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Theorem List for Metamath Proof Explorer - 26301-26400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrgrancvvdeqlem7 26301* Lemma 7 for frgrancvvdeq 26307. (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       ((𝜑𝑥𝐷) → {𝑥, (𝐴𝑥)} ∈ ran 𝐸)

TheoremfrgrancvvdeqlemA 26302* Lemma A for frgrancvvdeq 26307. This corresponds to statement 1 in [Huneke] p. 1: "This common neighbor cannot be x, as x and y are not adjacent.". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑 → ∀𝑥𝐷 (𝐴𝑥) ≠ 𝑋)

TheoremfrgrancvvdeqlemB 26303* Lemma B for frgrancvvdeq 26307. This corresponds to statement 2 in [Huneke] p. 1: "The map is one-to-one since z in N(x) is uniquely determined as the common neighbor of x and a(x)". (Contributed by Alexander van der Vekens, 23-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷1-1→ran 𝐴)

TheoremfrgrancvvdeqlemC 26304* Lemma C for frgrancvvdeq 26307. This corresponds to statement 3 in [Huneke] p. 1: "By symmetry the map is onto". (Contributed by Alexander van der Vekens, 24-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷onto𝑁)

Theoremfrgrancvvdeqlem8 26305* Lemma 8 for frgrancvvdeq 26307. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
𝐷 = (⟨𝑉, 𝐸⟩ Neighbors 𝑋)    &   𝑁 = (⟨𝑉, 𝐸⟩ Neighbors 𝑌)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)    &   (𝜑𝑌𝐷)    &   (𝜑𝑉 FriendGrph 𝐸)    &   𝐴 = (𝑥𝐷 ↦ (𝑦𝑁 {𝑥, 𝑦} ∈ ran 𝐸))       (𝜑𝐴:𝐷1-1-onto𝑁)

Theoremfrgrancvvdeqlem9 26306* Lemma 9 for frgrancvvdeq 26307. (Contributed by Alexander van der Vekens, 24-Dec-2017.)
(𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ∃𝑓 𝑓:(⟨𝑉, 𝐸⟩ Neighbors 𝑥)–1-1-onto→(⟨𝑉, 𝐸⟩ Neighbors 𝑦)))

Theoremfrgrancvvdeq 26307* In a finite friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.)
((𝑉 FriendGrph 𝐸𝐸 ∈ Fin) → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)))

Theoremfrgrancvvdgeq 26308* In a friendship graph, two vertices which are not connected by an edge have the same degree. This corresponds to claim 1 in [Huneke] p. 1: "If x,y, are elements of (the friendship graph) G and are not adjacent, then they have the same degree (i.e., the same number of adjacent vertices).". (Contributed by Alexander van der Vekens, 19-Dec-2017.) (Proof shortened by AV, 5-May-2021.)
(𝑉 FriendGrph 𝐸 → ∀𝑥𝑉𝑦 ∈ (𝑉 ∖ {𝑥})(𝑦 ∉ (⟨𝑉, 𝐸⟩ Neighbors 𝑥) → ((𝑉 VDeg 𝐸)‘𝑥) = ((𝑉 VDeg 𝐸)‘𝑦)))

Theoremfrgrawopreglem1 26309* Lemma 1 for frgrawopreg 26314. In a friendship graph, the classes A and B are sets. The definition of A and B corresponds to definition 3 in [Huneke] p. 2: "Let A be the set of all vertices of degree k, let B be the set of all vertices of degree different from k, ..." (Contributed by Alexander van der Vekens, 31-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝑉 FriendGrph 𝐸 → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Theoremfrgrawopreglem2 26310* Lemma 2 for frgrawopreg 26314. In a friendship graph with at least two vertices, the degree of a vertex must be at least 2. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝑉) ∧ 𝐴 ≠ ∅) → 1 < 𝐾)

Theoremfrgrawopreglem3 26311* Lemma 3 for frgrawopreg 26314. The vertices in the sets A and B have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑋𝐴𝑌𝐵) → ((𝑉 VDeg 𝐸)‘𝑋) ≠ ((𝑉 VDeg 𝐸)‘𝑌))

Theoremfrgrawopreglem4 26312* Lemma 4 for frgrawopreg 26314. In a friendship graph each vertex with degree K is connected with a vertex with degree other than K. This corresponds to statement 4 in [Huneke] p. 2: "By the first claim, every vertex in A is adjacent to every vertex in B.". (Contributed by Alexander van der Vekens, 30-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝑉 FriendGrph 𝐸 → ∀𝑎𝐴𝑏𝐵 {𝑎, 𝑏} ∈ ran 𝐸)

Theoremfrgrawopreglem5 26313* Lemma 5 for frgrawopreg 26314. If A as well as B contain at least two vertices in a friendship graph, there is a 4-cycle in the graph. This corresponds to statement 6 in [Huneke] p. 2: "... otherwise, there are two different vertices in A, and they have two common neighbors in B, ...". (Contributed by Alexander van der Vekens, 31-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑉 FriendGrph 𝐸 ∧ 1 < (#‘𝐴) ∧ 1 < (#‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑏𝑦𝑎𝑥) ∧ ({𝑎, 𝑏} ∈ ran 𝐸 ∧ {𝑥, 𝑏} ∈ ran 𝐸) ∧ ({𝑎, 𝑦} ∈ ran 𝐸 ∧ {𝑥, 𝑦} ∈ ran 𝐸)))

Theoremfrgrawopreg 26314* In a friendship graph there are either no vertices or exactly one vertex having degree K, or all or all except one vertices have degree K. (Contributed by Alexander van der Vekens, 31-Dec-2017.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝑉 FriendGrph 𝐸 → (((#‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((#‘𝐵) = 1 ∨ 𝐵 = ∅)))

Theoremfrgrawopreg1 26315* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐴) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

Theoremfrgrawopreg2 26316* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
𝐴 = {𝑥𝑉 ∣ ((𝑉 VDeg 𝐸)‘𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       ((𝑉 FriendGrph 𝐸 ∧ (#‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

Theoremfrgraregorufr0 26317* In a friendship graph there are either no vertices having degree 𝐾, or all vertices have degree 𝐾 for any (nonnegative integer) 𝐾, unless there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "... all vertices have degree k, unless there is a universal friend." (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(𝑉 FriendGrph 𝐸 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸))

Theoremfrgraregorufr 26318* If there is a vertex having degree 𝐾 for each (nonnegative integer) 𝐾 in a friendship graph, then either all vertices have degree 𝐾 or there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". (Contributed by Alexander van der Vekens, 1-Jan-2018.)
(𝑉 FriendGrph 𝐸 → (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝐾 → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))

Theoremfrgraeu 26319* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
(𝑉 FriendGrph 𝐸 → ((𝐴𝑉𝐶𝑉𝐴𝐶) → ∃!𝑏({𝐴, 𝑏} ∈ ran 𝐸 ∧ {𝑏, 𝐶} ∈ ran 𝐸)))

Theoremfrg2woteu 26320* For two different vertices in a friendship graph, there is exactly one third vertex being the middle vertex of a (simple) path/walk of length 2 between the two vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑐𝑉𝐴, 𝑐, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵))

Theoremfrg2wotn0 26321 In a friendship graph, there is always a path/walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ≠ ∅)

Theoremfrg2wot1 26322 In a friendship graph, there is exactly one walk of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(𝑉 2WalksOnOt 𝐸)𝐵)) = 1)

Theoremfrg2spot1 26323 In a friendship graph, there is exactly one simple path of length 2 between two different vertices as ordered triple. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
((𝑉 FriendGrph 𝐸 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (#‘(𝐴(𝑉 2SPathOnOt 𝐸)𝐵)) = 1)

Theoremfrg2woteqm 26324 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 20-Feb-2018.)
((𝑉 FriendGrph 𝐸𝐴𝐵) → ((⟨𝐴, 𝑃, 𝐵⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ ⟨𝐵, 𝑄, 𝐴⟩ ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → 𝑄 = 𝑃))

Theoremfrg2woteq 26325 There is a (simple) path of length 2 from one vertex to another vertex in a friendship graph if and only if there is a (simple) path of length 2 from the other vertex back to the first vertex. (Contributed by Alexander van der Vekens, 14-Feb-2018.)
((𝑉 FriendGrph 𝐸𝐴𝐵) → ((𝑃 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) ∧ 𝑄 ∈ (𝐵(𝑉 2WalksOnOt 𝐸)𝐴)) → ((1st ‘(1st𝑃)) = (2nd𝑄) ∧ (2nd ‘(1st𝑃)) = (2nd ‘(1st𝑃)) ∧ (1st ‘(1st𝑄)) = (2nd𝑃))))

Theorem2spotdisj 26326* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ 𝐴𝑉) → Disj 𝑏 ∈ (𝑉 ∖ {𝐴})(𝐴(𝑉 2SPathOnOt 𝐸)𝑏))

Theorem2spotiundisj 26327* All simple paths of length 2 as ordered triple from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.)
((𝑉𝑋𝐸𝑌) → Disj 𝑎𝑉 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(𝑉 2SPathOnOt 𝐸)𝑏))

Theoremfrghash2spot 26328 The number of simple paths of length 2 is n*(n-1) in a friendship graph with 𝑛 vertices. This corresponds to the proof of claim 3 in [Huneke] p. 2: "... the paths of length two in G: by assumption there are ( n 2 ) such paths.". However, the order of vertices is not respected by Huneke, so he only counts half of the paths which are existing when respecting the order as it is the case for simple paths represented by ordered triples. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
((𝑉 FriendGrph 𝐸 ∧ (𝑉 ∈ Fin ∧ 𝑉 ≠ ∅)) → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · ((#‘𝑉) − 1)))

Theorem2spot0 26329 If there are no vertices, then there are no paths (of length 2), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
((𝑉 = ∅ ∧ 𝐸𝑋) → (𝑉 2SPathsOt 𝐸) = ∅)

Theoremusg2spot2nb 26330* The set of paths of length 2 with a given vertex in the middle for a finite graph is the union of all paths of length 2 from one neighbor to another neighbor of this vertex via this vertex. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) 𝑦 ∈ ((⟨𝑉, 𝐸⟩ Neighbors 𝑁) ∖ {𝑥}){⟨𝑥, 𝑁, 𝑦⟩})

Theoremusgreghash2spotv 26331* According to statement 7 in [Huneke] p. 2: "For each vertex v, there are exactly ( k 2 ) paths with length two having v in the middle, ..." in a finite k-regular graph. For simple paths of length 2 represented by ordered triples, we have again k*(k-1) such paths. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → ∀𝑣𝑉 (((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))

Theoremusgreg2spot 26332* In a finite k-regular graph the set of all paths of length two is the union of all the paths of length 2 over the vertices which are in the middle of such a path. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (𝑉 2SPathsOt 𝐸) = 𝑥𝑉 (𝑀𝑥)))

Theorem2spotmdisj 26333* The sets of paths of length 2 with a given vertex in the middle are distinct for different vertices in the middle. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 17-Sep-2021.)
𝑀 = (𝑎𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ (𝑡 ∈ (𝑉 2SPathsOt 𝐸) ∧ (2nd ‘(1st𝑡)) = 𝑎)})       (𝑉𝑊Disj 𝑥𝑉 (𝑀𝑥))

Theoremusgreghash2spot 26334* In a finite k-regular graph with N vertices there are N times "𝑘 choose 2 " paths with length 2, according to statement 8 in [Huneke] p. 2: "... giving n * ( k 2 ) total paths of length two.", if the direction of traversing the path is not respected. For simple paths of length 2 represented by ordered triples, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘(𝑉 2SPathsOt 𝐸)) = ((#‘𝑉) · (𝐾 · (𝐾 − 1)))))

Theoremfrgregordn0 26335* If a nonempty friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". (Contributed by Alexander van der Vekens, 11-Mar-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣𝑉 ((𝑉 VDeg 𝐸)‘𝑣) = 𝐾 → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theoremfrrusgraord 26336 If a nonempty finite friendship graph is k-regular, its order is k(k-1)+1. This corresponds to claim 3 in [Huneke] p. 2: "Next we claim that the number n of vertices in G is exactly k(k-1)+1.". Variant of frgregordn0 26335, using the definition RegUSGrph (df-rusgra 26190). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (#‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theoremfrgraregorufrg 26337* If there is a vertex having degree 𝑘 for each nonnegative integer 𝑘 in a friendship graph, then there is a universal friend. This corresponds to claim 2 in [Huneke] p. 2: "Suppose there is a vertex of degree k > 1. ... all vertices have degree k, unless there is a universal friend. ... It follows that G is k-regular, i.e., the degree of every vertex is k". Variant of frgraregorufr 26318 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.)
(𝑉 FriendGrph 𝐸 → ∀𝑘 ∈ ℕ0 (∃𝑎𝑉 ((𝑉 VDeg 𝐸)‘𝑎) = 𝑘 → (⟨𝑉, 𝐸⟩ RegUSGrph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)))

Theoremnumclwlk3lem3 26338 Lemma 3 for numclwwlk3 26374. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
((𝐾 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑁 ∈ (ℤ‘2)) → (((𝐾↑(𝑁 − 2)) − 𝑌) + (𝐾 · 𝑌)) = (((𝐾 − 1) · 𝑌) + (𝐾↑(𝑁 − 2))))

Theoremextwwlkfablem1 26339 Lemma 1 for extwwlkfab 26355. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
((((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))

Theoremextwwlkfablem2lem 26340 Lemma for extwwlkfablem2 26343. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 𝑁𝑁 ∈ (ℤ‘2)) → (#‘(𝑤 substr ⟨0, (𝑁 − 2)⟩)) = (𝑁 − 2))

Theoremclwwlkextfrlem1 26341 Lemma for numclwwlk2lem1 26367. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(((𝑋𝑉𝑁 ∈ ℕ ∧ 𝑍𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))

Theoremnumclwwlkfvc 26342* Value of function 𝐶, mapping a nonnegative number n to the closed walks having length n. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       (𝑁 ∈ ℕ0 → (𝐶𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremextwwlkfablem2 26343* Lemma 2 for extwwlkfab 26355. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       ((((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝐶‘(𝑁 − 2)))

Theoremnumclwwlkun 26344* The set of closed walks in an undirected simple graph is the union of the numbers of closed walks starting at each of the vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → (𝐶𝑁) = 𝑥𝑉 {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑥})

Theoremnumclwwlkdisj 26345* The sets of closed walks starting at different vertices in an undirected simple graph are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       Disj 𝑥𝑉 {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑥}

Theoremnumclwwlkovf 26346* Value of operation 𝐹, mapping a vertex v and a nonnegative integer n to the "(For a fixed vertex v, let f(n) be the number of) walks from v to v of length n" according to definition 5 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})

Theoremnumclwwlkffin 26347* In a finite graph, the value of operation 𝐹 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((𝑉 ∈ Fin ∧ 𝐸𝑈) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (𝑋𝐹𝑁) ∈ Fin)

Theoremnumclwwlkovfel2 26348* Properties of an element of the value of operation 𝐹. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑋𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋)))

Theoremnumclwwlkovf2 26349* Value of operation 𝐹 for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑋𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ (𝑤‘0) = 𝑋)})

Theoremnumclwwlkovf2num 26350* In a k regular graph, therere are k closed walks of length 2 starting at a fixed vertex. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑋𝑉) → (#‘(𝑋𝐹2)) = 𝐾)

Theoremnumclwwlkovf2ex 26351* Extending a closed walk starting at a fixed vertex by an additional edge (forth and back). (Contributed by AV, 22-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) ∧ 𝑄 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ 𝑃 ∈ (𝑋𝐹(𝑁 − 2))) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝐶𝑁))

Theoremnumclwwlkovg 26352* Value of operation 𝐺, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))})

Theoremnumclwwlkovgel 26353* Properties of an element of the value of operation 𝐺. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) ↔ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))

Theoremnumclwwlkovgelim 26354* Properties of an element of the value of operation 𝐺. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))

Theoremextwwlkfab 26355* The set of closed walks (having a fixed length greater than 1 and starting at a fixed vertex) with the last but 2 vertex is identical with the first (and therefore last) vertex can be constructed from the set of closed walks with length smaller by 2 than the fixed length appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). 3 ≤ 𝑁 is required since for 𝑁 = 2: (𝑋𝐹(𝑁 − 2)) = (𝑋𝐹0) = ∅, see clwwlkgt0 26037 stating that a walk of length 0 is not represented as word, at least not for an undirected simple graph.) (Contributed by Alexander van der Vekens, 18-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝑋𝐹(𝑁 − 2)) ∧ (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)})

Theoremnumclwlk1lem2foa 26356* Going forth and back form the end of a (closed) walk. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑃 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑄 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝑋𝐺𝑁)))

Theoremnumclwlk1lem2f 26357* T is a function. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2fv 26358* Value of the function T. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑃 ∈ (𝑋𝐺𝑁) → (𝑇𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩))

Theoremnumclwlk1lem2f1 26359* T is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–1-1→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2fo 26360* T is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2f1o 26361* T is a 1-1 onto function. (Contributed by Alexander van der Vekens, 26-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2 26362* There is a bijection between the set of closed walks (having a fixed length greater than 2 and starting at a fixed vertex) with the last but 2 vertex identical with the first (and therefore last) vertex and the set of closed walks (having a fixed length less by 2 and starting at the same vertex) and the neighbors of this vertex. (Contributed by Alexander van der Vekens, 6-Jul-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwwlk1 26363* Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝑉, 𝐸 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlkovq 26364* Value of operation Q, mapping a vertex v and a nonnegative integer n to the not closed walks v(0) ... v(n) of length n from a fixed vertex v = v(0). "Not closed" means v(n) =/= v(0). (Contributed by Alexander van der Vekens, 27-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})

Theoremnumclwwlkqhash 26365* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))

Theoremnumclwwlkovh 26366* Value of operation H, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})

Theoremnumclwwlk2lem1 26367* In a friendship graph, for each walk of length n starting with a fixed vertex and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation H. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation H, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem only generally holds for Friendship Graphs, because these guarantee that for the first and last vertex there is a third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Theoremnumclwlk2lem2f 26368* R is a function. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))

Theoremnumclwlk2lem2fv 26369* Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))

Theoremnumclwlk2lem2f1o 26370* R is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Theoremnumclwwlk2lem3 26371* In a friendship graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex equals the size of the set of all closed walks of length (n+2) starting with this vertex and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2))))

Theoremnumclwwlk2 26372* Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgranumwlkg 26223, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlk3lem 26373* Lemma for numclwwlk3 26374. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑋𝑉) ∧ 𝑁 ∈ (ℤ‘2)) → (#‘(𝑋𝐹𝑁)) = ((#‘(𝑋𝐻𝑁)) + (#‘(𝑋𝐺𝑁))))

Theoremnumclwwlk3 26374* Statement 12 in [Huneke] p. 2: "Thus f(n) = (k - 1)f(n - 2) + k^(n-2)." - the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) is the sum of the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) with v(n-2) = v(n) (see numclwwlk1 26363) and with v(n-2) =/= v(n) ( see numclwwlk2 26372): f(n) = kf(n-2) + k^(n-2) - f(n-2) = (k - 1)f(n - 2) + k^(n-2). (Contributed by Alexander van der Vekens, 26-Aug-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Theoremnumclwwlk4 26375* The total number of closed walks in a finite undirected simple graph is the sum of the numbers of closed walks starting at each of its vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (#‘(𝐶𝑁)) = Σ𝑥𝑉 (#‘(𝑥𝐹𝑁)))

Theoremnumclwwlk5lem 26376* Lemma for numclwwlk5 26377. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 2 ∥ (𝐾 − 1) ∧ 𝑋𝑉) → ((#‘(𝑋𝐹2)) mod 2) = 1)

Theoremnumclwwlk5 26377* Statement 13 in [Huneke] p. 2: "Let p be a prime divisor of k-1; then f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸𝑉 ∈ Fin) ∧ (𝑋𝑉𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑋𝐹𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk6 26378* For a prime divisor p of k-1, the total number of closed walks of length p in an undirected simple graph with m vertices mod p is equal to the number of vertices mod p. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝐶𝑃)) mod 𝑃) = ((#‘𝑉) mod 𝑃))

Theoremnumclwwlk7 26379 Statement 14 in [Huneke] p. 2: "The total number of closed walks of length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)", since the number of vertices in a friendship graph is (k(k-1)+1), see frgregordn0 26335 or frrusgraord 26336, and p divides (k-1), i.e. (k-1) mod p = 0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the empty graph is a friendship graph, see frgra0 26259, as well as k-regular (for any k), see 0vgrargra 26202, but has no closed walk, see clwlk0 26028, this theorem would be false: ((#‘(𝐶𝑃)) mod 𝑃) = 0 ≠ 1, so this case must be excluded. ( (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk8 26380 The size of the set of closed walks of length p, p prime, is divisible by p. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p", see also clwlkndivn 26118. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑃 ∈ ℙ) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 0)

Theoremfrgrareggt1 26381 If a finite friendship graph is k-regular with k > 1, then k must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))

Theoremfrgrareg 26382 If a finite friendship graph is k-regular, then k must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Theoremfrgraregord013 26383 If a finite friendship graph is k-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Theoremfrgraregord13 26384 If a nonempty finite friendship graph is k-regular, then it must have order 1 or 3. Special case of frgraregord013 26383. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
(((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Theoremfrgraogt3nreg 26385* If a finite friendship graph has an order greater than 3, it cannot be k-regular for any k. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑘)

Theoremfriendshipgt3 26386* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

Theoremfriendship 26387* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

PART 17  GUIDES AND MISCELLANEA

17.1  Guides (conventions, explanations, and examples)

17.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. For the general conventions, see conventions 26388, and for conventions related to labels, see conventions-label 26389. 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 26388

Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they are identical). For conventions related to labels, see conventions-label 26389.

• 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 14134) 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 14322). 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 195, df-cleq 2507, df-clel 2510, df-clab 2501) 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. 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 9749, proven by the preceding theorem ax1cn 9725. 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 " 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 9749 (not ax1cn 9725) and ax-1ne0 9760 (not ax1ne0 9736), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9725 and ax1ne0 9736 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 axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, we prefer proofs that do not use the axiom of choice (df-ac 8698) where such proofs can be found. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 7841) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 9016) or axiom of dependent choice (ax-dc 9027) can be used instead. Similarly, any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 1966 through ax-13 2137, by invoking ax10w 1954 through ax13w 1961. We encourage proving theorems *without* ax-10 1966 through ax-13 2137 and moving them up to the ax-4 1713 through ax-9 1947 section.
• Alternative (ALT) proofs. If a different proof is significantly shorter or clearer but uses more or stronger axioms, we prefer to make that proof an "alternative" proof (marked with an ALT label suffix), even if this alternative proof was formalized first. We then make the proof that requires fewer axioms the main proof. This has the effect of reducing (over time) the number and strength of axioms used by any particular proof. There can be multiple alternatives if it makes sense to do so. Alternative (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternative and main statement comments should use hyperlinks to refer to each other (so that a reader of one will become easily aware of the other).
• Alternative (ALTV) versions. If a theorem or definition is an alternative/variant of an already existing theorem resp. definition, its label should have the same name with suffix ALTV. Such alternatives should be temporary only, until it is decided which alternative should be used in the future. Alternative (*ALTV) theorems or definitions are usually contained in mathboxes. Their comments need not to contain "(Proof modification is discouraged.) (New usage is discouraged.)". Alternative statements should follow the main statement, so that people reading the text in order will see the main statement first.
• Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first. This typically happens when a shorter proof to an existing theorem is found: the existing theorem is kept as an *OLD statement for one year. When a proof is shortened automatically (using Metamath's minimize_with command), then it is not necessary to keep the old proof, nor to add credit for the shortening.
• Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are normally used in this order: 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 normally used in this order: 𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡. Variables that represent classes are often represented by uppercase Latin letters: 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g., 0 for poset zero (see p0val 16756) and connective symbols such as + for some group addition operation. (See prdsplusgval 15840 for an example of the use of +). Class variables are selected in alphabetical order starting from 𝐴 if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.
• Turnstile. "", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff ¬ 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 218 or mpbir 219. 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 204 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 956, sylbir 223, or 3imtr4i 279.
• Substitution. "[𝑦 / 𝑥]𝜑" should be read "the wff that results from the proper substitution of 𝑦 for 𝑥 in wff 𝜑." See df-sb 1831 and the related df-sbc 3307 and df-csb 3404.
• 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 3079 and isset 3084. 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 3089 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 4635). 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 4940. This can be used to define a subset, e.g., df-tan 14510 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 14588). 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 5697. 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 6429). For example, the + in (2 + 2); see 2p2e4 10899. 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 383, df-an 384, and df-bi 195 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 1939), equality 𝐴 = 𝐵 (see df-cleq 2507), subset 𝐴𝐵 (see df-ss 3458), and less-than 𝐴 < 𝐵 (see df-lt 9704). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 4482. For example, 0 < 1 (see 0lt1 10299) 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 10018.
• 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 14503). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 14509). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 14565), its function application form labeled NAMEval (e.g., cosval 14561), and at least one simple value (e.g., cos0 14588).
• 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 12791 and fac4 12798).
• 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 9698), while "0g" is the group identity element (df-0g 15809), "0." is the poset zero (df-p0 16754), "0𝑝" is the zero polynomial (df-0p 23118), "0vec" is the zero vector in a normed complex vector space (df-0v 26593), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 16756). 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 capital vs. lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).
• Typography conventions. Class symbols for functions (e.g., abs, sin) should usually not have leading or trailing blanks in their HTML/Latex representation. This is in contrast to class symbols for operations (e.g., gcd, sadd, eval), which usually do include leading and trailing blanks in their representation. If a class symbol is used for a function as well as an operation (according to the definition df-ov 6429, 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.
• 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 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 (except for construction independence and number theorem domains). In most cases we try to minimize hypotheses, that is, we eliminate or reduce what must be true to prove something, so that the proof is more general and easier to use. There are exceptions. For example, we intentionally add hypotheses if they help make proofs independent of a particular construction (e.g., the contruction of complex numbers ). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they aren't strictly needed. For example, we could show that (𝐴 < 𝐵𝐵𝐴) without any other hypotheses, but in practice we also require proving at least some domains (e.g., see ltnei 9912). Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ:
1. Having the hypotheses immediately shows the intended domain of applicability (is it , *, ω, or something else?), without having to trace back to definitions.
2. Having the hypotheses forces its use in the intended domain, which generally is desirable.
3. The behavior is dependent on accidental behavior of definitions outside of their domains, so the theorems are non-portable and "brittle".
4. Only a few theorems can have their hypotheses removed in this fashion due to happy coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who hasn't traced back the set-theoretical foundations of the definitions, it is seemingly random and isn't intuitive at all.
5. The consensus of opinion of people on this group seemed to be against doing this.
• Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 10776) for the set of positive integers starting from 1, and 0 (df-n0 11048) 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 11234, e.g., 4001 (see 4001prm 15574 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 3653). The term "tautology" is also used, especially in propositional calculus. This form was formerly called "theorem form" or "closed theorem form".
• A theorem is in "deduction form" (or is a "deduction") if it has one or more \$e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually 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 23458) 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 3655.
• 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 3654).
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 148 is the inference associated with con3 147). 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 80 because it is the inference associated with imim1i 60 which is itself the inference associated with imim1 80.
"Deduction form" is the preferred form for theorems because this form allows us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other forms of the same theorem (e.g., pm2.43d 50). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 49). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 53). When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the closed form with "g" (for "more general") as in uniex 6727 vs. uniexg 6729.
• 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. 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 3992, which works in certain cases in set theory. We also sometimes use dedhb 3247. 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; a list of translations for common natural deduction rules is given in natded 26390.
• 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 7231, rec(𝐹, 𝐼) in df-rdg 7269, seq𝜔(𝐹, 𝐼) in df-seqom 7306, and seq𝑀( + , 𝐹) in df-seq 12532. These have characteristic function 𝐹 and initial value 𝐼. (Σg in df-gsum 15810 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7231, but for the "average user" the most useful one is probably df-seq 12532- 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 15581.
• 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 5674 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 5997 on the argument being in the domain of the function. It is impossible to derive a contradiction from sound definitions (i.e. that pass the definition check), assuming ZFC is consistent, and he doesn't see the point of all the extra busy work and huge increase in set.mm size that would result from restricting *all* definitions. So instead of implementing a complex system to counter a problem that does not appear to occur in practice, we use a significantly simpler set of approaches.
• Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 7831 is the first lemma for sbth 7841. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
• Limit proof size. It is often preferable to break longer proofs into smaller parts, just as you would do with traditional proofs. One reason is that humans have trouble understanding long proofs. Another reason is that it's generally best to prove reusable results separately, so that others will be able to easily reuse them. Finally, the "minimize" routine can take much longer with very long proofs. We encourage proofs to be no more than 200 essential steps, and generally no more than 500 essential steps, though these are simply guidelines and not hard-and-fast rules. Much smaller proofs are fine! We also acknowledge that some proofs, especially autogenerated ones, should sometimes not be broken up (e.g., because breaking them up might be useless and inefficient due to many interconnections and reused terms within the proof). In Metamath, breaking up longer proofs is done by creating multiple separate proofs of separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 7831 is the first lemma for sbth 7841.
• 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 15100"). 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.)
• Acceptable shorter proofs Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem's description. However, in some cases a proof may be "shorter" or not depending on how it is formatted. This section provides general guidelines.

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

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

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

• Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
• Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.
• Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.
• Language and spelling. It is preferred to use American English for comments and symbols, e.g. we use "neighborhood" instead of the British English "neighbourhood". An exception is the word "analog", which can be either a noun or an adjective. Furthermore, "analog" has the confounding meaning "not digital", whereas "analogue" is often used in the sense something that bears analogy to something else also in American English. Therefore, "analogue" is used for the noun and "analogous" for the adjective in set.mm.

The challenge of varying mathematical conventions

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

Distinctness or freeness

Here are some conventions that address distinctness or freeness of a variable:
• 𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑"; see df-nf 1699 (whose description has some important technical details). Similarly, 𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2644.
• "\$d x y \$." should be read "Assume x and y are distinct variables."
• "\$d x 𝜑 \$." should be read "Assume x does not occur in phi \$." Sometimes a theorem is proved using 𝑥𝜑 (df-nf 1699) in place of "\$d 𝑥𝜑 \$." when a more general result is desired; ax-5 1793 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 2370 from df-eu 2366.
• "\$d x A \$." should be read "Assume x is not a variable occurring in class A."
• "\$d x A \$. \$d x ps \$. \$e |- (𝑥 = 𝐴 → (𝜑𝜓)) \$." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
• " (¬ ∀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 , 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 219 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 2162. The section mmtheorems32.html#mm3146s also describes the metatheorem that underlies this.

Standard Metamath verifiers do not distinguish between axioms and definitions (both are \$a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions (\$a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 195, df-clab 2501, df-cleq 2507, and df-clel 2510); 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 1029 or wsb 1830).
2. The defined expression must not appear in any statement between its syntax axiom () and its definition, and the defined expression must not be used in its definiens. See df-3an 1032 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 3078. The definition df-v 3079 V = {𝑥𝑥 = 𝑥} is justified by the justification theorem vjust 3078 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}. Another example of a justification theorem is trujust 1476; the definition df-tru 1477 (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1476 ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).

• Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
• Maximum text line length is 79 characters. You can fix comment line length by running the commands scripts/rewrap or metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' quit . As a general rule, a math string in a comment should be surrounded by backquotes on the same line, and if it is too long it should be broken into multiple adjacent mathstrings on multiple lines. Those commands don't modify the math content of statements. In statements we try to break before the outermost important connective (not including the typecode and perhaps not the antecedent). For examples, see sqrtmulii 13833 and absmax 13776.
• 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

(Contributed by DAW, 27-Dec-2016.) (New usage is discouraged.)

𝜑       𝜑

Theoremconventions-label 26389

The following explains some of the label conventions in use in the Metamath Proof Explorer ("set.mm"). For the general conventions, see conventions 26388.

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 2810"rgen.1 \$e |- ( x e. A -> ph ) \$." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 20134: "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 2455 and stirling 38885.
• Principia Mathematica. Proofs of theorems from Principia Mathematica often use a special naming convention: "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 40.
• 19.x series of theorems. Similar to the conventions for the theorems from Principia Mathematica, theorems from Section 19 of [Margaris] p. 90 often use a special naming convention: "19." resp. "r19." (for corresponding restricted quantifier versions) followed by its identifier. For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled 19.38 1745, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 2843.
• 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... 14320. Furthermore, the underscore "_" should not be used.
• Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct (𝐴𝐵) is defined in df-dif 3447, 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 3458. 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 3603. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4029), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4031). 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 4163. An "n" is often used for negation (¬), e.g., nan 601.
• 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 9697) and "re" represents real numbers ( definition df-r 9701). The empty set often uses fragment 0, even though it is defined in df-nul 3778. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 9702), 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 10899.
• 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 14588 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 14509) we have value cosval 14561 and closure coscl 14565.
• 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 26390 for a list), and that is about all you need for most things. As for the rest, you can just assume that if it involves at most three connectives, then it is probably already proved in set.mm, and searching for it will give you the label.
• Suffixes. Suffixes are used to indicate the form of a theorem (see above). Additionally, we sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as 𝑥𝜑 in 19.21 2036 via the use of disjoint variable conditions combined with nfv 1796. If two (or three) such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used, e.g. exlimivv 1813. 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 2370 derived from df-eu 2366. 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 4742. 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 2172 (cbval 2162 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). Here is a non-exhaustive list of common suffixes:
• a : theorem having a conjunction as antecedent
• b : theorem expressing a logical equivalence
• c : contraction (e.g., sylc 62, syl2anc 690), commutes (e.g., biimpac 501)
• 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)
• v : theorem with one (main) disjoint variable condition
• vv : theorem with two (main) disjoint variable conditions
• w : weak(er) form of a theorem
• ALT : alternate proof of a theorem
• ALTV : alternate version of a theorem or definition
• OLD : old/obsolete version of a theorem/definition/proof
• Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic, the source theorem or the assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.
AbbreviationMnenomicSource ExpressionSyntax?Example(s)
aand (suffix) No biimpa 499, rexlimiva 2914
ablAbelian group df-abl 17927 Abel Yes ablgrp 17929, zringabl 19545
absabsorption No ressabs 15650
absabsolute value (of a complex number) df-abs 13683 (abs‘𝐴) Yes absval 13685, absneg 13724, abs1 13744
al"for all" 𝑥𝜑 No alim 1714, alex 1731
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 384 (𝜑𝜓) Yes anor 508, iman 438, imnan 436
assassociative No biass 372, orass 544, mulass 9779
asymasymmetric, antisymmetric No intasym 5321, asymref 5322, posasymb 16667
axaxiom No ax6dgen 1953, ax1cn 9725
bas, base base (set of an extensible structure) df-base 15584 (Base‘𝑆) Yes baseval 15630, ressbas 15641, cnfldbas 19475
b, bibiconditional ("iff", "if and only if") df-bi 195 (𝜑𝜓) Yes impbid 200, sspwb 4742
brbinary relation df-br 4482 𝐴𝑅𝐵 Yes brab1 4528, brun 4531
cbvchange bound variable No cbvalivw 1884, cbvrex 3048
clclosure No ifclda 3973, ovrcl 6461, zaddcl 11158
cncomplex numbers df-c 9697 Yes nnsscn 10780, nncn 10783
cnfldfield of complex numbers df-cnfld 19472 fld Yes cnfldbas 19475, cnfldinv 19500
cntzcentralizer df-cntz 17465 (Cntz‘𝑀) Yes cntzfval 17468, dprdfcntz 18144
cnvconverse df-cnv 4940 𝐴 Yes opelcnvg 5116, f1ocnv 5946
cocomposition df-co 4941 (𝐴𝐵) Yes cnvco 5122, fmptco 6187
comcommutative No orcom 400, bicomi 212, eqcomi 2523
concontradiction, contraposition No condan 830, con2d 127
csbclass substitution df-csb 3404 𝐴 / 𝑥𝐵 Yes csbid 3411, csbie2g 3434
cygcyclic group df-cyg 18010 CycGrp Yes iscyg 18011, zringcyg 19562
ddeduction form (suffix) No idd 24, impbid 200
df(alternate) definition (prefix) No dfrel2 5392, dffn2 5845
di, distrdistributive No andi 906, imdi 376, ordi 903, difindi 3743, ndmovdistr 6597
difclass difference df-dif 3447 (𝐴𝐵) Yes difss 3603, difindi 3743
divdivision df-div 10434 (𝐴 / 𝐵) Yes divcl 10440, divval 10436, divmul 10437
dmdomain df-dm 4942 dom 𝐴 Yes dmmpt 5437, iswrddm0 13043
e, eq, equequals df-cleq 2507 𝐴 = 𝐵 Yes 2p2e4 10899, uneqri 3621, equtr 1898
elelement of 𝐴𝐵 Yes eldif 3454, eldifsn 4163, elssuni 4301
eu"there exists exactly one" df-eu 2366 ∃!𝑥𝜑 Yes euex 2386, euabsn 4108
exexists (i.e. is a set) No brrelex 4974, 0ex 4617
ex"there exists (at least one)" df-ex 1695 𝑥𝜑 Yes exim 1739, alex 1731
expexport No expt 166, expcom 449
f"not free in" (suffix) No equs45f 2242, sbf 2272
ffunction df-f 5693 𝐹:𝐴𝐵 Yes fssxp 5858, opelf 5863
falfalse df-fal 1480 Yes bifal 1487, falantru 1498
fifinite intersection df-fi 8076 (fi‘𝐵) Yes fival 8077, inelfi 8083
fi, finfinite df-fin 7721 Fin Yes isfi 7741, snfi 7799, onfin 7912
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 32851) df-field 18480 Field Yes isfld 18486, fldidom 19030
fnfunction with domain df-fn 5692 𝐴 Fn 𝐵 Yes ffn 5843, fndm 5789
frgpfree group df-frgp 17854 (freeGrp‘𝐼) Yes frgpval 17902, frgpadd 17907
fsuppfinitely supported function df-fsupp 8035 𝑅 finSupp 𝑍 Yes isfsupp 8038, fdmfisuppfi 8043, fsuppco 8066
funfunction df-fun 5691 Fun 𝐹 Yes funrel 5706, ffun 5846
fvfunction value df-fv 5697 (𝐹𝐴) Yes fvres 6001, swrdfv 13135
fzfinite set of sequential integers df-fz 12066 (𝑀...𝑁) Yes fzval 12067, eluzfz 12076
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 12174, fz0tp 12177
fzohalf-open integer range df-fzo 12203 (𝑀..^𝑁) Yes elfzo 12209, elfzofz 12222
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 6729
gragraph No uhgrav 25563, isumgra 25582, usgrares 25636
grpgroup df-grp 17140 Grp Yes isgrp 17143, tgpgrp 21595
gsumgroup sum df-gsum 15810 (𝐺 Σg 𝐹) Yes gsumval 16986, gsumwrev 17511
hashsize (of a set) df-hash 12848 (#‘𝐴) Yes hashgval 12850, hashfz1 12861, hashcl 12874
hbhypothesis builder (prefix) No hbxfrbi 1727, hbald 1977, hbequid 33102
hm(monoid, group, ring) homomorphism No ismhm 17052, isghm 17375, isrhm 18451
iinference (suffix) No eleq1i 2583, tcsni 8378
iimplication (suffix) No brwdomi 8232, infeq5i 8292
ididentity No biid 249
idmidempotent No anidm 673, tpidm13 4138
im, impimplication (label often omitted) df-im 13548 (𝐴𝐵) Yes iman 438, imnan 436, impbidd 198
imaimage df-ima 4945 (𝐴𝐵) Yes resima 5242, imaundi 5354
impimport No biimpa 499, impcom 444
inintersection df-in 3451 (𝐴𝐵) Yes elin 3662, incom 3670
infinfimum df-inf 8108 inf(ℝ+, ℝ*, < ) Yes fiinfcl 8166, infiso 8172
is...is (something a) ...? No isring 18281
jjoining, disjoining No jc 157, jaoi 392
lleft No olcd 406, simpl 471
mapmapping operation or set exponentiation df-map 7622 (𝐴𝑚 𝐵) Yes mapvalg 7630, elmapex 7640
matmatrix df-mat 19936 (𝑁 Mat 𝑅) Yes matval 19939, matring 19971
mdetdeterminant (of a square matrix) df-mdet 20113 (𝑁 maDet 𝑅) Yes mdetleib 20115, mdetrlin 20130
mgmmagma df-mgm 16957 Magma Yes mgmidmo 16974, mgmlrid 16981, ismgm 16958
mgpmultiplicative group df-mgp 18220 (mulGrp‘𝑅) Yes mgpress 18230, ringmgp 18283
mndmonoid df-mnd 17010 Mnd Yes mndass 17017, mndodcong 17685
mo"there exists at most one" df-mo 2367 ∃*𝑥𝜑 Yes eumo 2391, moim 2411
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mptmodus ponendo tollens No mptnan 1683, mptxor 1684
mptmaps-to notation for a function df-mpt 4543 (𝑥𝐴𝐵) Yes fconstmpt 4979, resmpt 5260
mpt2maps-to notation for an operation df-mpt2 6431 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpt2mpt 6527, resmpt2 6533
mulmultiplication (see "t") df-mul 9703 (𝐴 · 𝐵) Yes mulcl 9775, divmul 10437, mulcom 9777, mulass 9779
n, notnot ¬ 𝜑 Yes nan 601, notnotr 123
nenot equaldf-ne 𝐴𝐵 Yes exmidne 2696, neeqtrd 2755
nelnot element ofdf-nel 𝐴𝐵 Yes neli 2789, nnel 2796
ne0not equal to zero (see n0) ≠ 0 No negne0d 10141, ine0 10216, gt0ne0 10242
nf "not free in" (prefix) No nfnd 2032
ngpnormed group df-ngp 22100 NrmGrp Yes isngp 22112, ngptps 22118
nmnorm (on a group or ring) df-nm 22099 (norm‘𝑊) Yes nmval 22106, subgnm 22143
nnpositive integers df-nn 10776 Yes nnsscn 10780, nncn 10783
nn0nonnegative integers df-n0 11048 0 Yes nnnn0 11054, nn0cn 11057
n0not the empty set (see ne0) ≠ ∅ No n0i 3782, vn0 3786, ssn0 3831
OLDold, obsolete (to be removed soon) No 19.43OLD 1781
opordered pair df-op 4035 𝐴, 𝐵 Yes dfopif 4235, opth 4769
oror df-or 383 (𝜑𝜓) Yes orcom 400, anor 508
otordered triple df-ot 4037 𝐴, 𝐵, 𝐶 Yes euotd 4795, fnotovb 6469
ovoperation value df-ov 6429 (𝐴𝐹𝐵) Yes fnotovb 6469, fnovrn 6583
pplus (see "add"), for all-constant theorems df-add 9702 (3 + 2) = 5 Yes 3p2e5 10915
pfxprefix df-pfx 40153 (𝑊 prefix 𝐿) Yes pfxlen 40162, ccatpfx 40180
pmPrincipia Mathematica No pm2.27 40
pmpartial mapping (operation) df-pm 7623 (𝐴pm 𝐵) Yes elpmi 7638, pmsspw 7654
prpair df-pr 4031 {𝐴, 𝐵} Yes elpr 4049, prcom 4114, prid1g 4142, prnz 4156
prm, primeprime (number) df-prm 15100 Yes 1nprm 15106, dvdsprime 15114
pssproper subset df-pss 3460 𝐴𝐵 Yes pssss 3568, sspsstri 3575
q rational numbers ("quotients") df-q 11531 Yes elq 11532
rright No orcd 405, simprl 789
rabrestricted class abstraction df-rab 2809 {𝑥𝐴𝜑} Yes rabswap 3002, df-oprab 6430
ralrestricted universal quantification df-ral 2805 𝑥𝐴𝜑 Yes ralnex 2879, ralrnmpt2 6550
rclreverse closure No ndmfvrcl 6013, nnarcl 7459
rereal numbers df-r 9701 Yes recn 9781, 0re 9795
relrelation df-rel 4939 Rel 𝐴 Yes brrelex 4974, relmpt2opab 7021
resrestriction df-res 4944 (𝐴𝐵) Yes opelres 5213, f1ores 5948
reurestricted existential uniqueness df-reu 2807 ∃!𝑥𝐴𝜑 Yes nfreud 2995, reurex 3041
rexrestricted existential quantification df-rex 2806 𝑥𝐴𝜑 Yes rexnal 2882, rexrnmpt2 6551
rmorestricted "at most one" df-rmo 2808 ∃*𝑥𝐴𝜑 Yes nfrmod 2996, nrexrmo 3044
rnrange df-rn 4943 ran 𝐴 Yes elrng 5128, rncnvcnv 5161
rng(unital) ring df-ring 18279 Ring Yes ringidval 18233, isring 18281, ringgrp 18282
rotrotation No 3anrot 1035, 3orrot 1036
seliminates need for syllogism (suffix) No ancoms 467
sb(proper) substitution (of a set) df-sb 1831 [𝑦 / 𝑥]𝜑 Yes spsbe 1834, sbimi 1836
sbc(proper) substitution of a class df-sbc 3307 [𝐴 / 𝑥]𝜑 Yes sbc2or 3315, sbcth 3321
scascalar df-sca 15668 (Scalar‘𝐻) Yes resssca 15738, mgpsca 18226
simpsimple, simplification No simpl 471, simp3r3 1163
snsingleton df-sn 4029 {𝐴} Yes eldifsn 4163
spspecialization No spsbe 1834, spei 2152
sssubset df-ss 3458 𝐴𝐵 Yes difss 3603
structstructure df-struct 15581 Struct Yes brstruct 15587, structfn 15592
subsubtract df-sub 10019 (𝐴𝐵) Yes subval 10023, subaddi 10119
supsupremum df-sup 8107 sup(𝐴, 𝐵, < ) Yes fisupcl 8134, supmo 8117
suppsupport (of a function) df-supp 7058 (𝐹 supp 𝑍) Yes ressuppfi 8060, mptsuppd 7080
swapswap (two parts within a theorem) No rabswap 3002, 2reuswap 3281
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 3709, cnvsym 5320
symgsymmetric group df-symg 17513 (SymGrp‘𝐴) Yes symghash 17520, pgrpsubgsymg 17543
t times (see "mul"), for all-constant theorems df-mul 9703 (3 · 2) = 6 Yes 3t2e6 10934
ththeorem No nfth 1707, sbcth 3321, weth 9076
tptriple df-tp 4033 {𝐴, 𝐵, 𝐶} Yes eltpi 4079, tpeq1 4124
trtransitive No bitrd 266, biantr 967
trutrue df-tru 1477 Yes bitru 1486, truanfal 1497
ununion df-un 3449 (𝐴𝐵) Yes uneqri 3621, uncom 3623
unitunit (in a ring) df-unit 18372 (Unit‘𝑅) Yes isunit 18387, nzrunit 18992
vdisjoint variable conditions used when a not-free hypothesis (suffix) No spimv 2148
vv2 disjoint variables (in a not-free hypothesis) (suffix) No 19.23vv 1853
wweak (version of a theorem) (suffix) No ax11w 1955, spnfw 1878
wrdword df-word 13013 Word 𝑆 Yes iswrdb 13025, wrdfn 13033, ffz0iswrd 13046
xpcross product (Cartesian product) df-xp 4938 (𝐴 × 𝐵) Yes elxp 4949, opelxpi 4966, xpundi 4988
xreXtended reals df-xr 9833 * Yes ressxr 9838, rexr 9840, 0xr 9841
z integers (from German "Zahlen") df-z 11119 Yes elz 11120, zcn 11123
zn ring of integers mod 𝑛 df-zn 19580 (ℤ/nℤ‘𝑁) Yes znval 19608, zncrng 19618, znhash 19632
zringring of integers df-zring 19542 ring Yes zringbas 19547, zringcrng 19543
0, z slashed zero (empty set) (see n0) df-nul 3778 Yes n0i 3782, vn0 3786; snnz 4155, prnz 4156

(Contributed by DAW, 27-Dec-2016.) (New usage is discouraged.)

𝜑       𝜑

17.1.2  Natural deduction

Theoremnatded 26390 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 Γ𝜓 => Γ𝜓 idi 2 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 552 jca 552, pm3.2i 469 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 473 simpld 473, adantr 479 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 477 simpr 475, adantl 480 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 448 ex 448 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 698, imp 443 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 406 olc 397, olci 404, olcd 406 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 405 orc 398, orci 403, orcd 405 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 822 mpjaodan 822, jaodan 821, jaod 393 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1493 pm2.01d 179
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 688 mtand 688 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 597 pm2.65da 597 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 456 pm2.01d 179, pm2.65da 597, pm2.65d 185 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1495 pm2.21dd 184
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 184 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 184 pm2.21dd 184, pm2.21d 116, pm2.21 118 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ a1tru 1490 tru 1478, a1tru 1490, trud 1483 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1489 falim 1488 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1808 alrimiv 1808, ralrimiva 2853 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3320 spcv 3176, rspcv 3182 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3392 spcev 3177, rspcev 3186 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1816 exlimddv 1816, exlimdd 2121, exlimdv 1814, rexlimdva 2917 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1494 efald 1494 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 457 pm2.18da 457, pm2.18d 122, pm2.18 120 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 126 notnotrd 126, notnotr 123 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 430 exmid 429 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2515 eqid 2514, eqidd 2515 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3312 sbceq1d 3311, 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 26391, ex-natded5.3 26394, ex-natded5.5 26397, ex-natded5.7 26398, ex-natded5.8 26400, ex-natded5.13 26402, ex-natded9.20 26404, and ex-natded9.26 26406.

(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 26390 and mmnatded.html 26390. 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 26391 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 552, 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 26392. A proof without context is shown in ex-natded5.2i 26393. (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 26392 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 26391 and ex-natded5.2i 26393. (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 26393 The same as ex-natded5.2 26391 and ex-natded5.2-2 26392 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 26394 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 26395. A proof without context is shown in ex-natded5.3i 26396. 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 479 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given \$e; adantr 479 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 475, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 479 was used to transform its dependency (we could also use imp 443 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 479 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 552, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 448, 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 26395 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 26394 and ex-natded5.3i 26396. (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 26396 The same as ex-natded5.3 26394 and ex-natded5.3-2 26395 but with no context. Identical to jccir 559, 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 26397 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 479 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given \$e; we'll use adantr 479 to move it into the ND hypothesis
31 ...| 𝜓 (𝜑𝜓) ND hypothesis assumption simpr 475
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 479 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 597 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 479; simpr 475 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 187; a proof without context is shown in mto 186.

(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 26398 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 26399. 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 479 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 475
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 405, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 475
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 473, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 406, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 822, 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 26399 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 26398. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremex-natded5.8 26400 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 479 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given \$e; adantr 479 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given \$e; adantr 479 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given \$e. adantr 479 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 475. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 552 (I), 6,8 (adantr 479 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 597 (¬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 479; simpr 475 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 26401.

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

(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)

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206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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