm), and how they correspond to typical textbook language (skipping the many cases where they are identical). For more specific conventions, see:

  • 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 Σ𝑘𝐴𝐵 (df-sum 15033) which denotes that index variable 𝑘 ranges over 𝐴 when evaluating 𝐵. Thus, Σ𝑘 ∈ ℕ (1 / (2↑𝑘)) = 1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 15228). 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 208, df-cleq 2814, df-clel 2893, df-clab 2800) 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 mmcomplex.html 2800. 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 the complex arithmetic axiom ax-1cn 10584, proven by the preceding theorem ax1cn 10560. The Metamath 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 displayed in summarized form such as in the "Theorem List" page (to get to it, click on "Nearby theorems" on the ax-mp 5 page), the hypotheses are connected with an ampersand and separated from the conclusion with a double right arrow, such as in " 𝜑 & (𝜑𝜓) ⇒ 𝜓". 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 10584 (not ax1cn 10560) and ax-1ne0 10595 (not ax1ne0 10571), as these are proven axioms for complex arithmetic. Thus, both ax1cn 10560 and ax1ne0 10571 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 · 𝐴)) / i) instead of (sinh‘𝐴) for the hyperbolic sine.
  • Minimizing axiom dependencies.

    We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, because of the non-constructive nature of the axiom of choice df-ac 9531, we prefer proofs that do not use it, or use weaker versions like countable choice ax-cc 9846 or dependent choice ax-dc 9857. An example is our proof of the Schroeder-Bernstein Theorem sbth 8626, which does not use the axiom of choice. Similarly, any theorem in first-order logic (FOL) that contains only setvar variables that are all mutually distinct, and has no wff variables, can be proved without using ax-10 2136 through ax-13 2383, by using ax10w 2124 through ax13w 2131 instead.

    We do not try to similarly reduce dependencies on definitions, since definitions are conservative (they do not increase the proving power of a deductive system), and are introduced in order to be used to increase readability). An exception is made for the definitions df-clab 2800, df-cleq 2814, df-clel 2893, since they can be considered as axioms under some definitions of what a definition is exactly (see their comments).

  • Alternate proofs (ALT). If a different proof is shorter or clearer but uses more or stronger axioms, we make that proof an "alternate" proof (marked with an ALT label suffix), even if this alternate proof was formalized first. We then make the proof that requires fewer axioms the main proof. Alternate proofs can also occur in other cases when an alternate proof gives some particular insight. Their comment should begin with "Alternate proof of ~ xxx " followed by a description of the specificity of that alternate proof. There can be multiple alternates. Alternate (*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 alternate and main statement comments should use hyperlinks to refer to each other.
  • Alternate versions (ALTV). The suffix ALTV is reserved for theorems (or definitions) which are alternate versions, or variants, of an existing theorem. This is reserved to statements in mathboxes and is typically used temporarily, when it is not clear yet which variant to use. If it is decided that both variants should be kept and moved to the main part of set.mm, then a label for the variant should be found with a more explicit suffix indicating how it is a variant (e.g., commutation of some subformula, antecedent replaced with hypothesis, (un)curried variant, biconditional instead of implication, etc.). There is no requirement to add discouragement tags, but their comment should have a link to the main version of the statement and describe how it is a variant of it.
  • 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 the Metamath program "MM-PA> 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 generally used in this order: 𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta, 𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma. Individual setvar variables are represented with lowercase Latin letters and are generally 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 a zero element (e.g., fsuppcor 8856) and connective symbols such as + for some group addition operation (e.g., grprinvd ). 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 ¬ 𝜑".
  • 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 231 or mpbir 232. 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 217 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 707, sylbir 236, or 3imtr4i 293.
  • Quantifiers. The quantifiers are named as follows:
    • : universal quantifier (wal 1526);
    • : existential quantifier (df-ex 1772);
    • ∃*: at-most-one quantifier (df-mo 2618);
    • ∃!: unique existential quantifier (df-eu 2650);
    the phrase "uniqueness quantifier" is avoided since it is ambiguous: it can be understood as claiming either uniqueness (∃*) or unique existence (∃!).
  • Substitution. The expression "[𝑦 / 𝑥]𝜑" should be read "the formula that results from the proper substitution of 𝑦 for 𝑥 in the formula 𝜑". See df-sb 2061 and the related df-sbc 3772 and df-csb 3883.
  • Is-a-set. "𝐴 ∈ 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 3497 and isset 3507. However, instead of using 𝐼 ∈ V in the antecedent of a theorem for some variable 𝐼, we now prefer to use 𝐼𝑉 (or another variable if 𝑉 is not available) to make it more general. That way we can often avoid needing extra uses of elex 3513 and syl 17 in the common case where 𝐼 is already a member of something. For hypotheses ($e statement) of theorems (mostly in inference form), however, 𝐴 ∈ V is used rather than 𝐴𝑉 (e.g., difexi 5224). This is because 𝐴 ∈ V is almost always satisfied using an existence theorem stating "... ∈ V", and a hard-coded V in the $e statement saves a couple of syntax building steps that substitute V into 𝑉. Notice that this does not hold for hypotheses of theorems in deduction form: Here still (𝜑𝐴𝑉) should be used rather than (𝜑𝐴 ∈ V).
  • 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 5557. This can be used to define a subset, e.g., df-tan 15415 notates "the set of values whose cosine is a nonzero complex number" as (cos “ (ℂ ∖ {0})).
  • Function application. "(𝐹𝑥)" 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 15493). 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 6357. 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 7148). For example, the + in (2 + 2); see 2p2e4 11761. 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 (𝜑𝜓), (𝜑𝜓), (𝜑𝜓), and (𝜑𝜓) (see wi 4, df-or 842, df-an 397, and df-bi 208 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 𝐴𝐵 (see wel 2106), equality 𝐴 = 𝐵 (see df-cleq 2814), subset 𝐴𝐵 (see df-ss 3951), and less-than 𝐴 < 𝐵 (see df-lt 10539). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 5059. For example, 0 < 1 (see 0lt1 11151) does not use parentheses.
  • Unary minus. The symbol - is used to indicate a unary minus, e.g., -1. It is specially defined because it is so commonly used. See cneg 10860.
  • 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 15408). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 15414). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 15470), its function application form labeled NAMEval (e.g., cosval 15466), and at least one simple value (e.g., cos0 15493).
  • Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g., (!‘4) = 24 (df-fac 13624 and fac4 13631).
  • 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 10533), while "0g" is the group identity element (df-0g 16705), "0." is the poset zero (df-p0 17639), "0𝑝" is the zero polynomial (df-0p 24200), "0vec" is the zero vector in a normed subcomplex vector space (df-0v 28303), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 17641). 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 , are often represented by that letter followed by a period. For example, "A." is used to represent , "e." is used to represent , and "E." is used to represent . 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 upper versus 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 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 7148, 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 (𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸.
  • LaTeX definitions. Each token has a "LaTeX definition" which is used by the Metamath program to output tex files. When writing LaTeX definitions, contributors should favor simplicity over perfection of the display, and should only use core LaTeX symbols or symbols from standard packages; if packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic, graphicx are needed, this should be discussed. A useful resource is The Comprehensive LaTeX Symbol List.
  • 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.html 7148 for details. Thus, for example, we don't allow the use of ∅ ∉ ℂ, as handy as that would be, because that would be construction-specific. We want proofs about to be independent of whether or not ∅ ∈ ℂ.
  • Minimize hypotheses. In most cases we try to minimize hypotheses, so that the statement be 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 the complex numbers ). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they are not needed. For example, we could show that (𝐴 < 𝐵𝐵𝐴) without any hypotheses, but we require that theorems using this result prove that 𝐴 and 𝐵 are real numbers, so that the statement we use is ltnei 10753. Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 10753:
    1. Having the hypotheses immediately shows the intended domain of applicability (is it , *, ω, or something else?), without having to trace back to definitions.
    2. Having the hypotheses forces the intended use of the statement, which generally is desirable.
    3. Many out-of-domain values are dependent on contingent details of definitions, so hypothesis-free theorems would be non-portable and "brittle".
    4. Only a few theorems can have their hypotheses removed in this fashion, due to coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning, e.g., real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who has not traced back the set-theoretical foundations of the definitions, it is seemingly random and is not intuitive at all.
    5. Ultimately, this is a matter of consensus, and the consensus in the group was in favor of keeping sometimes redundant hypotheses.
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 11628) for the set of positive integers starting from 1, and 0 (df-n0 11887) for the set of nonnegative integers starting at zero.
  • Decimal numbers. Numbers larger than nine are often expressed in base 10 using the decimal constructor df-dec 12088, e.g., 4001 (see 4001prm 16468 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 4159). 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 zero or more $e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually 𝜑), and every hypothesis ($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 24542) 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 𝜑) or is a conjunction (𝜑 ∩ ...) including that wff variable (𝜑). E.g. a1d 25, unssd 4161. Although they are no real deductions, theorems without $e hypotheses, but in the form (𝜑 → ...), are also said to be in "deduction form". Such theorems usually have a two step proof, applying a1i 11 to a given theorem, and are used as convenience theorems to shorten many proofs. E.g. eqidd 2822, which is used more than 1500 times.
    • 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 4160).
    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 157 is the inference associated with con3 156). 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 83 because it is the inference associated with imim1i 63 which is itself the inference associated with imim1 83.
    "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 53). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 56) or, if the non-suffixed label is already used, then we add the suffix "t" (for "theorem" or "tautology", e.g., ancomst 465 or nfimt 1887). When an inference with an "is a set" hypothesis (e.g., 𝐴 ∈ V) is converted to a theorem (in closed form) by replacing the hypothesis with an antecedent of the form (𝐴𝑉, we sometimes suffix the closed form with "g" (for "more general") as in uniex 7454 versus uniexg 7456. In this case, the inference often has no suffix "i". When submitting a new theorem, a revision of a theorem, or an upgrade of a theorem from a Mathbox to the Main database, please use the general form to be the default form of the theorem, without the suffix "g" . For example, "brresg" lost its suffix "g" when it was revised for some other reason, and now it is brres 5854. Its inference form which was the original "brres", now is brresi 5856. The same holds for the suffix "t".
  • 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 mmdeduction.html 5856. 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 4521, which works in certain cases in set theory. We also sometimes use dedhb 3694. 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 𝜑 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 mmnatded.html 3694; a list of translations for common natural deduction rules is given in natded 28110.
  • 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 7999, rec(𝐹, 𝐼) in df-rdg 8037, seqω(𝐹, 𝐼) in df-seqom 8075, and seq𝑀( + , 𝐹) in df-seq 13360. These have characteristic function 𝐹 and initial value 𝐼. (Σg in df-gsum 16706 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7999, but for the "average user" the most useful one is probably df-seq 13360- 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 16475.
  • 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 (℩𝑥𝜑) is meant to be "the 𝑥 such that 𝜑" (where 𝜑 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 ∈ (𝐹𝐵) → 𝐵 ∈ 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 𝑦𝑥 $. $d 𝑦𝜑 $. df-iota $a (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) $. $}. 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 6333 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 6677 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 label followed by "lem", and a number if there is more than one. E.g., sbthlem1 8616 is the first lemma for sbth 8626. The comment should begin with "Lemma for", followed by the final proof label, so that it can be suppressed in theorem lists (see the Metamath program "MM> WRITE THEOREM_LIST" command). 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 Metamath program "MM-PA> MINIMIZE__WITH *" command 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 8616 is the first lemma for sbth 8626.
  • 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 16006"). When the Metamath program 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 assertions.
  • 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 16006 entries) have an anchor in mmtheorems.html 16006 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 cedgf 26702. To reference it we link to the anchor using a space-separated tilde followed by the space-separated link mmtheorems.html#cedgf, which will become the hyperlink mmtheorems.html#cedgf 26702. Note that no theorem in set.mm is allowed to begin with "mm" (this is enforced by the Metamath program "MM> VERIFY MARKUP" command). Whenever the program 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 26702.
  • 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 program "MM> WRITE BIBLIOGRAPHY" command.)
  • Acceptable shorter proofs. Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem 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.

  • Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see mmdefinitions.html 23. 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.

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:

  • 𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑"; see df-nf 1776 (whose description has some important technical details). Similarly, 𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2963.
  • "$d 𝑥𝑦 $." should be read "Assume 𝑥 and 𝑦 are distinct variables."
  • "$d 𝜑𝑥 $." should be read "Assume 𝑥 does not occur in ϕ." Sometimes a theorem is proved using 𝑥𝜑 (df-nf 1776) in place of "$d 𝜑𝑥 $." when a more general result is desired; ax-5 1902 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2657 from eu6 2655.
  • "$d 𝐴𝑥 $." should be read "Assume 𝑥 is not a variable occurring in class 𝐴."
  • "$d 𝐴𝑥 $. $d 𝜓𝑥 $. $e |- (𝑥 = 𝐴 → (𝜑𝜓)) $." is an idiom often used instead of explicit substitution, meaning "Assume ψ results from the proper substitution of 𝐴 for 𝑥 in ϕ." Therefore, we often use the term "implicit substitution" for such a hypothesis.
  • Class and wff variables should appear at the beginning of distinct variable conditions, and setvars should be in alphabetical order. E.g., "$d 𝑍𝑥𝑦 $.", "$d 𝜓𝑎𝑥 $.". This convention should be applied for new theorems (formerly, the class and wff variables mostly appear at the end) and will be assured by a formatter in the future.
  • " (¬ ∀𝑥𝑥 = 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 𝑥𝐴 or 𝑥𝜑; here it is. T[x, A] where $d 𝑥𝐴, and you wish to prove 𝑥𝐴 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 232 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 2410. The section mmtheorems32.html#mm3146s 2410 also describes the metatheorem that underlies this.

Additional rules for definitions

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 208, df-clab 2800, df-cleq 2814, and df-clel 2893); 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 1078 or wsb 2060).
  2. The defined expression must not appear in any statement between its syntax axiom ($a wff ) and its definition, and the defined expression must not be used in its definiens. See df-3an 1081 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 (𝜑 → ∀𝑥𝜑) for each setvar dummy variable 𝑥 where 𝜑 is the definiens. We use two different tests for non-freeness; 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 (𝜑 → ∀𝑥𝜑) 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 3496. The definition df-v 3497 V = {𝑥𝑥 = 𝑥} is justified by the justification theorem vjust 3496 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}. Another example of a justification theorem is trujust 1530; the definition df-tru 1531 (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1530 ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 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.
  • 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 1530.
  • Community. We encourage anyone interested in Metamath to join our mailing list: https://groups.google.com/g/metamath 1530.

(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision. (Revised by the Metamath team, 22-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑
 
Theoremconventions-labels 28108

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 28107 and links therein.

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 3148"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 21145: "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 2746 and stirling 42255.
  • 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 42.
  • 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 1830, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 3215.
  • 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... 15227. Furthermore, the underscore "_" should not be used. Finally, only lower case characters should be used (except the special suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at least in main set.mm (exceptions are tolerated in mathboxes).
  • 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 (𝐴𝐵) is defined in df-dif 3938, and thus its syntax label fragment is "dif". Similarly, the subclass relation 𝐴𝐵 has syntax label fragment "ss" because it is defined in df-ss 3951. Most theorem names follow from these fragments, for example, the theorem proving (𝐴𝐵) ⊆ 𝐴 involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 4107. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4560), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4562). 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 𝐴𝐵 does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses is-element-of ("el") of a class difference ("dif") of a singleton ("sn"), it is labeled eldifsn 4713. An "n" is often used for negation (¬), e.g., nan 825.
  • 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 (even though its definition is in df-c 10532) and "re" represents real numbers ( definition df-r 10536). The empty set often uses fragment 0, even though it is defined in df-nul 4291. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 10537), 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 11761.
  • 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 15493 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 15414) we have value cosval 15466 and closure coscl 15470.
  • 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 28110 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 (inference, deduction, or closed form, see above). Additionally, we sometimes suffix with "v" the label of a theorem adding a disjoint variable condition, as in 19.21v 1931 versus 19.21 2198. This often permits to prove the result using fewer axioms, and/or to eliminate a nonfreeness hypothesis (such as 𝑥𝜑 in 19.21 2198). If no constraint is put on axiom use, then the v-version can be proved from the original theorem using nfv 1906. If two (resp. three) such disjoint variable conditions are added, then the suffix "vv" (resp. "vvv") is used, e.g., exlimivv 1924. 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 2657 derived from eu6 2655. 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 5333. 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 2424 (cbval 2410 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). As a general rule, the suffixes for the theorem forms ("i", "d" or "g") should be the first of multiple suffixes, as for example in vtocldf 3556. 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 65, syl2anc 584), commutes (e.g., biimpac 479)
    • d : theorem in deduction form
    • f : theorem with a hypothesis such as 𝑥𝜑
    • 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)
    • t : theorem in closed form (not having an "is a set" antecedent)
    • 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 (mathbox only)
    • OLD : old/obsolete version of a theorem (or proof) or definition
  • 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 477, rexlimiva 3281
ablAbelian group df-abl 18840 Abel Yes ablgrp 18842, zringabl 20551
absabsorption No ressabs 16553
absabsolute value (of a complex number) df-abs 14585 (abs‘𝐴) Yes absval 14587, absneg 14627, abs1 14647
adadding No adantr 481, ad2antlr 723
addadd (see "p") df-add 10537 (𝐴 + 𝐵) Yes addcl 10608, addcom 10815, addass 10613
al"for all" 𝑥𝜑 No alim 1802, alex 1817
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 397 (𝜑𝜓) Yes anor 976, iman 402, imnan 400
antantecedent No adantr 481
assassociative No biass 386, orass 915, mulass 10614
asymasymmetric, antisymmetric No intasym 5969, asymref 5970, posasymb 17552
axaxiom No ax6dgen 2123, ax1cn 10560
bas, base base (set of an extensible structure) df-base 16479 (Base‘𝑆) Yes baseval 16532, ressbas 16544, cnfldbas 20479
b, bibiconditional ("iff", "if and only if") df-bi 208 (𝜑𝜓) Yes impbid 213, sspwb 5333
brbinary relation df-br 5059 𝐴𝑅𝐵 Yes brab1 5106, brun 5109
cbvchange bound variable No cbvalivw 2005, cbvrex 3447
clclosure No ifclda 4499, ovrcl 7186, zaddcl 12011
cncomplex numbers df-c 10532 Yes nnsscn 11632, nncn 11635
cnfldfield of complex numbers df-cnfld 20476 fld Yes cnfldbas 20479, cnfldinv 20506
cntzcentralizer df-cntz 18387 (Cntz‘𝑀) Yes cntzfval 18390, dprdfcntz 19068
cnvconverse df-cnv 5557 𝐴 Yes opelcnvg 5745, f1ocnv 6621
cocomposition df-co 5558 (𝐴𝐵) Yes cnvco 5750, fmptco 6884
comcommutative No orcom 864, bicomi 225, eqcomi 2830
concontradiction, contraposition No condan 814, con2d 136
csbclass substitution df-csb 3883 𝐴 / 𝑥𝐵 Yes csbid 3895, csbie2g 3922
cygcyclic group df-cyg 18928 CycGrp Yes iscyg 18929, zringcyg 20568
ddeduction form (suffix) No idd 24, impbid 213
df(alternate) definition (prefix) No dfrel2 6040, dffn2 6510
di, distrdistributive No andi 1001, imdi 391, ordi 999, difindi 4257, ndmovdistr 7326
difclass difference df-dif 3938 (𝐴𝐵) Yes difss 4107, difindi 4257
divdivision df-div 11287 (𝐴 / 𝐵) Yes divcl 11293, divval 11289, divmul 11290
dmdomain df-dm 5559 dom 𝐴 Yes dmmpt 6088, iswrddm0 13878
e, eq, equequals (equ for setvars, eq for classes) df-cleq 2814 𝐴 = 𝐵 Yes 2p2e4 11761, uneqri 4126, equtr 2019
edgedge df-edg 26761 (Edg‘𝐺) Yes edgopval 26764, usgredgppr 26906
elelement of 𝐴𝐵 Yes eldif 3945, eldifsn 4713, elssuni 4861
enequinumerous df-en 𝐴𝐵 Yes domen 8511, enfi 8723
eu"there exists exactly one" eu6 2655 ∃!𝑥𝜑 Yes euex 2658, euabsn 4656
exexists (i.e. is a set) ∈ V No brrelex1 5599, 0ex 5203
ex, e"there exists (at least one)" df-ex 1772 𝑥𝜑 Yes exim 1825, alex 1817
expexport No expt 178, expcom 414
f"not free in" (suffix) No equs45f 2477, sbf 2262
ffunction df-f 6353 𝐹:𝐴𝐵 Yes fssxp 6528, opelf 6533
falfalse df-fal 1541 Yes bifal 1544, falantru 1563
fifinite intersection df-fi 8864 (fi‘𝐵) Yes fival 8865, inelfi 8871
fi, finfinite df-fin 8502 Fin Yes isfi 8522, snfi 8583, onfin 8698
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 35153) df-field 19436 Field Yes isfld 19442, fldidom 20008
fnfunction with domain df-fn 6352 𝐴 Fn 𝐵 Yes ffn 6508, fndm 6449
frgpfree group df-frgp 18767 (freeGrp‘𝐼) Yes frgpval 18815, frgpadd 18820
fsuppfinitely supported function df-fsupp 8823 𝑅 finSupp 𝑍 Yes isfsupp 8826, fdmfisuppfi 8831, fsuppco 8854
funfunction df-fun 6351 Fun 𝐹 Yes funrel 6366, ffun 6511
fvfunction value df-fv 6357 (𝐹𝐴) Yes fvres 6683, swrdfv 14000
fzfinite set of sequential integers df-fz 12883 (𝑀...𝑁) Yes fzval 12884, eluzfz 12893
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 12995, fz0tp 12998
fzohalf-open integer range df-fzo 13024 (𝑀..^𝑁) Yes elfzo 13030, elfzofz 13043
gmore general (suffix); eliminates "is a set" hypotheses No uniexg 7456
grgraph No uhgrf 26775, isumgr 26808, usgrres1 27025
grpgroup df-grp 18046 Grp Yes isgrp 18049, tgpgrp 22616
gsumgroup sum df-gsum 16706 (𝐺 Σg 𝐹) Yes gsumval 17877, gsumwrev 18434
hashsize (of a set) df-hash 13681 (♯‘𝐴) Yes hashgval 13683, hashfz1 13696, hashcl 13707
hbhypothesis builder (prefix) No hbxfrbi 1816, hbald 2165, hbequid 35927
hm(monoid, group, ring) homomorphism No ismhm 17948, isghm 18298, isrhm 19404
iinference (suffix) No eleq1i 2903, tcsni 9174
iimplication (suffix) No brwdomi 9021, infeq5i 9088
ididentity No biid 262
iedgindexed edge df-iedg 26712 (iEdg‘𝐺) Yes iedgval0 26753, edgiedgb 26767
idmidempotent No anidm 565, tpidm13 4686
im, impimplication (label often omitted) df-im 14450 (𝐴𝐵) Yes iman 402, imnan 400, impbidd 211
imaimage df-ima 5562 (𝐴𝐵) Yes resima 5881, imaundi 6002
impimport No biimpa 477, impcom 408
inintersection df-in 3942 (𝐴𝐵) Yes elin 4168, incom 4177
infinfimum df-inf 8896 inf(ℝ+, ℝ*, < ) Yes fiinfcl 8954, infiso 8961
is...is (something a) ...? No isring 19232
jjoining, disjoining No jc 164, jaoi 851
lleft No olcd 870, simpl 483
mapmapping operation or set exponentiation df-map 8398 (𝐴m 𝐵) Yes mapvalg 8406, elmapex 8417
matmatrix df-mat 20947 (𝑁 Mat 𝑅) Yes matval 20950, matring 20982
mdetdeterminant (of a square matrix) df-mdet 21124 (𝑁 maDet 𝑅) Yes mdetleib 21126, mdetrlin 21141
mgmmagma df-mgm 17842 Magma Yes mgmidmo 17860, mgmlrid 17867, ismgm 17843
mgpmultiplicative group df-mgp 19171 (mulGrp‘𝑅) Yes mgpress 19181, ringmgp 19234
mndmonoid df-mnd 17902 Mnd Yes mndass 17910, mndodcong 18601
mo"there exists at most one" df-mo 2618 ∃*𝑥𝜑 Yes eumo 2659, moim 2622
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mpomaps-to notation for an operation df-mpo 7150 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpompt 7255, resmpo 7261
mptmodus ponendo tollens No mptnan 1760, mptxor 1761
mptmaps-to notation for a function df-mpt 5139 (𝑥𝐴𝐵) Yes fconstmpt 5608, resmpt 5899
mpt2maps-to notation for an operation (deprecated). We are in the process of replacing mpt2 with mpo in labels. df-mpo 7150 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpompt 7255, resmpo 7261
mulmultiplication (see "t") df-mul 10538 (𝐴 · 𝐵) Yes mulcl 10610, divmul 11290, mulcom 10612, mulass 10614
n, notnot ¬ 𝜑 Yes nan 825, notnotr 132
nenot equaldf-ne 𝐴𝐵 Yes exmidne 3026, neeqtrd 3085
nelnot element ofdf-nel 𝐴𝐵 Yes neli 3125, nnel 3132
ne0not equal to zero (see n0) ≠ 0 No negne0d 10984, ine0 11064, gt0ne0 11094
nf "not free in" (prefix) No nfnd 1849
ngpnormed group df-ngp 23122 NrmGrp Yes isngp 23134, ngptps 23140
nmnorm (on a group or ring) df-nm 23121 (norm‘𝑊) Yes nmval 23128, subgnm 23171
nnpositive integers df-nn 11628 Yes nnsscn 11632, nncn 11635
nn0nonnegative integers df-n0 11887 0 Yes nnnn0 11893, nn0cn 11896
n0not the empty set (see ne0) ≠ ∅ No n0i 4298, vn0 4303, ssn0 4353
OLDold, obsolete (to be removed soon) No 19.43OLD 1875
onordinal number df-on 6189 𝐴 ∈ On Yes elon 6194, 1on 8100 onelon 6210
opordered pair df-op 4566 𝐴, 𝐵 Yes dfopif 4794, opth 5360
oror df-or 842 (𝜑𝜓) Yes orcom 864, anor 976
otordered triple df-ot 4568 𝐴, 𝐵, 𝐶 Yes euotd 5395, fnotovb 7195
ovoperation value df-ov 7148 (𝐴𝐹𝐵) Yes fnotovb 7195, fnovrn 7312
pplus (see "add"), for all-constant theorems df-add 10537 (3 + 2) = 5 Yes 3p2e5 11777
pfxprefix df-pfx 14023 (𝑊 prefix 𝐿) Yes pfxlen 14035, ccatpfx 14053
pmPrincipia Mathematica No pm2.27 42
pmpartial mapping (operation) df-pm 8399 (𝐴pm 𝐵) Yes elpmi 8415, pmsspw 8431
prpair df-pr 4562 {𝐴, 𝐵} Yes elpr 4582, prcom 4662, prid1g 4690, prnz 4706
prm, primeprime (number) df-prm 16006 Yes 1nprm 16013, dvdsprime 16021
pssproper subset df-pss 3953 𝐴𝐵 Yes pssss 4071, sspsstri 4078
q rational numbers ("quotients") df-q 12338 Yes elq 12339
rright No orcd 869, simprl 767
rabrestricted class abstraction df-rab 3147 {𝑥𝐴𝜑} Yes rabswap 3489, df-oprab 7149
ralrestricted universal quantification df-ral 3143 𝑥𝐴𝜑 Yes ralnex 3236, ralrnmpo 7278
rclreverse closure No ndmfvrcl 6695, nnarcl 8232
rereal numbers df-r 10536 Yes recn 10616, 0re 10632
relrelation df-rel 5556 Rel 𝐴 Yes brrelex1 5599, relmpoopab 7780
resrestriction df-res 5561 (𝐴𝐵) Yes opelres 5853, f1ores 6623
reurestricted existential uniqueness df-reu 3145 ∃!𝑥𝐴𝜑 Yes nfreud 3373, reurex 3432
rexrestricted existential quantification df-rex 3144 𝑥𝐴𝜑 Yes rexnal 3238, rexrnmpo 7279
rmorestricted "at most one" df-rmo 3146 ∃*𝑥𝐴𝜑 Yes nfrmod 3374, nrexrmo 3436
rnrange df-rn 5560 ran 𝐴 Yes elrng 5756, rncnvcnv 5798
rng(unital) ring df-ring 19230 Ring Yes ringidval 19184, isring 19232, ringgrp 19233
rotrotation No 3anrot 1092, 3orrot 1084
seliminates need for syllogism (suffix) No ancoms 459
sb(proper) substitution (of a set) df-sb 2061 [𝑦 / 𝑥]𝜑 Yes spsbe 2079, sbimi 2070
sbc(proper) substitution of a class df-sbc 3772 [𝐴 / 𝑥]𝜑 Yes sbc2or 3780, sbcth 3786
scascalar df-sca 16571 (Scalar‘𝐻) Yes resssca 16640, mgpsca 19177
simpsimple, simplification No simpl 483, simp3r3 1275
snsingleton df-sn 4560 {𝐴} Yes eldifsn 4713
spspecialization No spsbe 2079, spei 2406
sssubset df-ss 3951 𝐴𝐵 Yes difss 4107
structstructure df-struct 16475 Struct Yes brstruct 16482, structfn 16490
subsubtract df-sub 10861 (𝐴𝐵) Yes subval 10866, subaddi 10962
supsupremum df-sup 8895 sup(𝐴, 𝐵, < ) Yes fisupcl 8922, supmo 8905
suppsupport (of a function) df-supp 7822 (𝐹 supp 𝑍) Yes ressuppfi 8848, mptsuppd 7844
swapswap (two parts within a theorem) No rabswap 3489, 2reuswap 3736
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 4218, cnvsym 5968
symgsymmetric group df-symg 18436 (SymGrp‘𝐴) Yes symghash 18443, pgrpsubgsymg 18468
t times (see "mul"), for all-constant theorems df-mul 10538 (3 · 2) = 6 Yes 3t2e6 11792
th, t theorem No nfth 1793, sbcth 3786, weth 9906, ancomst 465
tptriple df-tp 4564 {𝐴, 𝐵, 𝐶} Yes eltpi 4619, tpeq1 4672
trtransitive No bitrd 280, biantr 802
tru, t true, truth df-tru 1531 Yes bitru 1537, truanfal 1562, biimt 362
ununion df-un 3940 (𝐴𝐵) Yes uneqri 4126, uncom 4128
unitunit (in a ring) df-unit 19323 (Unit‘𝑅) Yes isunit 19338, nzrunit 19970
v setvar (especially for specializations of theorems when a class is replaced by a setvar variable) x Yes cv 1527, vex 3498, velpw 4545, vtoclf 3559
v disjoint variable condition used in place of nonfreeness hypothesis (suffix) No spimv 2402
vtx vertex df-vtx 26711 (Vtx‘𝐺) Yes vtxval0 26752, opvtxov 26718
vv two disjoint variable conditions used in place of nonfreeness hypotheses (suffix) No 19.23vv 1935
wweak (version of a theorem) (suffix) No ax11w 2125, spnfw 1975
wrdword df-word 13852 Word 𝑆 Yes iswrdb 13857, wrdfn 13866, ffz0iswrd 13881
xpcross product (Cartesian product) df-xp 5555 (𝐴 × 𝐵) Yes elxp 5572, opelxpi 5586, xpundi 5614
xreXtended reals df-xr 10668 * Yes ressxr 10674, rexr 10676, 0xr 10677
z integers (from German "Zahlen") df-z 11971 Yes elz 11972, zcn 11975
zn ring of integers mod 𝑁 df-zn 20584 (ℤ/nℤ‘𝑁) Yes znval 20612, zncrng 20621, znhash 20635
zringring of integers df-zring 20548 ring Yes zringbas 20553, zringcrng 20549
0, z slashed zero (empty set) df-nul 4291 Yes n0i 4298, vn0 4303; snnz 4705, prnz 4706

(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision. (Revised by the Metamath team, 22-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑
 
Theoremconventions-comments 28109

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding comments, and more generally nonmathematical conventions. For other conventions, see conventions 28107 and links therein.

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    The input format is ASCII. Tab characters are not allowed. If non-ASCII characters have to be displayed in comments, use embedded mathematical symbols when they have been defined (e.g., "` -> `" for " ") or HTML entities (e.g., "&eacute;" for "é"). Default indentation is by two spaces. Lines are hard-wrapped to be at most 79-character long, excluding the newline character (this can be achieved, except currently for section comments, by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command or by running the script scripts/rewrap). The file ends with an empty line. There are no trailing spaces. As for line wrapping in statements, we try to break lines before the most important token.

  • Language and spelling.

    The MPE uses American English, e.g., we write "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"); therefore, "analogue" is used for the noun and "analogous" for the adjective. We favor regular plurals, e.g., "formulas" instead of "formulae", "lemmas" instead of "lemmata".

    Since comments may contain many space-separated symbols, we use the older convention of two spaces after a period ending a sentence, to better separate sentences (this is also achieved by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command).

    When compound words have several variants, we prefer the concatenated variant (e.g., nonempty, nontrivial, nonpositive, nonzero, nonincreasing, nondegenerate...).

  • Quotation style.

    We use the "logical quotation style", which means that when a quoted text is followed by punctuation not pertaining to the quote, then the quotation mark precedes the punctuation (like at the beginning of this sentence). We use the double quote as default quotation mark (since the single quote also serves as apostrophe), and the single quote in the case of a nested quotation.

  • Sectioning and section headers.

    The database set.mm has a sectioning system with four levels of titles, signaled by "decoration lines" which are 79-character long repetitions of ####, #*#*, =-=-, and -.-. (in descending order of sectioning level). Sections of any level are separated by two blank lines (if there is a "@( Begin $[ ... $] @)" comment (where "@" is actually "$") before a section header, then the double blank line should go before that comment, which is considered as belonging to that section). The format of section headers is best seen in the source file (set.mm); it is as follows:

    • a line with "@(" (with the "@" replaced by "$");
    • a decoration line;
    • section title indented with two spaces;
    • a (matching) decoration line;
    • [blank line; header comment indented with two spaces; blank line;]
    • a line with "@)" (with the "@" replaced by "$");
    • one blank line.
    As everywhere else, lines are hard-wrapped to be 79-character long. It is expected that in a future version, the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will reformat section headers to automatically conform with this format.

  • Comments.

    As for formatting of the file set.mm, and in particular formatting and layout of the comments, the foremost rule is consistency. The first sections of set.mm, in particular Part 1 "Classical first-order logic with equality" can serve as a model for contributors. Some formatting rules are enforced when using the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command. Here are a few other rules, which are not enforced, but that we try to follow:

      A math string in a comment should be surrounded by space-separated backquotes on the same line, and if it is too long it should be broken into multiple adjacent math strings on multiple lines.
    • The file set.mm should have a double blank line between sections, and at no other places. In particular, there are no triple blank lines.
    • The header comments should be spaced as those of Part 1, namely, with a blank line before and after the comment, and an indentation of two spaces.
    • As of 20-Sep-2022, section comments are not rewrapped by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command, though this is expected in a future version. Similar spacing and wrapping should be used as for other comments: double spaces after a period ending a sentence, line wrapping with line width of 79, and no trailing spaces at the end of lines.

  • Contributors.

    Each assertion (theorem, definition or axiom) has a contribution tag of the form "(Contributed by xxx, dd-Mmm-yyyy.)" (see Metamath Book, p. 142). The date cannot serve as a proof of anteriority since there is currently no formal guarantee that the date is correct (a claim of anterioty can be backed, for instance, by the uploading of a result to a public repository with verifiable date). The contributor is the first person who proved (or stated, in the case of a definition or axiom) the statement. The list of contributors appears at the beginning of set.mm.

    An exception should be made if a theorem is essentially an extract or a variant of an already existing theorem, in which case the contributor should be that of the statement from which it is derived, with the modification signaled by a "(Revised by xxx, dd-Mmm-yyyy.)" tag.

  • Usage of parentheticals.

    Usually, the comment of a theorem should contain at most one of the "Revised by" and "Proof shortened by" parentheticals, see Metamath Book, pp. 142-143 (there must always be a "Contributed by" parenthetical for every theorem). Exceptions for "Proof shortened by" parentheticals are essential additional shortenings by a different person. If a proof is shortened by the same person, the date within the "Proof shortened by" parenthetical should be updated only. This also holds for "Revised by" parentheticals, except that also more than one of such parentheticals for the same person are acceptable (if there are good reasons for this). A revision tag is optionally preceded by a short description of the revision. Since this is somewhat subjective, judgment and intellectual honesty should be applied, with collegial settlement in case of dispute.

  • Explaining new labels.

    A comment should explain the first use of an abbreviation within a label. This is often in a definition (e.g., the definition df-an 397 introduces the abbreviation "an" for conjunction ("and")), but not always (e.g., the theorem alim 1802 introduces the abbreviation "al" for the universal quantifier ("for all")). See conventions-labels 28108 for a table of abbreviations.

(Contributed by the Metamath team, 27-Dec-2016.) Date of last revision. (Revised by the Metamath team, 22-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑       𝜑
 
17.1.2  Natural deduction
 
Theoremnatded 28110 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 set.mm). 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 Γ𝜓 => Γ𝜓 idi 1 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
I Γ𝜓 & Γ𝜒 => Γ𝜓𝜒 jca 512 jca 512, pm3.2i 471 Definition I in [Pfenning] p. 18, definition Im,n in [Clemente] p. 10, and definition I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
EL Γ𝜓𝜒 => Γ𝜓 simpld 495 simpld 495, adantr 481 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 Γ𝜓𝜒 => Γ𝜒 simprd 496 simpr 485, adantl 482 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 Γ, 𝜓𝜒 => Γ𝜓𝜒 ex 413 ex 413 Definition I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition I in [Indrzejczak] p. 33.
E Γ𝜓𝜒 & Γ𝜓 => Γ𝜒 mpd 15 ax-mp 5, mpd 15, mpdan 683, imp 407 Definition E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition E in [Indrzejczak] p. 33.
IL Γ𝜓 => Γ𝜓𝜒 olcd 870 olc 862, olci 860, olcd 870 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 869 orc 861, orci 859, orcd 869 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 952 mpjaodan 952, jaodan 951, jaod 853 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1548 pm2.01d 191
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 812 mtand 812 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 813 pm2.65da 813 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 795 pm2.01d 191, pm2.65da 813, pm2.65d 197 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1550 pm2.21dd 196
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 196 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 196 pm2.21dd 196, pm2.21d 121, pm2.21 123 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ trud 1538 tru 1532, trud 1538, mptru 1535 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1546 falim 1545 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1919 alrimiv 1919, ralrimiva 3182 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3785 spcv 3605, rspcv 3617 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3865 spcev 3606, rspcev 3622 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1927 exlimddv 1927, exlimdd 2211, exlimdv 1925, rexlimdva 3284 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1549 efald 1549 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 796 pm2.18da 796, pm2.18d 127, pm2.18 128 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 135 notnotrd 135, notnotr 132 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 889 exmid 888 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2822 eqid 2821, eqidd 2822 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3777 sbceq1d 3776, 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 Γ represents the set of (current) hypotheses. We use wff variable names beginning with 𝜓 to provide a closer representation of the Metamath equivalents (which typically use the antedent 𝜑 to represent the context Γ).

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 28111, ex-natded5.3 28114, ex-natded5.5 28117, ex-natded5.7 28118, ex-natded5.8 28120, ex-natded5.13 28122, ex-natded9.20 28124, and ex-natded9.26 28126.

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

𝜑       𝜑
 
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 28110 and mmnatded.html 28110. 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 28111 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 ((𝜓𝜒) → 𝜃) (𝜑 → ((𝜓𝜒) → 𝜃)) Given $e.
22 (𝜒𝜓) (𝜑 → (𝜒𝜓)) Given $e.
31 𝜒 (𝜑𝜒) Given $e.
43 𝜓 (𝜑𝜓) E 2,3 mpd 15, the MPE equivalent of E, 1,2
54 (𝜓𝜒) (𝜑 → (𝜓𝜒)) I 4,3 jca 512, the MPE equivalent of I, 3,1
66 𝜃 (𝜑𝜃) 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 𝜑 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 28112. A proof without context is shown in ex-natded5.2i 28113. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → ((𝜓𝜒) → 𝜃))    &   (𝜑 → (𝜒𝜓))    &   (𝜑𝜒)       (𝜑𝜃)
 
Theoremex-natded5.2-2 28112 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 28111 and ex-natded5.2i 28113. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → 𝜃))    &   (𝜑 → (𝜒𝜓))    &   (𝜑𝜒)       (𝜑𝜃)
 
Theoremex-natded5.2i 28113 The same as ex-natded5.2 28111 and ex-natded5.2-2 28112 but with no context. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒) → 𝜃)    &   (𝜒𝜓)    &   𝜒       𝜃
 
Theoremex-natded5.3 28114 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 28115. A proof without context is shown in ex-natded5.3i 28116. 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 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e; adantr 481 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given $e; adantr 481 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 485, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 481 was used to transform its dependency (we could also use imp 407 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 481 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 512, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 413, 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 𝜑 and uses the Metamath equivalents of the natural deduction rules. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremex-natded5.3-2 28115 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 28114 and ex-natded5.3i 28116. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))
 
Theoremex-natded5.3i 28116 The same as ex-natded5.3 28114 and ex-natded5.3-2 28115 but with no context. Identical to jccir 522, which should be used instead. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   (𝜒𝜃)       (𝜓 → (𝜒𝜃))
 
Theoremex-natded5.5 28117 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 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e; adantr 481 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given $e; we'll use adantr 481 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 485
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 481 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 813 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 𝜑 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 481; simpr 485 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 199; a proof without context is shown in mto 198.

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

(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ 𝜓)
 
Theoremex-natded5.7 28118 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 28119. 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 (𝜓 ∨ (𝜒𝜃)) (𝜑 → (𝜓 ∨ (𝜒𝜃))) Given $e. No need for adantr 481 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 485
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 869, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 485
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 495, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 870, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 952, 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 𝜑 and uses the Metamath equivalents of the natural deduction rules. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))
 
Theoremex-natded5.7-2 28119 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 28118. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))
 
Theoremex-natded5.8 28120 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 ((𝜓𝜒) → ¬ 𝜃) (𝜑 → ((𝜓𝜒) → ¬ 𝜃)) Given $e; adantr 481 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given $e; adantr 481 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given $e; adantr 481 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given $e. adantr 481 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 485. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 512 (I), 6,8 (adantr 481 to bring in scope)
75 ... ¬ 𝜃 ((𝜑𝜓) → ¬ 𝜃) E 1,6 mpd 15 (E), 2,4
812 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 ¬ 𝜓 (𝜑 → ¬ 𝜓) ¬I 5,7,8 pm2.65da 813 (¬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 𝜑 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 481; simpr 485 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 28121.

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

(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)
 
Theoremex-natded5.8-2 28121 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 28120. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)
 
Theoremex-natded5.13 28122 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 28123. 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 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given $e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given $e. adantr 481 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given $e. ad2antrr 722 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 485
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 869 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 485
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 485
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 481 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 813 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 135 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 870 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 952 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 𝜑 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 481; simpr 485 is useful when you want to depend directly on the new assumption). (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))
 
Theoremex-natded5.13-2 28123 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 28122. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))
 
Theoremex-natded9.20 28124 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 (𝜓 ∧ (𝜒𝜃)) (𝜑 → (𝜓 ∧ (𝜒𝜃))) Given $e
22 𝜓 (𝜑𝜓) EL 1 simpld 495 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 496 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 485
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 512 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 869 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 485
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 512 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 870 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 952 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 𝜑 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 481; simpr 485 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 28125. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 
Theoremex-natded9.20-2 28125 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 28124. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))
 
Theoremex-natded9.26 28126* 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 𝑥𝑦𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝑦𝜓) Given $e.
26 ...| 𝑦𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) ND hypothesis assumption simpr 485. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3785 (E), 5,6. To use it we need a1i 11 and vex 3498. This could be immediately done with 19.21bi 2178, but we want to show the general approach for substitution.
412;8,9,10,11 ... 𝑥𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) I 3,a spesbcd 3865 (I), 11. To use it we need sylibr 235, which in turn requires sylib 219 and two uses of sbcid 3788. This could be more immediately done using 19.8a 2170, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2211 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1906 and nfe1 2145 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1919 (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 𝜑 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, 𝜓(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 28127.

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

(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)
 
Theoremex-natded9.26-2 28127* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 28126. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)
 
17.1.4  Definitional examples
 
Theoremex-or 28128 Example for df-or 842. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 ∨ 4 = 4)
 
Theoremex-an 28129 Example for df-an 397. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 ∧ 3 = 3)
 
Theoremex-dif 28130 Example for df-dif 3938. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∖ {1, 8}) = {3}
 
Theoremex-un 28131 Example for df-un 3940. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∪ {1, 8}) = {1, 3, 8}
 
Theoremex-in 28132 Example for df-in 3942. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∩ {1, 8}) = {1}
 
Theoremex-uni 28133 Example for df-uni 4833. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
{{1, 3}, {1, 8}} = {1, 3, 8}
 
Theoremex-ss 28134 Example for df-ss 3951. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊆ {1, 2, 3}
 
Theoremex-pss 28135 Example for df-pss 3953. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊊ {1, 2, 3}
 
Theoremex-pw 28136 Example for df-pw 4539. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))
 
Theoremex-pr 28137 Example for df-pr 4562. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐴 ∈ {1, -1} → (𝐴↑2) = 1)
 
Theoremex-br 28138 Example for df-br 5059. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)
 
Theoremex-opab 28139* Example for df-opab 5121. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)
 
Theoremex-eprel 28140 Example for df-eprel 5459. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
5 E {1, 5}
 
Theoremex-id 28141 Example for df-id 5454. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(5 I 5 ∧ ¬ 4 I 5)
 
Theoremex-po 28142 Example for df-po 5468. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
( < Po ℝ ∧ ¬ ≤ Po ℝ)
 
Theoremex-xp 28143 Example for df-xp 5555. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩})
 
Theoremex-cnv 28144 Example for df-cnv 5557. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}
 
Theoremex-co 28145 Example for df-co 5558. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((exp ∘ cos)‘0) = e
 
Theoremex-dm 28146 Example for df-dm 5559. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3})
 
Theoremex-rn 28147 Example for df-rn 5560. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})
 
Theoremex-res 28148 Example for df-res 5561. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {⟨2, 6⟩})
 
Theoremex-ima 28149 Example for df-ima 5562. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6})
 
Theoremex-fv 28150 Example for df-fv 6357. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9)
 
Theoremex-1st 28151 Example for df-1st 7680. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(1st ‘⟨3, 4⟩) = 3
 
Theoremex-2nd 28152 Example for df-2nd 7681. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(2nd ‘⟨3, 4⟩) = 4
 
Theorem1kp2ke3k 28153 Example for df-dec 12088, 1000 + 2000 = 3000.

This proof disproves (by counterexample) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with (2 + 1) = 3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 12088 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.)

(1000 + 2000) = 3000
 
Theoremex-fl 28154 Example for df-fl 13152. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)
 
Theoremex-ceil 28155 Example for df-ceil 13153. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)
 
Theoremex-mod 28156 Example for df-mod 13228. (Contributed by AV, 3-Sep-2021.)
((5 mod 3) = 2 ∧ (-7 mod 2) = 1)
 
Theoremex-exp 28157 Example for df-exp 13420. (Contributed by AV, 4-Sep-2021.)
((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))
 
Theoremex-fac 28158 Example for df-fac 13624. (Contributed by AV, 4-Sep-2021.)
(!‘5) = 120
 
Theoremex-bc 28159 Example for df-bc 13653. (Contributed by AV, 4-Sep-2021.)
(5C3) = 10
 
Theoremex-hash 28160 Example for df-hash 13681. (Contributed by AV, 4-Sep-2021.)
(♯‘{0, 1, 2}) = 3
 
Theoremex-sqrt 28161 Example for df-sqrt 14584. (Contributed by AV, 4-Sep-2021.)
(√‘25) = 5
 
Theoremex-abs 28162 Example for df-abs 14585. (Contributed by AV, 4-Sep-2021.)
(abs‘-2) = 2
 
Theoremex-dvds 28163 Example for df-dvds 15598: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
3 ∥ 6
 
Theoremex-gcd 28164 Example for df-gcd 15834. (Contributed by AV, 5-Sep-2021.)
(-6 gcd 9) = 3
 
Theoremex-lcm 28165 Example for df-lcm 15924. (Contributed by AV, 5-Sep-2021.)
(6 lcm 9) = 18
 
Theoremex-prmo 28166 Example for df-prmo 16358: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.)
(#p10) = 210
 
17.1.5  Other examples
 
Theoremaevdemo 28167* Proof illustrating the comment of aev2 2054. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))
 
Theoremex-ind-dvds 28168 Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
(𝑁 ∈ ℕ0 → 3 ∥ ((4↑𝑁) + 2))
 
Theoremex-fpar 28169 Formalized example provided in the comment for fpar 7802. (Contributed by AV, 3-Jan-2024.)
𝐻 = (((1st ↾ (V × V)) ∘ (𝐹 ∘ (1st ↾ (V × V)))) ∩ ((2nd ↾ (V × V)) ∘ (𝐺 ∘ (2nd ↾ (V × V)))))    &   𝐴 = (0[,)+∞)    &   𝐵 = ℝ    &   𝐹 = (√ ↾ 𝐴)    &   𝐺 = (sin ↾ 𝐵)       ((𝑋𝐴𝑌𝐵) → (𝑋( + ∘ 𝐻)𝑌) = ((√‘𝑋) + (sin‘𝑌)))
 
17.2  Humor
 
17.2.1  April Fool's theorem
 
Theoremavril1 28170 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 mmnotes.txt, under the 1-Apr-2006 entry.

¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))
 
Theorem2bornot2b 28171 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 · 𝐵 ∨ ¬ 2 · 𝐵)
 
Theoremhelloworld 28172 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.)
¬ ( ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑))
 
Theorem1p1e2apr1 28173 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 28174 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 28175 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 28176 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.) (Revised by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ (0...+∞) → +∞ < 𝐵)
 
Theorem9p10ne21 28177 9 + 10 is not equal to 21. This disproves a popular meme which asserts that 9 + 10 does equal 21. See https://www.quora.com/Can-someone-try-to-prove-to-me-that-9+10-21 for attempts to prove that 9 + 10 = 21, and see https://tinyurl.com/9p10e21 for the history of the 9 + 10 = 21 meme. (Contributed by BTernaryTau, 25-Aug-2023.)
(9 + 10) ≠ 21
 
Theorem9p10ne21fool 28178 9 + 10 equals 21. This astonishing thesis lives as a meme on the internet, and may be believed by quite some people. At least repeated requests to falsify it are a permanent part of the story. Prof. Loof Lirpa did not rest until he finally came up with a computer verifiable mathematical proof, that only a fool can think so. (Contributed by Prof. Loof Lirpa, 26-Aug-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
((9 + 10) = 21 → 𝐹∅(0 · 1))
 
17.3  (Future - to be reviewed and classified)
 
17.3.1  Planar incidence geometry
 
Syntaxcplig 28179 Extend class notation with the class of all planar incidence geometries.
class Plig
 
Definitiondf-plig 28180* Define the class of planar incidence geometries. We use Hilbert's axioms and adapt them to planar geometry. We use for the incidence relation. We could have used a generic binary relation, but using allows us to reuse previous results. Much of what follows is directly borrowed from Aitken, Incidence-Betweenness Geometry, 2008, http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf.

The class Plig is the class of planar incidence geometries, where a planar incidence geometry is defined as a set of lines satisfying three axioms. In the definition below, 𝑥 denotes a planar incidence geometry, so 𝑥 denotes the union of its lines, that is, the set of points in the plane, 𝑙 denotes a line, and 𝑎, 𝑏, 𝑐 denote points. Therefore, the axioms are: 1) for all pairs of (distinct) points, there exists a unique line containing them; 2) all lines contain at least two points; 3) there exist three non-collinear points. (Contributed by FL, 2-Aug-2009.)

Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}
 
Theoremisplig 28181* The predicate "is a planar incidence geometry" for sets. (Contributed by FL, 2-Aug-2009.)
𝑃 = 𝐺       (𝐺𝐴 → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
 
Theoremispligb 28182* The predicate "is a planar incidence geometry". (Contributed by BJ, 2-Dec-2021.)
𝑃 = 𝐺       (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))
 
Theoremtncp 28183* In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.)
𝑃 = 𝐺       (𝐺 ∈ Plig → ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))
 
Theoreml2p 28184* For any line in a planar incidence geometry, there exist two different points on the line. (Contributed by AV, 28-Nov-2021.)
𝑃 = 𝐺       ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝐿𝑏𝐿))
 
Theoremlpni 28185* For any line in a planar incidence geometry, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
𝑃 = 𝐺       ((𝐺 ∈ Plig ∧ 𝐿𝐺) → ∃𝑎𝑃 𝑎𝐿)
 
Theoremnsnlplig 28186 There is no "one-point line" in a planar incidence geometry. (Contributed by BJ, 2-Dec-2021.) (Proof shortened by AV, 5-Dec-2021.)
(𝐺 ∈ Plig → ¬ {𝐴} ∈ 𝐺)
 
TheoremnsnlpligALT 28187 Alternate version of nsnlplig 28186 using the predicate instead of ¬ ∈ and whose proof is shorter. (Contributed by AV, 5-Dec-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐺 ∈ Plig → {𝐴} ∉ 𝐺)
 
Theoremn0lplig 28188 There is no "empty line" in a planar incidence geometry. (Contributed by AV, 28-Nov-2021.) (Proof shortened by BJ, 2-Dec-2021.)
(𝐺 ∈ Plig → ¬ ∅ ∈ 𝐺)
 
Theoremn0lpligALT 28189 Alternate version of n0lplig 28188 using the predicate instead of ¬ ∈ and whose proof bypasses nsnlplig 28186. (Contributed by AV, 28-Nov-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐺 ∈ Plig → ∅ ∉ 𝐺)
 
Theoremeulplig 28190* Through two distinct points of a planar incidence geometry, there is a unique line. (Contributed by BJ, 2-Dec-2021.)
𝑃 = 𝐺       ((𝐺 ∈ Plig ∧ ((𝐴𝑃𝐵𝑃) ∧ 𝐴𝐵)) → ∃!𝑙𝐺 (𝐴𝑙𝐵𝑙))
 
Theorempliguhgr 28191 Any planar incidence geometry 𝐺 can be regarded as a hypergraph with its points as vertices and its lines as edges. See incistruhgr 26792 for a generalization of this case for arbitrary incidence structures (planar incidence geometries are such incidence structures). (Proposed by Gerard Lang, 24-Nov-2021.) (Contributed by AV, 28-Nov-2021.)
(𝐺 ∈ Plig → ⟨ 𝐺, ( I ↾ 𝐺)⟩ ∈ UHGraph)
 
17.3.2  Aliases kept to prevent broken links

This section contains a few aliases that we temporarily keep to prevent broken links. If you land on any of these, please let the originating site and/or us know that the link that made you land here should be changed.

 
Theoremdummylink 28192 Alias for a1ii 2 that may be referenced in some older works, and kept here to prevent broken links.

If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to a1ii 2.

(Contributed by NM, 7-Feb-2006.) (Proof modification is discouraged.) (New usage is discouraged.)

𝜑    &   𝜓       𝜑
 
Theoremid1 28193 Alias for idALT 23 that may be referenced in some older works, and kept here to prevent broken links.

If you landed here, please let the originating site and/or us know that the link that made you land here should be changed to a link to idALT 23.

(Contributed by NM, 30-Sep-1992.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑𝜑)
 
PART 18  COMPLEX TOPOLOGICAL VECTOR SPACES (DEPRECATED)

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 is not useful), and if so, delete it. Then, repeat this recursively. One way to search for terminal theorems is to log the output ("MM> OPEN LOG xxx.txt") of "MM> SHOW USAGE <label-match>" in the Metamath program and search for "(None)".

 
18.1  Additional material on group theory (deprecated)

This section contains an earlier development of groups that was defined before extensible structures were introduced.

The intent is for this deprecated section to be deleted once the corresponding definitions and theorems for complex topological vector spaces, which are using them, are revised accordingly.

 
18.1.1  Definitions and basic properties for groups
 
Syntaxcgr 28194 Extend class notation with the class of all group operations.
class GrpOp
 
Syntaxcgi 28195 Extend class notation with a function mapping a group operation to the group's identity element.
class GId
 
Syntaxcgn 28196 Extend class notation with a function mapping a group operation to the inverse function for the group.
class inv
 
Syntaxcgs 28197 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
class /𝑔
 
Definitiondf-grpo 28198* 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 ∣ ∃𝑡(𝑔:(𝑡 × 𝑡)⟶𝑡 ∧ ∀𝑥𝑡𝑦𝑡𝑧𝑡 ((𝑥𝑔𝑦)𝑔𝑧) = (𝑥𝑔(𝑦𝑔𝑧)) ∧ ∃𝑢𝑡𝑥𝑡 ((𝑢𝑔𝑥) = 𝑥 ∧ ∃𝑦𝑡 (𝑦𝑔𝑥) = 𝑢))}
 
Definitiondf-gid 28199* 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 ∈ V ↦ (𝑢 ∈ ran 𝑔𝑥 ∈ ran 𝑔((𝑢𝑔𝑥) = 𝑥 ∧ (𝑥𝑔𝑢) = 𝑥)))
 
Definitiondf-ginv 28200* 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 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔 ↦ (𝑧 ∈ ran 𝑔(𝑧𝑔𝑥) = (GId‘𝑔))))
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