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Theorem List for Metamath Proof Explorer - 21701-21800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremvdgrfif 21701 The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgr0 21702 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
VDeg

Theoremvdgrun 21703 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.)
UMGrph        UMGrph               VDeg VDeg VDeg

Theoremvdgrfiun 21704 The degree of a vertex in the union of two graphs (of finite size) on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
UMGrph        UMGrph               VDeg VDeg VDeg

Theoremvdgr1d 21705 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1b 21706 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1c 21707 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdgr1a 21708 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
VDeg

Theoremvdusgraval 21709* The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph VDeg

Theoremvdusgra0nedg 21710* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
USGrph VDeg

Theoremvdgrnn0pnf 21711 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
USGrph VDeg

Theoremhashnbgravd 21712 The size of the set of the neighbors of a vertex is the vertex degree of this vertex. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
USGrph Neighbors VDeg

Theoremhashnbgravdg 21713 The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 21712. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph Neighbors VDeg

Theoremusgravd0nedg 21714* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge, analogous to vdusgra0nedg 21710. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
USGrph VDeg

14.2  Eulerian paths and the Konigsberg Bridge problem

14.2.1  Eulerian paths

Syntaxceup 21715 Extend class notation with Eulerian paths.
EulPaths

Definitiondf-eupa 21716* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Theoremreleupa 21717 The set EulPaths of all Eulerian paths on is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremiseupa 21718* The property " is an Eulerian path on the graph ". An Eulerian path is defined as bijection from the edges to a set a function into the vertices such that for each , is an edge from to . (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths UMGrph

Theoremeupagra 21719 If an eulerian path exists, then is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths UMGrph

Theoremeupai 21720* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupatrl 21721* An Eulerian path is a trail.

Unfortunately, the edge function of an Eulerian path has the domain , whereas the edge functions of all kinds of walks defined here have the domain ..^ (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 21652, fargshiftfv 21653, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

..^        EulPaths Trails

Theoremeupacl 21722 An Eulerian path has length , which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaf1o 21723 The function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupafi 21724 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupapf 21725 The function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupaseg 21726 The -th edge in an eulerian path is the edge from to . (Contributed by Mario Carneiro, 12-Mar-2015.)
EulPaths

Theoremeupa0 21727 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths

Theoremeupares 21728 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
EulPaths                                    EulPaths

Theoremeupap1 21729 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths                                    EulPaths

Theoremeupath2lem1 21730 Lemma for eupath2 21733. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem2 21731 Lemma for eupath2 21733. (Contributed by Mario Carneiro, 8-Apr-2015.)

Theoremeupath2lem3 21732* Lemma for eupath2 21733. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths                             VDeg        VDeg

Theoremeupath2 21733* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
EulPaths        VDeg

Theoremeupath 21734* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
EulPaths VDeg

14.2.2  The Konigsberg Bridge problem

Theoremvdeg0i 21735 The base case for the induction for calculating the degree of a vertex. The degree of in the empty graph is . (Contributed by Mario Carneiro, 12-Mar-2015.)
VDeg

Theoremumgrabi 21736* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Theoremvdegp1ai 21737* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                                    concat        VDeg

Theoremvdegp1bi 21738* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             concat        VDeg

Theoremvdegp1ci 21739* The induction step for a vertex degree calculation. If the degree of in the edge set is , then adding to the edge set, where , yields degree . (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
Word               VDeg                             concat        VDeg

Theoremkonigsberg 21740 The Konigsberg Bridge problem. If is the graph on four vertices , with edges , then vertices each have degree three, and has degree five, so there are four vertices of odd degree and thus by eupath 21734 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
EulPaths

PART 15  GUIDES AND MISCELLANEA

15.1  Guides (conventions, explanations, and examples)

15.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. 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 21741 Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they're identical):

• Notation. Where possible, the notation attempts to conform to modern conventions, with variations due to our choice of the axiom system or to make proofs shorter. However, our notation is strictly sequential (left-to-right). For example, summation is written in the form (df-sum 12511) which denotes that index variable ranges over when evaluating . Thus, means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12690). Also, the method of definition, the axioms for predicate calculus, and the development of substitution are somewhat different from those found in standard texts. For example, the expressions for the axioms were designed for direct derivation of standard results without excessive use of metatheorems. (See Theorem 9.7 of [Megill] p. 448 for a rigorous justification.) The notation is usually explained in more detail when first introduced.
• Axiomatic assertions (\$a). All axiomatic assertions (\$a statements) starting with " " have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats \$a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound, except for 4 (df-bi, df-cleq, df-clel, df-clab) that require a more complex metalogical justification by hand.
• Proven axioms. In some cases we wish to treat an expression as an axiom in later theorems, even though it can be proved. For example, we derive the postulates or axioms of complex arithmetic as theorems of ZFC set theory. For convenience, after deriving the postulates we re-introduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. For more, see http://us.metamath.org/mpeuni/mmcomplex.html. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is complex arithmetic axiom ax-1cn 9079, proven by the preceding theorem ax1cn 9055. The metamath.exe program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
• Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
• Statements with hypotheses. Many theorems and some axioms, such as ax-mp 5, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When presented in summarized form such as in the Theorem List (click on "Nearby theorems" on the ax-mp 5 page), the hypotheses are connected with an ampersand and separated from the conclusion with a big arrow, such as in " & => ". 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 9079 (not ax1cn 9055) and ax-1ne0 9090 (not ax1ne0 9066), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9055 and ax1ne0 9066 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 instead of sinh for the hyperbolic sine.
• Axiom of choice. The axiom of choice (df-ac 8028) is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 7256) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8346) or axiom of dependent choice (ax-dc 8357) can be used instead.
• Variables. Typically, Greek letters ( = phi, = psi, = chi, etc.),... are used for propositional (wff) variables; , , ,... for individual (set) variables; and , , ,... for class variables.
• Turnstile. "", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff ".
• 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 201 or mpbir 202. 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 188 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 888, sylbir 206, or 3imtr4i 259.
• Substitution. " " should be read "the wff that results from the proper substitution of for in wff ." See df-sb 1660 and the related df-sbc 3168 and df-csb 3268.
• Is-a set. " " should be read "Class is a set (i.e. exists)." This is a convenient convention based on Definition 2.9 of [Quine] p. 19. See df-v 2964 and isset 2966.
• Converse. "" should be read "converse of (relation) " and is the same as the more standard notation R^{-1} (the standard notation is ambiguous). See df-cnv 4915. This can be used to define a subset, e.g., df-tan 12705 notates "the set of values whose cosine is a nonzero complex number" as .
• Function application. "()" should be read "the value of function at " and has the same meaning as the more familiar but ambiguous notation F(x). For example, (see cos0 12782). 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 5491. 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 6113). For example, the in ; see 2p2e4 10129. 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 , , , and (see wi 4, df-or 361, df-an 362, and df-bi 179 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 (see wel 1728), equality (see df-cleq 2435), subset (see df-ss 3320), and less-than (see df-lt 9034). For the general definition of a binary relation in the form , see df-br 4238. For example, ( see 0lt1 9581) does not use parentheses.
• Unary minus. The symbol is used to indicate a unary minus, e.g., . It is specially defined because it is so commonly used. See cneg 9323.
• 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 12698). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 12704). Typically, there are other proofs such as its closure labelled NAMEcl (e.g., coscl 12759), its function application form labelled NAMEval (e.g., cosval 12755), and at least one simple value (e.g., cos0 12782).
• Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g., ; (df-fac 11598 and fac4 11605).
• 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 "" always means the value zero (df-0 9028), while "" is the group identity element (df-0g 13758), "" is the poset zero (df-p0 14499), "" is the zero polynomial (df-0p 19591), "" is the zero vector in a normed complex vector space (df-0v 22108), and "" is a class variable for use as a connective symbol (this is used, for example, in p0val 14501). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including "", "", "", and "". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.
• Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 10032) for the integer numbers starting from 1, and (df-n0 10253) for the set of nonnegative integers starting at zero.
• Decimal numbers. Numbers larger than ten are often expressed in base 10 using the decimal constructor df-dec 10414, e.g., ;;; (see 4001prm 13495 for a proof that 4001 is prime).
• Theorem forms. We often call a theorem a "deduction" whenever the conclusion and all hypotheses are each prefixed with the same antecedent . Deductions are often the preferred form for theorems because they allow us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used. See, for example, a1d 24. A deduction hypothesis can have an indirect antecedent via definitions, e.g., see lhop 19931. Deductions have a label suffix of "d" if there are other forms of the same theorem. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 11. Finally, a "tautology" would be the form with no hypotheses, and its label would have no suffix. For example, see pm2.43 50, pm2.43i 46, and pm2.43d 47.
• Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see http://us.metamath.org/mpeuni/mmdeduction.html. The Deduction Theorem is a metalogical theorem that cannot be applied directly in metamath, and the explosion of steps would be a problem anyway, so alternatives are used. One alternative we use sometimes is the "weak deduction theorem" dedth 3804, which works in certain cases in set theory. We also sometimes use dedhb 3110. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in the deduction theorem form (aka "deduction style") described earlier; the prefixed mimics the context in a deduction proof system. In practice this mechanism works very well. This approach is described in the deduction form and natural deduction page; a list of translations for common natural deduction rules is given in natded 21742.
• 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 in df-recs 6662, in df-rdg 6697, seq𝜔 in df-seqom 6734, and in df-seq 11355. These have characteristic function and initial value . (g in df-gsum 13759 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 6662, but for the "average user" the most useful one is probably df-seq 11355- 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 13502.
• Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 7246 is the first lemma for sbth 7256. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
• Hypertext links. We strongly encourage comments to have many links to related material, with accompanying text that explains the relationship. These can help readers understand the context. Links to other statements, or to HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a space-separated tilde. When metamath.exe is used to generate HTML it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertion.
• Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g. mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we require page numbers in comments for individual \$a or \$p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the metamath.exe "write bibliography" command.)
• Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
• Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see http://us.metamath.org/mpeuni/mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.

Naming conventions

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

• Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
• Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id.
• Common names. If a theorem has a well-known name, that name (or a short version of it) is sometimes used directly. Examples include barbara 2384 and stirling 27852.
• Syntax label fragments. Most theorems are named using syntax label fragments. Almost every syntactic construct has a definition labelled "df-NAME", and NAME normally is the syntax label fragment. For example, the difference construct is defined in df-dif 3309, and thus its syntax label fragment is "dif". Similarly, the singleton construct has syntax label fragment "sn" (because it is defined in df-sn 3844), the subclass (subset) relation has "ss" (because it is defined in df-ss 3320), and the pair construct has "pr" (df-pr 3845). Theorem names are often a concatenation of the syntax label fragments (omitting variables). For example, a theorem about involves a difference ("dif") of a subset ("ss"), and thus is named difss 3460. 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 does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with uses is-element-of ("el") of a difference ("dif") of a singleton ("sn"), it is named eldifsn 3951. An "n" is often used for negation (), e.g., nan 565.
• Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The table below attempts to list all such cases and marks them in bold. For example, label fragment "cn" represents complex numbers (even though its definition is in df-c 9027) and "re" represents real numbers . The empty set often uses fragment 0, even though it is defined in df-nul 3614. Syntax construct usually uses the fragment "add" (which is consistent with df-add 9032), but "p" is used as the fragment for constant theorems. Equality often uses "e" as the fragment. As a result, "two plus two equals four" is named 2p2e4 10129.
• Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
• Principia Mathematica. Proofs of theorems from Principia Mathematica often use a different naming convention. They are instead often named "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 38.
• Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value named "NAMEval" and its closure named "NAMEcl". E.g., for cosine (df-cos 12704) we have value cosval 12755 and closure coscl 12759.
• Special cases. Sometimes syntax and related markings are insufficient to distinguish different theorems. For example, there are over 100 different implication-exclusive theorems. These are then grouped in a more ad-hoc way that attempts to make their distinctions clearer. These often use abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and "id" for "identity". It's especially hard to give good names in the propositional calculus section because there are so few primitives. However, in most cases this is not a serious problem. There are a few very common theorems like ax-mp 5 and syl 16 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 21742 for a list), and that's about all you need for most things. As for the rest, you can just assume that if it involves three or less connectives we probably already have a proof, and searching for it will give you the name.
• Suffixes. We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as in 19.21 1816 via the use of distinct variable conditions combined with nfv 1630. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 2293 derived from df-eu 2291. The "f" stands for "not free in" which is less restrictive than "does not occur in." We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 16) -type inference in a proof. When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the theorem form with "g" (for "more general") as in uniex 4734 vs. uniexg 4735. A theorem name is suffixed with "ALT" if it's an alternative less-preferred proof of a theorem.
• Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly-used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
ALTalternative/less preferred (suffix) No
anand df-an 362 Yes anor 477, iman 415, imnan 413
assassociative No biass 350, orass 512, mulass 9109
bibiconditional df-bi 179 Yes impbid 185
cncomplex numbers df-c 9027 Yes nnsscn 10036, nncn 10039
comcommutative No orcom 378, bicomi 195, eqcomi 2446
ddeduction form No idd 23, impbid 185
di, distrdistributive No andi 839, imdi 354, ordi 836, difindi 3580, ndmovdistr 6265
difdifference df-dif 3309 Yes difss 3460, difindi 3580
divdivision df-div 9709 Yes divcl 9715, divval 9711, divmul 9712
e, eqequals df-cleq 2435 Yes 2p2e4 10129, uneqri 3475
elelement of Yes eldif 3316, eldifsn 3951, elssuni 4067
f"not free in" (suffix) No
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 4735
ididentity No
idmidempotent No anidm 627, tpidm13 3930
im, impimplication (label often omitted) df-im 11937 Yes iman 415, imnan 413, impbidd 183
inintersection df-in 3313 Yes elin 3516, incom 3519
is...is (something a) ...? No isrng 15699
mpmodus ponens ax-mp 5 No mpd 15, mpi 17
mulmultiplication (see "t") df-mul 9033 Yes mulcl 9105, divmul 9712, mulcom 9107, mulass 9109
n, notnot Yes nan 565, notnot2 107
ne0not equal to zero (see n0) No negne0d 9440, ine0 9500, gt0ne0 9524
nnnatural numbers df-nn 10032 Yes nnsscn 10036, nncn 10039
n0not the empty set (see ne0) No n0i 3618, vn0 3620, ssn0 3645
oror df-or 361 Yes orcom 378, anor 477
pmPrincipia Mathematica No pm2.27 38
prpair df-pr 3845 Yes elpr 3856, prcom 3906, prid1g 3934, prnz 3947
q (quotients) df-q 10606 Yes elq 10607
rereal numbers df-r 9031 Yes recn 9111, 0re 9122
rngring df-rng 15694 Yes rngidval 15697, isrng 15699, rnggrp 15700
rotrotation No 3anrot 942, 3orrot 943
seliminates need for syllogism (suffix) No
snsingleton df-sn 3844 Yes eldifsn 3951
sssubset df-ss 3320 Yes difss 3460
subsubtract df-sub 9324 Yes subval 9328, subaddi 9418
sylsyllogism syl 16 No 3syl 19
t times (see "mul"), for all-constant theorems df-mul 9033 Yes 3t2e6 10159
tptriple df-tp 3846 Yes eltpi 3876, tpeq1 3916
ununion df-un 3311 Yes uneqri 3475, uncom 3477
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 1966
xreXtended reals df-xr 9155 Yes ressxr 9160, rexr 9161, 0xr 9162
z (integers, from German Zahlen) df-z 10314 Yes elz 10315, zcn 10318
0, z slashed zero (empty set) (see n0) df-nul 3614 Yes n0i 3618, vn0 3620; snnz 3946, prnz 3947

Distinctness or freeness

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

• is read " is not free in (wff) "; see df-nf 1555 (whose description has some important technical details). Similarly, is read is not free in (class) , see df-nfc 2567.
• "\$d x y \$." should be read "Assume x and y are distinct variables."
• "\$d x \$." should be read "Assume x does not occur in phi \$." Sometimes a theorem is proved using (df-nf 1555) in place of "\$d \$." when a more general result is desired; ax-17 1627 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 2293 from df-eu 2291.
• "\$d x A \$." should be read "Assume x is not a variable occurring in class A."
• "\$d x A \$. \$d x ps \$. \$e |- \$." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
• " " 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.

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

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

15.1.2  Natural deduction

Theoremnatded 21742 Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

IT => idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
I & => jca 520 jca 520, pm3.2i 443 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 => simpld 447 simpld 447, adantr 453 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 => simprd 451 simpr 449, adantl 454 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 => ex 425 ex 425 Definition I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition I in [Indrzejczak] p. 33.
E & => mpd 15 ax-mp 5, mpd 15, mpdan 651, imp 420 Definition E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition E in [Indrzejczak] p. 33.
IL => olcd 384 olc 375, olci 382, olcd 384 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR => orcd 383 orc 376, orci 381, orcd 383 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E & & => mpjaodan 763 mpjaodan 763, jaodan 762, jaod 371 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
I => inegd 1343 pm2.01d 164
I & => mtand 642 mtand 642 definition Im,n,p in [Clemente] p. 13.
I & => pm2.65da 561 pm2.65da 561 Contradiction.
I => pm2.01da 431 pm2.01d 164, pm2.65da 561, pm2.65d 169 For an alternative falsum-free natural deduction ruleset
E & => pm2.21fal 1345 pm2.21dd 102
E => pm2.21dd 102 definition E in [Indrzejczak] p. 33.
E & => pm2.21dd 102 pm2.21dd 102, pm2.21d 101, pm2.21 103 For an alternative falsum-free natural deduction ruleset. Definition E in [Pfenning] p. 18.
I a1tru 1340 tru 1331, a1tru 1340, trud 1333 Definition I in [Pfenning] p. 18.
E falimd 1339 falim 1338 Definition E in [Pfenning] p. 18.
I => alrimiv 1642 alrimiv 1642, ralrimiva 2795 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E => spsbcd 3180 spcv 3048, rspcv 3054 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I => spesbcd 3259 spcev 3049, rspcev 3058 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E & => exlimddv 1649 exlimddv 1649, exlimdd 1915, exlimdv 1647, rexlimdva 2836 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C => efald 1344 efald 1344 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C => pm2.18da 432 pm2.18da 432, pm2.18d 106, pm2.18 105 For an alternative falsum-free natural deduction ruleset
C => notnotrd 108 notnotrd 108, notnot2 107 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition En in [Clemente] p. 14.
EM exmidd 407 exmid 406 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
I eqidd 2443 eqid 2442, eqidd 2443 Introduce equality, definition =I in [Pfenning] p. 127.
E & => sbceq1dd 3173 sbceq1d 3172, 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 represents the set of (current) hypotheses. We use wff variable names beginning with to provide a closer representation of the Metamath equivalents (which typically use the antedent to represent the context ).

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 21743, ex-natded5.3 21746, ex-natded5.5 21749, ex-natded5.7 21750, ex-natded5.8 21752, ex-natded5.13 21754, ex-natded9.20 21756, and ex-natded9.26 21758.

(Contributed by DAW, 4-Feb-2017.)

15.1.3  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 21742 and http://us.metamath.org/mpeuni/mmnatded.html.

Theoremex-natded5.2 21743 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 Given \$e.
22 Given \$e.
31 Given \$e.
43 E 2,3 mpd 15, the MPE equivalent of E, 1,2
54 I 4,3 jca 520, the MPE equivalent of I, 3,1
66 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 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 21744. A proof without context is shown in ex-natded5.2i 21745. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.2-2 21744 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 21743 and ex-natded5.2i 21745. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.2i 21745 The same as ex-natded5.2 21743 and ex-natded5.2-2 21744 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3 21746 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 21747. A proof without context is shown in ex-natded5.3i 21748. 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 Given \$e; adantr 453 to move it into the ND hypothesis
25;6 Given \$e; adantr 453 to move it into the ND hypothesis
31 ...| ND hypothesis assumption simpr 449, to access the new assumption
44 ... E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 453 was used to transform its dependency (we could also use imp 420 to get this directly from 1)
57 ... E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 453 was used to transform its dependency
68 ... I 4,5 jca 520, the MPE equivalent of I, 4,7
79 I 3,6 ex 425, 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 and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3-2 21747 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 21746 and ex-natded5.3i 21748. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.3i 21748 The same as ex-natded5.3 21746 and ex-natded5.3-2 21747 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.5 21749 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 Given \$e; adantr 453 to move it into the ND hypothesis
25 Given \$e; we'll use adantr 453 to move it into the ND hypothesis
31 ...| ND hypothesis assumption simpr 449
44 ... E 1,3 mpd 15 1,3
56 ... IT 2 adantr 453 5
67 I 3,4,5 pm2.65da 561 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 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 453; simpr 449 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 171; a proof without context is shown in mto 170.

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

Theoremex-natded5.7 21750 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 21751. 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 Given \$e. No need for adantr 453 because we do not move this into an ND hypothesis
21 ...| ND hypothesis assumption (new scope) simpr 449
32 ... IL 2 orcd 383, the MPE equivalent of IL, 1
43 ...| ND hypothesis assumption (new scope) simpr 449
54 ... EL 4 simpld 447, the MPE equivalent of EL, 3
66 ... IR 5 olcd 384, the MPE equivalent of IR, 4
77 E 1,3,6 mpjaodan 763, 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 and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.7-2 21751 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 21750. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.8 21752 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 Given \$e; adantr 453 to move it into the ND hypothesis
23;4 Given \$e; adantr 453 to move it into the ND hypothesis
37;8 Given \$e; adantr 453 to move it into the ND hypothesis
41;2 Given \$e. adantr 453 to move it into the ND hypothesis
56 ...| ND Hypothesis/Assumption simpr 449. New ND hypothesis scope, each reference outside the scope must change antedent to .
69 ... I 5,3 jca 520 (I), 6,8 (adantr 453 to bring in scope)
75 ... E 1,6 mpd 15 (E), 2,4
812 ... E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 I 5,7,8 pm2.65da 561 (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 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 453; simpr 449 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 21753.

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

Theoremex-natded5.8-2 21753 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 21752. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.13 21754 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 21755. 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 Given \$e.
2;32 Given \$e. adantr 453 to move it into the ND hypothesis
39 Given \$e. ad2antrr 708 to move it into the ND sub-hypothesis
41 ...| ND hypothesis assumption simpr 449
54 ... E 2,4 mpd 15 1,3
65 ... I 5 orcd 383 4
76 ...| ND hypothesis assumption simpr 449
88 ... ...| (sub) ND hypothesis assumption simpr 449
911 ... ... E 3,8 mpd 15 8,10
107 ... ... IT 7 adantr 453 6
1112 ... I 8,9,10 pm2.65da 561 7,11
1213 ... E 11 notnotrd 108 12
1314 ... I 12 olcd 384 13
1416 E 1,6,13 mpjaodan 763 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 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 453; simpr 449 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded5.13-2 21755 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 21754. (Contributed by Mario Carneiro, 9-Feb-2017.)

Theoremex-natded9.20 21756 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 Given \$e
22 EL 1 simpld 447 1
311 ER 1 simprd 451 1
44 ...| ND hypothesis assumption simpr 449
55 ... I 2,4 jca 520 3,4
66 ... IR 5 orcd 383 5
78 ...| ND hypothesis assumption simpr 449
89 ... I 2,7 jca 520 7,8
910 ... IL 8 olcd 384 9
1012 E 3,6,9 mpjaodan 763 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 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 453; simpr 449 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 21757. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Theoremex-natded9.20-2 21757 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 21756. (Proof modification is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

Theoremex-natded9.26 21758* 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 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 Given \$e.
26 ...| ND hypothesis assumption simpr 449. Later statements will have this scope.
37;5,4 ... E 2,y spsbcd 3180 (E), 5,6. To use it we need a1i 11 and vex 2965. This could be immediately done with 19.21bi 1776, but we want to show the general approach for substitution.
412;8,9,10,11 ... I 3,a spesbcd 3259 (I), 11. To use it we need sylibr 205, which in turn requires sylib 190 and two uses of sbcid 3183. This could be more immediately done using 19.8a 1764, but we want to show the general approach for substitution.
513;1,2 E 1,2,4,a exlimdd 1915 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1630 and nfe1 1749 (MPE# 1,2)
614 I 5 alrimiv 1642 (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 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, 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 21759.

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

Theoremex-natded9.26-2 21759* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 21758. (Contributed by Mario Carneiro, 9-Feb-2017.)

15.1.4  Definitional examples

Theoremex-or 21760 Example for df-or 361. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremex-an 21761 Example for df-an 362. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)

Theoremex-dif 21762 Example for df-dif 3309. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-un 21763 Example for df-un 3311. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-in 21764 Example for df-in 3313. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-uni 21765 Example for df-uni 4040. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremex-ss 21766 Example for df-ss 3320. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-pss 21767 Example for df-pss 3322. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-pw 21768 Example for df-pw 3825. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)

Theoremex-pr 21769 Example for df-pr 3845. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-br 21770 Example for df-br 4238. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-opab 21771* Example for df-opab 4292. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-eprel 21772 Example for df-eprel 4523. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-id 21773 Example for df-id 4527. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-po 21774 Example for df-po 4532. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-xp 21775 Example for df-xp 4913. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-cnv 21776 Example for df-cnv 4915. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)

Theoremex-co 21777 Example for df-co 4916. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-dm 21778 Example for df-dm 4917. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-rn 21779 Example for df-rn 4918. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-res 21780 Example for df-res 4919. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-ima 21781 Example for df-ima 4920. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-fv 21782 Example for df-fv 5491. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)

Theoremex-1st 21783 Example for df-1st 6378. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-2nd 21784 Example for df-2nd 6379. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theorem1kp2ke3k 21785 Example for df-dec 10414, 1000 + 2000 = 3000.

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

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

The proof here starts with , 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 10414 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.)

;;; ;;; ;;;

Theoremex-fl 21786 Example for df-fl 11233. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)

Theoremex-dvds 21787 3 divides into 6. A demonstration of df-dvds 12884. (Contributed by David A. Wheeler, 19-May-2015.)

15.2  Humor

15.2.1  April Fool's theorem

Theoremavril1 21788 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 equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theorem2bornot2b 21789 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.)

Theoremhelloworld 21790 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.)

Theorem1p1e2apr1 21791 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel 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.)

Theoremeqid1 21792 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.)

Theorem1div0apr 21793 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.)

15.3  (Future - to be reviewed and classified)

15.3.1  Planar incidence geometry

Syntaxcplig 21794 Extend class notation with the class of all planar incidence geometries.

Definitiondf-plig 21795* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)

Theoremisplig 21796* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)

Theoremtncp 21797* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)

Theoremlpni 21798* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)

15.3.2  Algebra preliminaries

Syntaxcrpm 21799 Ring primes.
RPrime

Definitiondf-rprm 21800* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 13139. (Contributed by Mario Carneiro, 17-Feb-2015.)
RPrime Unit r

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