P. 305 of Theorem List - Metamath Proof Explorer

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Type Label Description

Statement

Theorem frgrwopreglem3 30401* Lemma 3 for frgrwopreg 30410. The vertices in the sets 𝐴 and 𝐵 have different degrees. (Contributed by Alexander van der Vekens, 30-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 2-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) ⇒ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵) → (𝐷‘𝑋) ≠ (𝐷‘𝑌))

Theorem frgrwopreglem4 30402* Lemma 4 for frgrwopreg 30410. In a friendship graph each vertex with degree 𝐾 is connected with any 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.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 4-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ (𝐺 ∈ FriendGraph → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐵 {𝑎, 𝑏} ∈ 𝐸)

Theorem frgrwopregasn 30403* According to statement 5 in [Huneke] p. 2: "If A ... is a singleton, then that singleton is a universal friend". This version of frgrwopreg1 30405 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by Alexander van der Vekens, 1-Jan-2018.) (Revised by AV, 4-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐴 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)

Theorem frgrwopregbsn 30404* According to statement 5 in [Huneke] p. 2: "If ... B is a singleton, then that singleton is a universal friend". This version of frgrwopreg2 30406 is stricter (claiming that the singleton itself is a universal friend instead of claiming the existence of a universal friend only) and therefore closer to Huneke's statement. This strict variant, however, is not required for the proof of the friendship theorem. (Contributed by AV, 4-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝐵 = {𝑋}) → ∀𝑤 ∈ (𝑉 ∖ {𝑋}){𝑋, 𝑤} ∈ 𝐸)

Theorem frgrwopreg1 30405* 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.) (Proof shortened by AV, 4-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐴) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theorem frgrwopreg2 30406* 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.) (Proof shortened by AV, 4-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (♯‘𝐵) = 1) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theorem frgrwopreglem5lem 30407* Lemma for frgrwopreglem5 30408. (Contributed by AV, 5-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ (((𝑎 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ (𝑏 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝐷‘𝑎) = (𝐷‘𝑥) ∧ (𝐷‘𝑎) ≠ (𝐷‘𝑏) ∧ (𝐷‘𝑥) ≠ (𝐷‘𝑦)))

Theorem frgrwopreglem5 30408* Lemma 5 for frgrwopreg 30410. If 𝐴 as well as 𝐵 contain at least two vertices, there is a 4-cycle in a friendship 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.) (Proof shortened by AV, 5-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) & 𝐸 = (Edg‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎 ∈ 𝐴 ∃𝑥 ∈ 𝐴 ∃𝑏 ∈ 𝐵 ∃𝑦 ∈ 𝐵 ((𝑎 ≠ 𝑥 ∧ 𝑏 ≠ 𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Theorem frgrwopreglem5ALT 30409* Alternate direct proof of frgrwopreglem5 30408, not using frgrwopreglem5a 30398. This proof would be even a little bit shorter than the proof of frgrwopreglem5 30408 without using frgrwopreglem5lem 30407. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 3-Jan-2022.) (Proof shortened by AV, 5-Feb-2022.) (New usage is discouraged.) (Proof modification is discouraged.)

Theorem frgrwopreg 30410* In a friendship graph there are either no vertices (𝐴 = ∅) or exactly one vertex ((♯‘𝐴) = 1) having degree 𝐾, or all (𝐵 = ∅) or all except one vertices ((♯‘𝐵) = 1) have degree 𝐾. (Contributed by Alexander van der Vekens, 31-Dec-2017.) (Revised by AV, 10-May-2021.) (Proof shortened by AV, 3-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝐷 = (VtxDeg‘𝐺) & 𝐴 = {𝑥 ∈ 𝑉 ∣ (𝐷‘𝑥) = 𝐾} & 𝐵 = (𝑉 ∖ 𝐴) ⇒ (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Theorem frgrregorufr0 30411* 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.) (Revised by AV, 11-May-2021.) (Proof shortened by AV, 3-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝐸 = (Edg‘𝐺) & 𝐷 = (VtxDeg‘𝐺) ⇒ (𝐺 ∈ FriendGraph → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∀𝑣 ∈ 𝑉 (𝐷‘𝑣) ≠ 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

Theorem frgrregorufr 30412* 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 = (Vtx‘𝐺) & 𝐸 = (Edg‘𝐺) & 𝐷 = (VtxDeg‘𝐺) ⇒ (𝐺 ∈ FriendGraph → (∃𝑎 ∈ 𝑉 (𝐷‘𝑎) = 𝐾 → (∀𝑣 ∈ 𝑉 (𝐷‘𝑣) = 𝐾 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theorem frgrregorufrg 30413* 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 frgrregorufr 30412 with generalization. (Contributed by Alexander van der Vekens, 6-Sep-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝐸 = (Edg‘𝐺) ⇒ (𝐺 ∈ FriendGraph → ∀𝑘 ∈ ℕ₀ (∃𝑎 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theorem frgr2wwlkeu 30414* 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. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 4-Jan-2022.) (Revised by Ender Ting, 29-Jan-2026.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → ∃!𝑐 ∈ 𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))

Theorem frgr2wwlkn0 30415 In a friendship graph, there is always a path/walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)

Theorem frgr2wwlk1 30416 In a friendship graph, there is exactly one walk of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Theorem frgr2wsp1 30417 In a friendship graph, there is exactly one simple path of length 2 between two different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 13-May-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (♯‘(𝐴(2 WSPathsNOn 𝐺)𝐵)) = 1)

Theorem frgr2wwlkeqm 30418 If there is a (simple) path of length 2 from one vertex to another vertex and a (simple) path of length 2 from the other vertex back to the first vertex in a friendship graph, then the middle vertex is the same. This is only an observation, which is not required to proof the friendship theorem. (Contributed by Alexander van der Vekens, 20-Feb-2018.) (Revised by AV, 13-May-2021.) (Proof shortened by AV, 7-Jan-2022.)

((𝐺 ∈ FriendGraph ∧ 𝐴 ≠ 𝐵 ∧ (𝑃 ∈ 𝑋 ∧ 𝑄 ∈ 𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Theorem frgrhash2wsp 30419 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, Huneke counts undirected paths, so obtains the result ((𝑛C2) = ((𝑛 · (𝑛 − 1)) / 2)), whereas we count directed paths, obtaining twice that number. (Contributed by Alexander van der Vekens, 6-Mar-2018.) (Revised by AV, 10-Jan-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · ((♯‘𝑉) − 1)))

Theorem fusgreg2wsplem 30420* Lemma for fusgreg2wsp 30423 and related theorems. (Contributed by AV, 8-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ (𝑁 ∈ 𝑉 → (𝑝 ∈ (𝑀‘𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Theorem fusgr2wsp2nb 30421* The set of paths of length 2 with a given vertex in the middle for a finite simple 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.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.)

𝑉 = (Vtx‘𝐺) & 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ 𝑉) → (𝑀‘𝑁) = ∪ 𝑥 ∈ (𝐺 NeighbVtx 𝑁)∪ 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Theorem fusgreghash2wspv 30422* 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 directed simple paths of length 2 represented by length 3 strings, we have again k*(k-1) such paths, see also comment of frgrhash2wsp 30419. (Contributed by Alexander van der Vekens, 10-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 12-Feb-2022.)

𝑉 = (Vtx‘𝐺) & 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ (𝐺 ∈ FinUSGraph → ∀𝑣 ∈ 𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀‘𝑣)) = (𝐾 · (𝐾 − 1))))

Theorem fusgreg2wsp 30423* In a finite simple graph, the set of all paths of length 2 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.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑀‘𝑥))

Theorem 2wspmdisj 30424* 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, 18-May-2021.) (Proof shortened by AV, 10-Jan-2022.)

𝑉 = (Vtx‘𝐺) & 𝑀 = (𝑎 ∈ 𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎}) ⇒ Disj 𝑥 ∈ 𝑉 (𝑀‘𝑥)

Theorem fusgreghash2wsp 30425* 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 length 3 strings, however, we have again n*k*(k-1) such paths. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 19-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FinUSGraph ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1)))))

Theorem frrusgrord0lem 30426* Lemma for frrusgrord0 30427. (Contributed by AV, 12-Jan-2022.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0))

Theorem frrusgrord0 30427* 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.". (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → (∀𝑣 ∈ 𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theorem frrusgrord 30428 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 frrusgrord0 30427, using the definition RegUSGraph (df-rusgr 29644). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 26-May-2021.) (Proof shortened by AV, 12-Jan-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theorem numclwwlk2lem1lem 30429 Lemma for numclwwlk2lem1 30463. (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 27-May-2021.) (Revised by AV, 15-Mar-2022.)

((𝑋 ∈ (Vtx‘𝐺) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (lastS‘𝑊) ≠ (𝑊‘0)) → (((𝑊 ++ ⟨“𝑋”⟩)‘0) = (𝑊‘0) ∧ ((𝑊 ++ ⟨“𝑋”⟩)‘𝑁) ≠ (𝑊‘0)))

Theorem 2clwwlklem 30430 Lemma for clwwnonrepclwwnon 30432 and extwwlkfab 30439. (Contributed by Alexander van der Vekens, 18-Sep-2018.) (Revised by AV, 10-May-2022.) (Revised by AV, 30-Oct-2022.)

((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑊 prefix (𝑁 − 2))‘0) = (𝑊‘0))

Theorem clwwnrepclwwn 30431 If the initial vertex of a closed walk occurs another time in the walk, the walk starts with a closed walk. Notice that 3 ≤ 𝑁 is required, because for 𝑁 = 2, (𝑤 prefix (𝑁 − 2)) = (𝑤 prefix 0) = ∅, but ∅ (and anything else) is not a representation of an empty closed walk as word, see clwwlkn0 30115. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 30-Oct-2022.)

((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊 prefix (𝑁 − 2)) ∈ ((𝑁 − 2) ClWWalksN 𝐺))

Theorem clwwnonrepclwwnon 30432 If the initial vertex of a closed walk occurs another time in the walk, the walk starts with a closed walk on this vertex. See also the remarks in clwwnrepclwwn 30431. (Contributed by AV, 24-Apr-2022.) (Revised by AV, 10-May-2022.) (Revised by AV, 30-Oct-2022.)

((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋) → (𝑊 prefix (𝑁 − 2)) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))

Theorem 2clwwlk2clwwlklem 30433 Lemma for 2clwwlk2clwwlk 30437. (Contributed by AV, 27-Apr-2022.)

((𝑁 ∈ (ℤ_≥‘3) ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = (𝑊‘0)) → (𝑊 substr ⟨(𝑁 − 2), 𝑁⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)2))

Theorem 2clwwlk 30434* Value of operation 𝐶, mapping a vertex v and an integer n greater than 1 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. Such closed walks are "double loops" consisting of a closed (n-2)-walk v = v(0) ... v(n-2) = v and a closed 2-walk v = v(n-2) v(n-1) v(n) = v, see 2clwwlk2clwwlk 30437. (𝑋𝐶𝑁) is called the "set of double loops of length 𝑁 on vertex 𝑋 " in the following. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})

Theorem 2clwwlk2 30435* The set (𝑋𝐶2) of double loops of length 2 on a vertex 𝑋 is equal to the set of closed walks with length 2 on 𝑋. Considered as "double loops", the first of the two closed walks/loops is degenerated, i.e., has length 0. (Contributed by AV, 18-Feb-2022.) (Revised by AV, 20-Apr-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ (𝑋 ∈ 𝑉 → (𝑋𝐶2) = (𝑋(ClWWalksNOn‘𝐺)2))

Theorem 2clwwlkel 30436* Characterization of an element of the value of operation 𝐶, i.e., of a word being a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 24-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 20-Apr-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋)))

Theorem 2clwwlk2clwwlk 30437* An element of the value of operation 𝐶, i.e., a word being a double loop of length 𝑁 on vertex 𝑋, is composed of two closed walks. (Contributed by AV, 28-Apr-2022.) (Proof shortened by AV, 3-Nov-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ ∃𝑎 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))∃𝑏 ∈ (𝑋(ClWWalksNOn‘𝐺)2)𝑊 = (𝑎 ++ 𝑏)))

Theorem numclwwlk1lem2foalem 30438 Lemma for numclwwlk1lem2foa 30441. (Contributed by AV, 29-May-2021.) (Revised by AV, 1-Nov-2022.)

(((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 2)) = 𝑋))

Theorem extwwlkfab 30439* The set (𝑋𝐶𝑁) of double loops of length 𝑁 on vertex 𝑋 can be constructed from the set 𝐹 of closed walks on 𝑋 with length smaller by 2 than the fixed length by 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: 𝐹 = (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 30179 stating that a closed walk of length 0 is not represented as word), which would result in an empty set on the right hand side, but (𝑋𝐶𝑁) needs not be empty, see 2clwwlk2 30435. (Contributed by Alexander van der Vekens, 18-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑤‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)})

Theorem extwwlkfabel 30440* Characterization of an element of the set (𝑋𝐶𝑁), i.e., a double loop of length 𝑁 on vertex 𝑋 with a construction from the set 𝐹 of closed walks on 𝑋 with length smaller by 2 than the fixed length by 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). (Contributed by AV, 22-Feb-2022.) (Revised by AV, 31-Oct-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ ((𝑊 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))))

Theorem numclwwlk1lem2foa 30441* Going forth and back from the end of a (closed) walk: 𝑊 represents the closed walk p₀, ..., p_(n-2), p₀ = p_(n-2). With 𝑋 = p_(n-2) = p₀ and 𝑌 = p_(n-1), ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) represents the closed walk p₀, ..., p_(n-2), p_(n-1), p_n = p₀ which is a double loop of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 22-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 2-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → ((𝑊 ∈ 𝐹 ∧ 𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋𝐶𝑁)))

Theorem numclwwlk1lem2f 30442* 𝑇 is a function, mapping a double loop of length 𝑁 on vertex 𝑋 to the ordered pair of the first loop and the successor of 𝑋 in the second loop, which must be a neighbor of 𝑋. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 23-Feb-2022.) (Revised by AV, 31-Oct-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))⟩) ⇒ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → 𝑇:(𝑋𝐶𝑁)⟶(𝐹 × (𝐺 NeighbVtx 𝑋)))

Theorem numclwwlk1lem2fv 30443* Value of the function 𝑇. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Oct-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) & 𝑇 = (𝑢 ∈ (𝑋𝐶𝑁) ↦ ⟨(𝑢 prefix (𝑁 − 2)), (𝑢‘(𝑁 − 1))⟩) ⇒ (𝑊 ∈ (𝑋𝐶𝑁) → (𝑇‘𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

Theorem numclwwlk1lem2f1 30444* 𝑇 is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 23-Feb-2022.) (Revised by AV, 31-Oct-2022.)

Theorem numclwwlk1lem2fo 30445* 𝑇 is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 29-May-2021.) (Proof shortened by AV, 13-Feb-2022.) (Revised by AV, 31-Oct-2022.)

Theorem numclwwlk1lem2f1o 30446* 𝑇 is a 1-1 onto function. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.)

Theorem numclwwlk1lem2 30447* The set of double loops of length 𝑁 on vertex 𝑋 and the set of closed walks of length less by 2 on 𝑋 combined with the neighbors of 𝑋 are equinumerous. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 31-Jul-2022.) (Proof shortened by AV, 3-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ ((𝐺 ∈ USGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))

Theorem numclwwlk1 30448* 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 𝐺 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 not for n=2, since 𝐹 = ∅, but (𝑋𝐶2), the set of closed walks with length 2 on 𝑋, see 2clwwlk2 30435, needs not be ∅ in this case. This is because of the special definition of 𝐹 and the usage of words to represent (closed) walks, and does not contradict Huneke's statement, which would read "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)", where f(0)=1 is the number of empty closed walks on v, see numclwlk1lem1 30456. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 30458. This case, however, is not required to prove the friendship theorem. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.) (Revised by AV, 6-Mar-2022.) (Proof shortened by AV, 31-Jul-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)) ⇒ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

Theorem clwwlknonclwlknonf1o 30449* 𝐹 is a bijection between the two representations of closed walks of a fixed positive length on a fixed vertex. (Contributed by AV, 26-May-2022.) (Proof shortened by AV, 7-Aug-2022.) (Revised by AV, 1-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} & 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) ⇒ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝐹:𝑊–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Theorem clwwlknonclwlknonen 30450* The sets of the two representations of closed walks of a fixed positive length on a fixed vertex are equinumerous. (Contributed by AV, 27-May-2022.) (Proof shortened by AV, 3-Nov-2022.)

((𝐺 ∈ USPGraph ∧ 𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Theorem dlwwlknondlwlknonf1olem1 30451 Lemma 1 for dlwwlknondlwlknonf1o 30452. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)

(((♯‘(1^st ‘𝑐)) = 𝑁 ∧ 𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))‘(𝑁 − 2)) = ((2^nd ‘𝑐)‘(𝑁 − 2)))

Theorem dlwwlknondlwlknonf1o 30452* 𝐹 is a bijection between the two representations of double loops of a fixed positive length on a fixed vertex. (Contributed by AV, 30-May-2022.) (Revised by AV, 1-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} & 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} & 𝐹 = (𝑐 ∈ 𝑊 ↦ ((2^nd ‘𝑐) prefix (♯‘(1^st ‘𝑐)))) ⇒ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝐹:𝑊–1-1-onto→𝐷)

Theorem dlwwlknondlwlknonen 30453* The sets of the two representations of double loops of a fixed length on a fixed vertex are equinumerous. (Contributed by AV, 30-May-2022.) (Proof shortened by AV, 3-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑊 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} & 𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋} ⇒ ((𝐺 ∈ USPGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → 𝑊 ≈ 𝐷)

Theorem wlkl0 30454* There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)

𝑉 = (Vtx‘𝐺) ⇒ (𝑋 ∈ 𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 0 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Theorem clwlknon2num 30455* There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 30192, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 2 ∧ ((2^nd ‘𝑤)‘0) = 𝑋)}) = 𝐾)

Theorem numclwlk1lem1 30456* Lemma 1 for numclwlk1 30458 (Statement 9 in [Huneke] p. 2 for n=2): "the number of closed 2-walks v(0) v(1) v(2) from v = v(0) = v(2) ... is kf(0)". (Contributed by AV, 23-May-2022.)

𝑉 = (Vtx‘𝐺) & 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = 𝑁 ∧ ((2^nd ‘𝑤)‘0) = 𝑋 ∧ ((2^nd ‘𝑤)‘(𝑁 − 2)) = 𝑋)} & 𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1^st ‘𝑤)) = (𝑁 − 2) ∧ ((2^nd ‘𝑤)‘0) = 𝑋)} ⇒ (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Theorem numclwlk1lem2 30457* Lemma 2 for numclwlk1 30458 (Statement 9 in [Huneke] p. 2 for n>2). This theorem corresponds to numclwwlk1 30448, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)

Theorem numclwlk1 30458* 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 𝐺 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. (Contributed by AV, 23-May-2022.)

Theorem numclwwlkovh0 30459* Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 greater than 1 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 AV, 1-May-2022.)

𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋})

Theorem numclwwlkovh 30460* Value of operation 𝐻, mapping a vertex 𝑣 and an integer 𝑛 greater than 1 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. Definition of ClWWalksNOn resolved. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 1-May-2022.)

𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})

Theorem numclwwlkovq 30461* Value of operation 𝑄, mapping a vertex 𝑣 and a positive 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). Remark: 𝑛 ∈ ℕ₀ would not be useful: numclwwlkqhash 30462 would not hold, because (𝐾↑0) = 1! (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 30-May-2021.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})

Theorem numclwwlkqhash 30462* In a 𝐾-regular graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set of closed walks of length 𝑁 on vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 5-Mar-2022.) (Proof shortened by AV, 7-Jul-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾↑𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))

Theorem numclwwlk2lem1 30463* In a friendship graph, for each walk of length 𝑛 starting at a fixed vertex 𝑣 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 generally holds only for friendship graphs, because these guarantee that for the first and last vertex there is a (unique) third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 30-May-2021.) (Revised by AV, 1-May-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣 ∈ 𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Theorem numclwlk2lem2f 30464* 𝑅 is a function mapping the "closed (n+2)-walks v(0) ... v(n-2) v(n-1) v(n) v(n+1) v(n+2) starting at 𝑋 = v(0) = v(n+2) with v(n) =/= X" to the words representing the prefix v(0) ... v(n-2) v(n-1) v(n) of the walk. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 31-May-2021.) (Proof shortened by AV, 23-Mar-2022.) (Revised by AV, 1-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) & 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1))) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))

Theorem numclwlk2lem2fv 30465* Value of the function 𝑅. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) & 𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1))) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅‘𝑊) = (𝑊 prefix (𝑁 + 1))))

Theorem numclwlk2lem2f1o 30466* 𝑅 is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Proof shortened by AV, 17-Mar-2022.) (Revised by AV, 1-Nov-2022.)

Theorem numclwwlk2lem3 30467* In a friendship graph, the size of the set of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex equals the size of the set of all closed walks of length (𝑁 + 2) starting at this vertex 𝑋 and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.) (Proof shortened by AV, 3-Nov-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (♯‘(𝑋𝑄𝑁)) = (♯‘(𝑋𝐻(𝑁 + 2))))

Theorem numclwwlk2 30468* 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 rusgrnumwlkg 30065, 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.) (Revised by AV, 31-May-2021.) (Revised by AV, 1-May-2022.)

𝑉 = (Vtx‘𝐺) & 𝑄 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))

Theorem numclwwlk3lem1 30469 Lemma 2 for numclwwlk3 30472. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Proof shortened by AV, 23-Jan-2022.)

((𝐾 ∈ ℂ ∧ 𝑌 ∈ ℂ ∧ 𝑁 ∈ (ℤ_≥‘2)) → (((𝐾↑(𝑁 − 2)) − 𝑌) + (𝐾 · 𝑌)) = (((𝐾 − 1) · 𝑌) + (𝐾↑(𝑁 − 2))))

Theorem numclwwlk3lem2lem 30470* Lemma for numclwwlk3lem2 30471: The set of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the union of the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by AV, 1-May-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)))

Theorem numclwwlk3lem2 30471* Lemma 1 for numclwwlk3 30472: The number of closed vertices of a fixed length 𝑁 on a fixed vertex 𝑉 is the sum of the number of closed walks of length 𝑁 at 𝑉 with the last but one vertex being 𝑉 and the set of closed walks of length 𝑁 at 𝑉 with the last but one vertex not being 𝑉. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 1-Jun-2021.) (Revised by AV, 1-May-2022.)

𝐶 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣}) & 𝐻 = (𝑣 ∈ 𝑉, 𝑛 ∈ (ℤ_≥‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣}) ⇒ (((𝐺 ∈ FinUSGraph ∧ 𝑋 ∈ 𝑉) ∧ 𝑁 ∈ (ℤ_≥‘2)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = ((♯‘(𝑋𝐻𝑁)) + (♯‘(𝑋𝐶𝑁))))

Theorem numclwwlk3 30472 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 30448) and with v(n-2) =/= v(n) (see numclwwlk2 30468): 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.) (Revised by AV, 6-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ_≥‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Theorem numclwwlk4 30473* The total number of closed walks in a finite 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.) (Revised by AV, 2-Jun-2021.) (Revised by AV, 7-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = Σ𝑥 ∈ 𝑉 (♯‘(𝑥(ClWWalksNOn‘𝐺)𝑁)))

Theorem numclwwlk5lem 30474 Lemma for numclwwlk5 30475. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.) (Revised by AV, 7-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ 𝑉 ∧ 𝐾 ∈ ℕ₀) → (2 ∥ (𝐾 − 1) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1))

Theorem numclwwlk5 30475 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.) (Revised by AV, 2-Jun-2021.) (Revised by AV, 7-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)𝑃)) mod 𝑃) = 1)

Theorem numclwwlk7lem 30476 Lemma for numclwwlk7 30478, frgrreggt1 30480 and frgrreg 30481: If a finite, nonempty friendship graph is 𝐾-regular, the 𝐾 is a nonnegative integer. (Contributed by AV, 3-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ₀)

Theorem numclwwlk6 30477 For a prime divisor 𝑃 of 𝐾 − 1, the total number of closed walks of length 𝑃 in a 𝐾-regular friendship graph is equal modulo 𝑃 to the number of vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) (Proof shortened by AV, 7-Mar-2022.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((♯‘𝑉) mod 𝑃))

Theorem numclwwlk7 30478 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 frrusgrord0 30427 or frrusgrord 30428, 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 null graph is a friendship graph, see frgr0 30352, as well as k-regular (for any k), see 0vtxrgr 29662, but has no closed walk, see 0clwlk0 30219, this theorem would be false for a null graph: ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0 ≠ 1, so this case must be excluded (by assuming 𝑉 ≠ ∅). (Contributed by Alexander van der Vekens, 1-Sep-2018.) (Revised by AV, 3-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 RegUSGraph 𝐾 ∧ 𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1)

Theorem numclwwlk8 30479 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 clwlksndivn 30173. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.) (Proof shortened by AV, 2-Mar-2022.)

((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)

Theorem frgrreggt1 30480 If a finite nonempty friendship graph is 𝐾-regular with 𝐾 > 1, then 𝐾 must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 RegUSGraph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))

Theorem frgrreg 30481 If a finite nonempty friendship graph is 𝐾-regular, then 𝐾 must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Theorem frgrregord013 30482 If a finite friendship graph is 𝐾-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Theorem frgrregord13 30483 If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 30482. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Theorem frgrogt3nreg 30484* If a finite friendship graph has an order greater than 3, it cannot be 𝑘-regular for any 𝑘. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ₀ ¬ 𝐺 RegUSGraph 𝑘)

Theorem friendshipgt3 30485* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

Theorem friendship 30486* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)

𝑉 = (Vtx‘𝐺) ⇒ ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣 ∈ 𝑉 ∀𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

PART 18 GUIDES AND MISCELLANEA

18.1 Guides (conventions, explanations, and examples)

18.1.1 Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. They are organized as follows:

conventions 30487: general conventions
conventions-labels 30488: conventions related to labels
conventions-comments 30489: conventions related to comments

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].

Theorem

conventions 30487

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

conventions-labels 30488: conventions related to labels
conventions-comments 30489: conventions related to comments

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 Σ𝑘 ∈ 𝐴𝐵 (df-sum 15622) 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 15817). 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 four of them (df-bi 207, df-clab 2716, df-cleq 2729, df-clel 2812) 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 2812. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is the complex arithmetic axiom ax-1cn 11096, proven by the preceding theorem ax1cn 11072. The Metamath program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
Statements with hypotheses. Many theorems and some axioms, such as ax-mp 5, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When displayed in summarized form such as in the "Theorem List" page (to get to it, click on "Nearby theorems" on the ax-mp 5 page), the hypotheses are connected with an ampersand and separated from the conclusion with a double right arrow, such as in " ⊢ 𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ 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 11096 (not ax1cn 11072) and ax-1ne0 11107 (not ax1ne0 11083), as these are proven axioms for complex arithmetic. Thus, both ax1cn 11072 and ax1ne0 11083 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".
New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use < and ≤ for inequality expressions, and use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴) for the hyperbolic sine.
Minimizing axiom dependencies. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, because of the non-constructive nature of the axiom of choice df-ac 10038, we prefer proofs that do not use it, or use weaker versions like countable choice ax-cc 10357 or dependent choice ax-dc 10368. An example is our proof of the Schroeder-Bernstein Theorem sbth 9037, which does not use the axiom of choice. Similarly, any theorem in first-order logic (FOL) that contains only setvar variables that are all mutually distinct, and has no wff variables, can be proved without using ax-10 2147 through ax-13 2377, by using ax10w 2135 through ax13w 2142 instead.
We do not try to similarly reduce dependencies on definitions, since definitions are conservative (they do not increase the proving power of a deductive system), and are introduced in order to be used to increase readability). An exception is made for Definitions df-clab 2716, df-cleq 2729, and df-clel 2812, since they can be considered as axioms under some definitions of what a definition is exactly (see their comments).
Alternate proofs (ALT). If a different proof is shorter or clearer but uses more or stronger axioms, we make that proof an "alternate" proof (marked with an ALT label suffix), even if this alternate proof was formalized first. We then make the proof that requires fewer axioms the main proof. Alternate proofs can also occur in other cases when an alternate proof gives some particular insight. Their comment should begin with "Alternate proof of ~ xxx " followed by a description of the specificity of that alternate proof. There can be multiple alternates. Alternate (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternate and main statement comments should use hyperlinks to refer to each other.
Alternate versions (ALTV). The suffix ALTV is reserved for theorems (or definitions) which are alternate versions, or variants, of an existing theorem. This is reserved to statements in mathboxes and is typically used temporarily, when it is not clear yet which variant to use. If it is decided that both variants should be kept and moved to the main part of set.mm, then a label for the variant should be found with a more explicit suffix indicating how it is a variant (e.g., commutation of some subformula, antecedent replaced with hypothesis, (un)curried variant, biconditional instead of implication, etc.). There is no requirement to add discouragement tags, but their comment should have a link to the main version of the statement and describe how it is a variant of it.
Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-Mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first. This typically happens when a shorter proof to an existing theorem is found: the existing theorem is kept as an *OLD statement for one year. When a proof is shortened automatically (using the Metamath program "MM-PA> MINIMIZE__WITH *" command), then it is not necessary to keep the old proof, nor to add credit for the shortening.
Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are generally used in this order: 𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta, 𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma. Individual setvar variables are represented with lowercase Latin letters and are generally used in this order: 𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡. In addition, the surreal number section uses subscripted lowercase Latin letters such as 𝑥_𝑂, 𝑥_𝐿, and 𝑥_𝑅. These match the conventional literature on surreal numbers. These variables should not be used outside of that section. Variables that represent classes are often represented by uppercase Latin letters: 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g., 0 for a zero element (e.g., fsuppcor 9319) and connective symbols such as + for some group addition operation (e.g., grpinva 18611). 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 230 or mpbir 231. 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 216 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 712, sylbir 235, or 3imtr4i 292.
Quantifiers. The quantifiers are named as follows:
- ∀: universal quantifier (wal 1540);
- ∃: existential quantifier (df-ex 1782);
- ∃*: at-most-one quantifier (df-mo 2540);
- ∃!: unique existential quantifier (df-eu 2570).
The phrase "uniqueness quantifier" is avoided since it is ambiguous: it can be understood as claiming either uniqueness (∃*) or unique existence (∃!).
Substitution. The expression "[𝑦 / 𝑥]𝜑" should be read "the formula that results from the proper substitution of 𝑦 for 𝑥 in the formula 𝜑". See df-sb 2069 and the related df-sbc 3743 and df-csb 3852.
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 3444 and isset 3456. However, instead of using 𝐼 ∈ V in the antecedent of a theorem for some variable 𝐼, we now prefer to use 𝐼 ∈ 𝑉 (or another variable if 𝑉 is not available) to make it more general. That way we can often avoid extra uses of elex 3463 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 ⊢ 𝐴 ∈ 𝑉 (e.g., difexi 5277). 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 ⊢ (𝜑 → 𝐴 ∈ 𝑉) should be used rather than ⊢ (𝜑 → 𝐴 ∈ V).
Converse. The symbol " ^◡" denotes the converse of a relation, so " ^◡𝑅" denotes the converse of the class 𝑅, which is typically a relation in that context (see df-cnv 5640). The converse of a relation 𝑅 is sometimes denoted by R^-1 in textbooks, especially when 𝑅 is a function, but we avoid this notation since it is generally not a genuine inverse (see f1cocnv1 6812 and funcocnv2 6807 for cases where it is a left or right-inverse). This can be used to define a subset, e.g., df-tan 16006 notates "the set of values whose cosine is a nonzero complex number" as (^◡cos “ (ℂ ∖ {0})).
Function application. The symbols "(𝐹‘𝑥)" should be read "the value of (function) 𝐹 at 𝑥" and has the same meaning as the more familiar but ambiguous notation F(x). For example, (cos‘0) = 1 (see cos0 16087). 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 6508. 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 7371). For example, the + in (2 + 2); see 2p2e4 12287. 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 (𝜑 → 𝜓), (𝜑 ∨ 𝜓), (𝜑 ∧ 𝜓), and (𝜑 ↔ 𝜓) (see wi 4, df-or 849, df-an 396, and df-bi 207 respectively). In contrast, a binary relation (which compares two _classes_ and produces a _wff_) applied in an infix expression is _not_ surrounded by parentheses. This includes set membership 𝐴 ∈ 𝐵 (see wel 2115), equality 𝐴 = 𝐵 (see df-cleq 2729), subset 𝐴 ⊆ 𝐵 (see df-ss 3920), and less-than 𝐴 < 𝐵 (see df-lt 11051). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 5101. For example, 0 < 1 (see 0lt1 11671) 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 11377.
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 15999). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 16005). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 16064), its function application form labeled NAMEval (e.g., cosval 16060), and at least one simple value (e.g., cos0 16087). Another way to define functions is to use recursion (for more details about recursion see below). For an example of how to define functions that aren't primitive recursive using recursion, see the Ackermann function definition df-ack 49014 (which is based on the sequence builder seq, see df-seq 13937).
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 14209 and fac4 14216).
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 11045), while "0_g" is the group identity element (df-0g 17373), "0." is the poset zero (df-p0 18358), "0_𝑝" is the zero polynomial (df-0p 25639), "0_vec" is the zero vector in a normed subcomplex vector space (df-0v 30685), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 18360). 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 have 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 cannot be an actual class since not all limit ordinals are sets. This initial upper versus lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).
Typography conventions. Class symbols for functions (e.g., abs, sin) should usually not have leading or trailing blanks in their HTML representation. This is in contrast to class symbols for operations (e.g., gcd, sadd, eval), which usually do include leading and trailing blanks in their representation. If a class symbol is used for a function as well as an operation (according to Definition df-ov 7371, each operation value can be written as function value of an ordered pair), the convention for its primary usage should be used, e.g., (iEdg‘𝐺) versus (𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸⟩.
LaTeX definitions. Each token has a "LaTeX definition" which is used by the Metamath program to output tex files. When writing LaTeX definitions, contributors should favor simplicity over perfection of the display, and should only use core LaTeX symbols or symbols from standard packages; if packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic are needed, this should be discussed. A useful resource is The Comprehensive LaTeX Symbol List.
Number construction independence. There are many ways to model complex numbers. After deriving the complex number postulates we reintroduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. This also lets us be independent of the specific construction, which we believe is valuable. See mmcomplex.html 7371 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 ℂ to be independent of whether or not ∅ ∈ ℂ.
Minimize hypotheses. In most cases we try to minimize hypotheses, so that the statement be more general and easier to use. There are exceptions. For example, we intentionally add hypotheses if they help make proofs independent of a particular construction (e.g., the contruction of the complex numbers ℂ). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they are not needed. For example, we could show that ⊢ (𝐴 < 𝐵 → 𝐵 ≠ 𝐴) without any hypotheses, but we require that theorems using this result prove that 𝐴 and 𝐵 are real numbers, so that the statement we use is ltnei 11269. Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 11269:
1. Having the hypotheses immediately shows the intended domain of applicability (is it ℝ, ℝ^*, ω, or something else?), without having to trace back to definitions.
2. Having the hypotheses forces the intended use of the statement, which generally is desirable.
3. Many out-of-domain values are dependent on contingent details of definitions, so hypothesis-free theorems would be non-portable and "brittle".
4. Only a few theorems can have their hypotheses removed in this fashion, due to coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning, e.g., real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who has not traced back the set-theoretical foundations of the definitions, it is seemingly random and is not intuitive at all.
5. Ultimately, this is a matter of consensus, and the consensus in the group was in favor of keeping sometimes redundant hypotheses.
Natural numbers. There are different definitions of "natural" numbers in the literature. We use ℕ (df-nn 12158) for the set of positive integers starting from 1, and ℕ₀ (df-n0 12414) 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 12620, e.g., ;;;4001 (see 4001prm 17084 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 4144). The term "tautology" is also used, especially in propositional calculus. This form was formerly called "theorem form" or "closed theorem form".
- A theorem is in "deduction form" (or is a "deduction") if it has zero or more $e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually 𝜑), and every hypothesis ($e statement) is either:
  1. an implication with the same antecedent as the conclusion, or
  2. a definition. A definition can be for a class variable (this is a class variable followed by =, e.g., the definition of 𝐷 in lhop 25989) 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 4146. Although they are no real deductions, theorems without $e hypotheses, but in the form (𝜑 → ...), are also said to be in "deduction form". Such theorems usually have a two step proof, applying a1i 11 to a given theorem, and are used as convenience theorems to shorten many proofs. E.g., eqidd 2738, which is used more than 1500 times.
- A theorem is in "inference form" (or is an "inference") if it has one or more $e hypotheses, but is not in deduction form, i.e., there is no common antecedent (e.g., unssi 4145).
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 154 is the inference associated with con3 153). The inference associated with a theorem is easily derivable from that theorem by a simple use of ax-mp 5. The other direction is the subject of the Deduction Theorem discussed below. We may also use the term "associated inference" when the above process is iterated. For instance, syl 17 is an inference associated with imim1 83 because it is the inference associated with imim1i 63 which is itself the inference associated with imim1 83.
"Deduction form" is the preferred form for theorems because this form allows to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that would end up requiring us to use the deduction theorem.
Deductions have a label suffix of "d", especially if there are other forms of the same theorem (e.g., pm2.43d 53). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 56) or, if the non-suffixed label is already used, then we add the suffix "t" (for "theorem" or "tautology", e.g., ancomst 464 or nfimt 1897). When an inference with an "is a set" hypothesis (e.g., 𝐴 ∈ V) is converted to a theorem (in closed form) by replacing the hypothesis with an antecedent of the form (𝐴 ∈ 𝑉 →, we sometimes suffix the closed form with "g" (for "more general") as in uniex 7696 versus uniexg 7695. In this case, the inference often has no suffix "i".
When submitting a new theorem, a revision of a theorem, or an upgrade of a theorem from a Mathbox to the Main database, please use the general form to be the default form of the theorem, without the suffix "g" . For example, "brresg" lost its suffix "g" when it was revised for some other reason, and now it is brres 5953. Its inference form which was the original "brres", now is brresi 5955. The same holds for the suffix "t".
Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction (its associated inference). See for instance Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see mmdeduction.html 5955. 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 4540, which works in certain cases in set theory. We also sometimes use dedhb 3663. 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 3663; a list of translations for common natural deduction rules is given in natded 30490.
Recursion. We define recursive functions using various "recursion constructors". These allow 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 8313, rec(𝐹, 𝐼) in df-rdg 8351, seq_ω(𝐹, 𝐼) in df-seqom 8389, and seq𝑀( + , 𝐹) in df-seq 13937. These have characteristic function 𝐹 and initial value 𝐼. (Σ_g in df-gsum 17374 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 8313, but for the "average user" the most useful one is probably df-seq 13937- 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 17086.
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 ∈ (𝐹‘𝐵) → 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) = ∪ {𝑥 ∣ 𝜑}) $. $}. 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 6484 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 6855 on the argument being in the domain of the function. It is impossible to derive a contradiction from sound definitions (i.e. that pass the definition check), assuming ZFC is consistent, and he doesn't see the point of all the extra busy work and huge increase in set.mm size that would result from restricting *all* definitions. So instead of implementing a complex system to counter a problem that does not appear to occur in practice, we use a significantly simpler set of approaches.
Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof label followed by "lem", and a number if there is more than one. E.g., sbthlem1 9027 is the first lemma for sbth 9037. The comment should begin with "Lemma for", followed by the final proof label, so that it can be suppressed in theorem lists (see the Metamath program "MM> WRITE THEOREM_LIST" command). Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
Limit proof size. It is often preferable to break longer proofs into smaller parts, just as you would do with traditional proofs. One reason is that humans have trouble understanding long proofs. Another reason is that it's generally best to prove reusable results separately, so that others will be able to easily reuse them. Finally, the Metamath program "MM-PA> MINIMIZE__WITH *" command can take much longer with very long proofs. We encourage proofs to be no more than 200 essential steps, and generally no more than 500 essential steps, though these are simply guidelines and not hard-and-fast rules. Much smaller proofs are fine! We also acknowledge that some proofs, especially autogenerated ones, should sometimes not be broken up (e.g., because breaking them up might be useless and inefficient due to many interconnections and reused terms within the proof). In Metamath, breaking up longer proofs is done by creating multiple separate proofs of separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 9027 is the first lemma for sbth 9037.
Proof stubs. It's sometimes useful to record partial proof results, e.g., incomplete proofs or proofs that depend on something else not fully proven. Some systems (like Lean) support a "sorry" axiom, which lets you assert anything is true, but this can quickly run into trouble, because the Metamath tooling is smart and may end up using it to prove everything. If you want to create a proof based on some other claim, without proving that claim, you can choose to define the claim as an axiom. If you temporarily define a claim as an axiom, we encourage you to include "Temporarily provided as axiom" in its comment. Such incomplete work will generally only be accepted in a mathbox until the rest of the work is complete. When you're working on your personal copy of the database you can use "?" in proofs to indicate an unknown step. However, since proofs with "?" will (obviously) fail verification, we don't accept proofs with unknown steps in the public database.
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 16611"). When the Metamath program is used to generate HTML, it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertions.
Hypertext links to section headers. Some section headers have text under them that describes or explains the section. However, they are not part of the description of axioms or theorems, and there is no way to link to them directly. To provide for this, section headers with accompanying text (indicated with "*" prefixed to mmtheorems.html#mmdtoc 16611 entries) have an anchor in mmtheorems.html 16611 whose name is the first $a or $p statement that follows the header. For example there is a glossary under the section heading called GRAPH THEORY. The first $a or $p statement that follows is cedgf 29073. To reference it we link to the anchor using a space-separated tilde followed by the space-separated link mmtheorems.html#cedgf, which will become the hyperlink mmtheorems.html#cedgf 29073. Note that no theorem in set.mm is allowed to begin with "mm" (this is enforced by the Metamath program "MM> VERIFY MARKUP" command). Whenever the program sees a tilde reference beginning with "http:", "https:", or "mm", the reference is assumed to be a link to something other than a statement label, and the tilde reference is used as is. This can also be useful for relative links to other pages such as mmcomplex.html 29073.
Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g., mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we _require_ page numbers in comments for individual $a or $p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the Metamath program "MM> WRITE BIBLIOGRAPHY" command.)
Acceptable shorter proofs. Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem description. However, in some cases a proof may be "shorter" or not depending on how it is formatted. This section provides general guidelines.
Usually we automatically accept shorter proofs that (1) shorten the set.mm file (with compressed proofs), (2) reduce the size of the HTML file generated with SHOW STATEMENT xx / HTML, (3) use only existing, unmodified theorems in the database (the order of theorems may be changed, though), and (4) use no additional axioms. Usually we will also automatically accept a _new_ theorem that is used to shorten multiple proofs, if the total size of set.mm (including the comment of the new theorem, not including the acknowledgment) decreases as a result.
In borderline cases, we typically place more importance on the number of compressed proof steps and less on the length of the label section (since the names are in principle arbitrary). If two proofs have the same number of compressed proof steps, we will typically give preference to the one with the smaller number of different labels, or if these numbers are the same, the proof with the smaller number of symbols in the proof display on an HTML page containing the theorem. If the difference in size is so insignificant that it is hardly detectable to a human reader, we prefer to keep the older proof, in honor of the first author coming up with a short proof. The community may decide to override these rules after a discussion, if non-technical reasons like aesthetics prefer a particular version.
Some theorems have a longer proof than necessary in order to avoid the use of certain axioms. To indicate this, a $j usage comment should be used just after the proof, for example $( $j usage '19.21v' avoids 'ax-6' 'ax-7' 'ax-12'; $).

Other theorems have a longer proof than necessary for pedagogical or other reasons. These theorems will (or should) have a "(Proof modification is discouraged.)" tag in their description. For example, idALT 23 shows a proof directly from axioms. Shorter proofs for such cases won't be accepted, of course, unless the criteria described continues to be satisfied.
Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see mmdefinitions.html 23. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.
Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.

The challenge of varying mathematical conventions

We try to follow mathematical conventions, but in many cases different texts use different conventions. In those cases we pick some reasonably common convention and stick to it. We have already mentioned that the term "natural number" has varying definitions (some start from 0, others start from 1), but that is not the only such case. A useful example is the set of metavariables used to represent arbitrary well-formed formulas (wffs). We use an open phi, φ, to represent the first arbitrary wff in an assertion with one or more wffs; this is a common convention and this symbol is easily distinguished from the empty set symbol. That said, it is impossible to please everyone or simply "follow the literature" because there are many different conventions for a variable that represents any arbitrary wff. To demonstrate the point, here are some conventions for variables that represent an arbitrary wff and some texts that use each convention:

open phi φ (and so on): Tarski's papers, Rasiowa & Sikorski's The Mathematics of Metamathematics (1963), Monk's Introduction to Set Theory (1969), Enderton's Elements of Set Theory (1977), Bell & Machover's A Course in Mathematical Logic (1977), Jech's Set Theory (1978), Takeuti & Zaring's Introduction to Axiomatic Set Theory (1982).
closed phi ϕ (and so on): Levy's Basic Set Theory (1979), Kunen's Set Theory (1980), Paulson's Isabelle: A Generic Theorem Prover (1994), Huth and Ryan's Logic in Computer Science (2004/2006).
Greek α, β, γ: Duffy's Principles of Automated Theorem Proving (1991).
Roman A, B, C: Kleene's Introduction to Metamathematics (1974), Smullyan's First-Order Logic (1968/1995).
script A, B, C: Hamilton's Logic for Mathematicians (1988).
italic A, B, C: Mendelson's Introduction to Mathematical Logic (1997).
italic P, Q, R: Suppes's Axiomatic Set Theory (1972), Gries and Schneider's A Logical Approach to Discrete Math (1993/1994), Rosser's Logic for Mathematicians (2008).
italic p, q, r: Quine's Set Theory and Its Logic (1969), Kuratowski & Mostowski's Set Theory (1976).
italic X, Y, Z: Dijkstra and Scholten's Predicate Calculus and Program Semantics (1990).
Fraktur letters: Fraenkel et. al's Foundations of Set Theory (1973).

Distinctness or freeness

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

Ⅎ𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑"; see df-nf 1786 (whose description has some important technical details). Similarly, Ⅎ𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2886.
"$d 𝑥𝑦 $." should be read "Assume 𝑥 and 𝑦 are distinct variables."
"$d 𝜑𝑥 $." should be read "Assume 𝑥 does not occur in ϕ." Sometimes a theorem is proved using Ⅎ𝑥𝜑 (df-nf 1786) in place of "$d 𝜑𝑥 $." when a more general result is desired; ax-5 1912 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2577 from eu6 2575.
"$d 𝐴𝑥 $." should be read "Assume 𝑥 is not a variable occurring in class 𝐴."
"$d 𝐴𝑥 $. $d 𝜓𝑥 $. $e |- (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) $." is an idiom often used instead of explicit substitution, meaning "Assume ψ results from the proper substitution of 𝐴 for 𝑥 in ϕ." Therefore, we often use the term "implicit substitution" for such a hypothesis.
Class and wff variables should appear at the beginning of distinct variable conditions, and setvars should be in alphabetical order. E.g., "$d 𝑍𝑥𝑦 $.", "$d 𝜓𝑎𝑥 $.". This convention should be applied for new theorems (formerly, the class and wff variables mostly appear at the end) and will be assured by a formatter in the future.
" ⊢ (¬ ∀𝑥𝑥 = 𝑦 → ...)" occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the $d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the $d eventually becomes a requirement, and after that this device is rarely used.

There is a general technique to replace a $d x A or $d x ph condition in a theorem with the corresponding Ⅎ𝑥𝐴 or Ⅎ𝑥𝜑; here it is. ⊢ T[x, A] where $d 𝑥𝐴, and you wish to prove ⊢ Ⅎ𝑥𝐴 ⇒ ⊢ 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 231 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 2403. The section mmtheorems32.html#mm3146s 2403 also describes the metatheorem that underlies this.

Additional rules for definitions

Standard Metamath verifiers do not distinguish between axioms and definitions (both are $a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions ($a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 207, df-clab 2716, df-cleq 2729, and df-clel 2812); 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):

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 1086 or wsb 2068).
The defined expression must not appear in any statement between its syntax axiom ($a wff ) and its definition, and the defined expression must not be used in its definiens. See df-3an 1089 for an example where the same symbol is used in different ways (this is allowed).
No two variables occurring in the LHS may share a disjoint variable (DV) condition.
All dummy variables are required to be disjoint from any other (dummy or not) variable occurring in this labeled expression.
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.
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 nonfreeness; one must succeed for each setvar dummy variable 𝑥. The first test requires that the setvar dummy variable 𝑥 be syntactically bound (this is sometimes called the "fast" test, and this implies that we must track binding operators). The second test requires a successful search for the directly-stated proof of (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 3443. Definition df-v 3444 ⊢ V = {𝑥 ∣ 𝑥 = 𝑥} is justified by the justification theorem vjust 3443 ⊢ {𝑥 ∣ 𝑥 = 𝑥} = {𝑦 ∣ 𝑦 = 𝑦}. Another example of a justification theorem is trujust 1544; Definition df-tru 1545 ⊢ (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1544 ⊢ ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).

Here is more information about our processes for checking and contributing to this work:

Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
Discouraged information. A separate file named "discouraged" lists all discouraged statements and uses of them, and this file is checked. If you change the use of discouraged things, you will need to change this file. This makes it obvious when there is a change to anything discouraged (triggering further review).
LRParser check. Metamath verifiers ensure that $p statements follow from previous $a and $p statements. However, by itself the Metamath language permits certain kinds of syntactic ambiguity that we choose to avoid in this database. Thus, we require that this database unambiguously parse using the "LRParser" check (implemented by at least mmj2). (For details, see mmj2 master file src/mmj/verify/LRParser.java). This check counters, for example, a devious ambiguous construct developed by saueran at oregonstate dot edu posted on Mon, 11 Feb 2019 17:32:32 -0800 (PST) based on creating definitions with mismatched parentheses.
Proposing specific changes. Please propose specific changes as pull requests (PRs) against the "develop" branch of set.mm, at: https://github.com/metamath/set.mm/tree/develop 1544.
Community. We encourage anyone interested in Metamath to join our mailing list: https://groups.google.com/g/metamath 1544.

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

𝜑 ⇒ 𝜑

Theorem

conventions-labels 30488

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 30487 and links therein.

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

Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id (these ids are usually consecutive integers starting with 1, e.g., for rgen 3054"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g., for mdet0 22562: "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 2664 and stirling 46441.
Principia Mathematica. Proofs of theorems from Principia Mathematica often use a special naming convention: "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 42.
19.x series of theorems. Similar to the conventions for the theorems from Principia Mathematica, theorems from Section 19 of [Margaris] p. 90 often use a special naming convention: "19." resp. "r19." (for corresponding restricted quantifier versions) followed by its identifier. For example, Theorem 38 from Section 19 of [Margaris] p. 90 is labeled 19.38 1841, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 3233.
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... 15816. Furthermore, the underscore "_" should not be used. Finally, only lower case characters should be used (except the special suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at least in main set.mm (exceptions are tolerated in mathboxes).
Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct (𝐴 ∖ 𝐵) is defined in df-dif 3906, and thus its syntax label fragment is "dif". Similarly, the subclass relation 𝐴 ⊆ 𝐵 has syntax label fragment "ss" because it is defined in df-ss 3920. Most theorem names follow from these fragments, for example, the theorem proving (𝐴 ∖ 𝐵) ⊆ 𝐴 involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 4090. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4583), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4585). Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct 𝐴 ∈ 𝐵 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 4744. An "n" is often used for negation (¬), e.g., nan 830.
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 ℂ (even though its definition is in df-c 11044) and "re" represents real numbers ℝ (Definition df-r 11048). The empty set ∅ often uses fragment 0, even though it is defined in df-nul 4288. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 11049), 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 12287.
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 16087 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 labeled "NAMEcl". E.g., for cosine (df-cos 16005) we have value cosval 16060 and closure coscl 16064.
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 30490 for a list), and that is about all you need for most things. As for the rest, you can just assume that if it involves at most three connectives, then it is probably already proved in set.mm, and searching for it will give you the label.
Suffixes. Suffixes are used to indicate the form of a theorem (inference, deduction, or closed form, see above). Additionally, we sometimes suffix with "v" the label of a theorem adding a disjoint variable condition, as in 19.21v 1941 versus 19.21 2215. This often permits to prove the result using fewer axioms, and/or to eliminate a nonfreeness hypothesis (such as Ⅎ𝑥𝜑 in 19.21 2215). If no constraint is put on axiom use, then the v-version can be proved from the original theorem using nfv 1916. If two (resp. three) such disjoint variable conditions are added, then the suffix "vv" (resp. "vvv") is used, e.g., exlimivv 1934. 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 2577 derived from eu6 2575. 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 5404. 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 2414 (cbval 2403 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). As a general rule, the suffixes for the theorem forms ("i", "d" or "g") should be the first of multiple suffixes, as for example in vtocldf 3519. Here is a non-exhaustive list of common suffixes:
- a : theorem having a conjunction as antecedent
- b : theorem expressing a logical equivalence
- c : contraction (e.g., sylc 65, syl2anc 585), commutes (e.g., biimpac 478)
- d : theorem in deduction form
- f : theorem with a hypothesis such as Ⅎ𝑥𝜑
- g : theorem in closed form having an "is a set" antecedent
- i : theorem in inference form
- l : theorem concerning something at the left
- r : theorem concerning something at the right
- r : theorem with something reversed (e.g., a biconditional)
- s : inference that manipulates an antecedent ("s" refers to an application of syl 17 that is eliminated)
- t : theorem in closed form (not having an "is a set" antecedent)
- v : theorem with one (main) disjoint variable condition
- vv : theorem with two (main) disjoint variable conditions
- w : weak(er) form of a theorem
- ALT : alternate proof of a theorem
- ALTV : alternate version of a theorem or definition (mathbox only)
- OLD : old/obsolete version of a theorem (or proof) or definition
Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

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

Abbreviation	Mnenomic	Source	Expression	Syntax?	Example(s)
a	and (suffix)			No	biimpa 476, rexlimiva 3131
abl	Abelian group	df-abl 19724	Abel	Yes	ablgrp 19726, zringabl 21418
abs	absorption			No	ressabs 17187
abs	absolute value (of a complex number)	df-abs 15171	(abs‘𝐴)	Yes	absval 15173, absneg 15212, abs1 15232
ad	adding			No	adantr 480, ad2antlr 728
add	add (see "p")	df-add 11049	(𝐴 + 𝐵)	Yes	addcl 11120, addcom 11331, addass 11125
al	"for all"		∀𝑥𝜑	No	alim 1812, alex 1828
ALT	alternative/less preferred (suffix)			No	idALT 23
an	and	df-an 396	(𝜑 ∧ 𝜓)	Yes	anor 985, iman 401, imnan 399
ant	antecedent			No	adantr 480
ass	associative			No	biass 384, orass 922, mulass 11126
asym	asymmetric, antisymmetric			No	intasym 6080, asymref 6081, posasymb 18254
ax	axiom			No	ax6dgen 2134, ax1cn 11072
bas, base	base (set of an extensible structure)	df-base 17149	(Base‘𝑆)	Yes	baseval 17150, ressbas 17175, cnfldbas 21325
b, bi	biconditional ("iff", "if and only if")	df-bi 207	(𝜑 ↔ 𝜓)	Yes	impbid 212, sspwb 5404
br	binary relation	df-br 5101	𝐴𝑅𝐵	Yes	brab1 5148, brun 5151
c	commutes, commuted (suffix)			No	biimpac 478
c	contraction (suffix)			No	sylc 65, syl2anc 585
cbv	change bound variable			No	cbvalivw 2009, cbvrex 3335
cdm	codomain			No	ffvelcdm 7035, focdmex 7910
cl	closure			No	ifclda 4517, ovrcl 7409, zaddcl 12543
cn	complex numbers	df-c 11044	ℂ	Yes	nnsscn 12162, nncn 12165
cnfld	field of complex numbers	df-cnfld 21322	ℂ_fld	Yes	cnfldbas 21325, cnfldinv 21369
cntz	centralizer	df-cntz 19258	(Cntz‘𝑀)	Yes	cntzfval 19261, dprdfcntz 19958
cnv	converse	df-cnv 5640	^◡𝐴	Yes	opelcnvg 5837, f1ocnv 6794
co	composition	df-co 5641	(𝐴 ∘ 𝐵)	Yes	cnvco 5842, fmptco 7084
com	commutative			No	orcom 871, bicomi 224, eqcomi 2746
con	contradiction, contraposition			No	condan 818, con2d 134
csb	class substitution	df-csb 3852	⦋𝐴 / 𝑥⦌𝐵	Yes	csbid 3864, csbie2g 3891
cyg	cyclic group	df-cyg 19819	CycGrp	Yes	iscyg 19820, zringcyg 21436
d	deduction form (suffix)			No	idd 24, impbid 212
df	(alternate) definition (prefix)			No	dfrel2 6155, dffn2 6672
di, distr	distributive			No	andi 1010, imdi 389, ordi 1008, difindi 4246, ndmovdistr 7557
dif	class difference	df-dif 3906	(𝐴 ∖ 𝐵)	Yes	difss 4090, difindi 4246
div	division	df-div 11807	(𝐴 / 𝐵)	Yes	divcl 11814, divval 11810, divmul 11811
dm	domain	df-dm 5642	dom 𝐴	Yes	dmmpt 6206, iswrddm0 14473
e, eq, equ	equals (equ for setvars, eq for classes)	df-cleq 2729	𝐴 = 𝐵	Yes	2p2e4 12287, uneqri 4110, equtr 2023
edg	edge	df-edg 29133	(Edg‘𝐺)	Yes	edgopval 29136, usgredgppr 29281
el	element of		𝐴 ∈ 𝐵	Yes	eldif 3913, eldifsn 4744, elssuni 4896
en	equinumerous	df-en	𝐴 ≈ 𝐵	Yes	domen 8910, enfi 9123
eu	"there exists exactly one"	eu6 2575	∃!𝑥𝜑	Yes	euex 2578, euabsn 4685
ex	exists (i.e. is a set)		∈ V	No	brrelex1 5685, 0ex 5254
ex, e	"there exists (at least one)"	df-ex 1782	∃𝑥𝜑	Yes	exim 1836, alex 1828
exp	export			No	expt 177, expcom 413
f	"not free in" (suffix)			No	equs45f 2464, sbf 2278
f	function	df-f 6504	𝐹:𝐴⟶𝐵	Yes	fssxp 6697, opelf 6703
fal	false	df-fal 1555	⊥	Yes	bifal 1558, falantru 1577
fi	finite intersection	df-fi 9326	(fi‘𝐵)	Yes	fival 9327, inelfi 9333
fi, fin	finite	df-fin 8899	Fin	Yes	isfi 8924, snfi 8992, onfin 9151
fld	field (Note: there is an alternative definition Fld of a field, see df-fld 38237)	df-field 20677	Field	Yes	isfld 20685, fldidom 20716
fn	function with domain	df-fn 6503	𝐴 Fn 𝐵	Yes	ffn 6670, fndm 6603
frgp	free group	df-frgp 19651	(freeGrp‘𝐼)	Yes	frgpval 19699, frgpadd 19704
fsupp	finitely supported function	df-fsupp 9277	𝑅 finSupp 𝑍	Yes	isfsupp 9280, fdmfisuppfi 9289, fsuppco 9317
fun	function	df-fun 6502	Fun 𝐹	Yes	funrel 6517, ffun 6673
fv	function value	df-fv 6508	(𝐹‘𝐴)	Yes	fvres 6861, swrdfv 14584
fz	finite set of sequential integers	df-fz 13436	(𝑀...𝑁)	Yes	fzval 13437, eluzfz 13447
fz0	finite set of sequential nonnegative integers		(0...𝑁)	Yes	nn0fz0 13553, fz0tp 13556
fzo	half-open integer range	df-fzo 13583	(𝑀..^𝑁)	Yes	elfzo 13589, elfzofz 13603
g	more general (suffix); eliminates "is a set" hypotheses			No	uniexg 7695
gr	graph			No	uhgrf 29147, isumgr 29180, usgrres1 29400
grp	group	df-grp 18878	Grp	Yes	isgrp 18881, tgpgrp 24034
gsum	group sum	df-gsum 17374	(𝐺 Σ_g 𝐹)	Yes	gsumval 18614, gsumwrev 19307
hash	size (of a set)	df-hash 14266	(♯‘𝐴)	Yes	hashgval 14268, hashfz1 14281, hashcl 14291
hb	hypothesis builder (prefix)			No	hbxfrbi 1827, hbald 2174, hbequid 39279
hm	(monoid, group, ring, ...) homomorphism			No	ismhm 18722, isghm 19156, isrhm 20426
i	inference (suffix)			No	eleq1i 2828, tcsni 9662
i	implication (suffix)			No	brwdomi 9485, infeq5i 9557
id	identity			No	biid 261
iedg	indexed edge	df-iedg 29084	(iEdg‘𝐺)	Yes	iedgval0 29125, edgiedgb 29139
idm	idempotent			No	anidm 564, tpidm13 4715
im, imp	implication (label often omitted)	df-im 15036	(𝐴 → 𝐵)	Yes	iman 401, imnan 399, impbidd 210
im	(group, ring, ...) isomorphism			No	isgim 19203, rimrcl 20429
ima	image	df-ima 5645	(𝐴 “ 𝐵)	Yes	resima 5982, imaundi 6115
imp	import			No	biimpa 476, impcom 407
in	intersection	df-in 3910	(𝐴 ∩ 𝐵)	Yes	elin 3919, incom 4163
inf	infimum	df-inf 9358	inf(ℝ⁺, ℝ^*, < )	Yes	fiinfcl 9418, infiso 9425
is...	is (something a) ...?			No	isring 20184
j	joining, disjoining			No	jc 161, jaoi 858
l	left			No	olcd 875, simpl 482
map	mapping operation or set exponentiation	df-map 8777	(𝐴 ↑_m 𝐵)	Yes	mapvalg 8785, elmapex 8797
mat	matrix	df-mat 22364	(𝑁 Mat 𝑅)	Yes	matval 22367, matring 22399
mdet	determinant (of a square matrix)	df-mdet 22541	(𝑁 maDet 𝑅)	Yes	mdetleib 22543, mdetrlin 22558
mgm	magma	df-mgm 18577	Magma	Yes	mgmidmo 18597, mgmlrid 18604, ismgm 18578
mgp	multiplicative group	df-mgp 20088	(mulGrp‘𝑅)	Yes	mgpress 20097, ringmgp 20186
mnd	monoid	df-mnd 18672	Mnd	Yes	mndass 18680, mndodcong 19483
mo	"there exists at most one"	df-mo 2540	∃*𝑥𝜑	Yes	eumo 2579, moim 2545
mp	modus ponens	ax-mp 5		No	mpd 15, mpi 20
mpo	maps-to notation for an operation	df-mpo 7373	(𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)	Yes	mpompt 7482, resmpo 7488
mpt	modus ponendo tollens			No	mptnan 1770, mptxor 1771
mpt	maps-to notation for a function	df-mpt 5182	(𝑥 ∈ 𝐴 ↦ 𝐵)	Yes	fconstmpt 5694, resmpt 6004
mul	multiplication (see "t")	df-mul 11050	(𝐴 · 𝐵)	Yes	mulcl 11122, divmul 11811, mulcom 11124, mulass 11126
n, not	not		¬ 𝜑	Yes	nan 830, notnotr 130
ne	not equal	df-ne	𝐴 ≠ 𝐵	Yes	exmidne 2943, neeqtrd 3002
nel	not element of	df-nel	𝐴 ∉ 𝐵	Yes	neli 3039, nnel 3047
ne0	not equal to zero (see n0)		≠ 0	No	negne0d 11502, ine0 11584, gt0ne0 11614
nf	"not free in" (prefix)	df-nf 1786	Ⅎ𝑥𝜑	Yes	nfnd 1860
ngp	normed group	df-ngp 24539	NrmGrp	Yes	isngp 24552, ngptps 24558
nm	norm (on a group or ring)	df-nm 24538	(norm‘𝑊)	Yes	nmval 24545, subgnm 24589
nn	positive integers	df-nn 12158	ℕ	Yes	nnsscn 12162, nncn 12165
nn0	nonnegative integers	df-n0 12414	ℕ₀	Yes	nnnn0 12420, nn0cn 12423
n0	not the empty set (see ne0)		≠ ∅	No	n0i 4294, vn0 4299, ssn0 4358
OLD	old, obsolete (to be removed soon)			No	19.43OLD 1885
on	ordinal number	df-on 6329	𝐴 ∈ On	Yes	elon 6334, 1on 8419 onelon 6350
op	ordered pair	df-op 4589	⟨𝐴, 𝐵⟩	Yes	dfopif 4828, opth 5432
or	or	df-or 849	(𝜑 ∨ 𝜓)	Yes	orcom 871, anor 985
ot	ordered triple	df-ot 4591	⟨𝐴, 𝐵, 𝐶⟩	Yes	euotd 5469, fnotovb 7420
ov	operation value	df-ov 7371	(𝐴𝐹𝐵)	Yes	fnotovb 7420, fnovrn 7543
p	plus (see "add"), for all-constant theorems	df-add 11049	(3 + 2) = 5	Yes	3p2e5 12303
pfx	prefix	df-pfx 14607	(𝑊 prefix 𝐿)	Yes	pfxlen 14619, ccatpfx 14636
pm	Principia Mathematica			No	pm2.27 42
pm	partial mapping (operation)	df-pm 8778	(𝐴 ↑_pm 𝐵)	Yes	elpmi 8795, pmsspw 8827
pr	pair	df-pr 4585	{𝐴, 𝐵}	Yes	elpr 4607, prcom 4691, prid1g 4719, prnz 4736
prm, prime	prime (number)	df-prm 16611	ℙ	Yes	1nprm 16618, dvdsprime 16626
pss	proper subset	df-pss 3923	𝐴 ⊊ 𝐵	Yes	pssss 4052, sspsstri 4059
q	rational numbers ("quotients")	df-q 12874	ℚ	Yes	elq 12875
r	reversed (suffix)			No	pm4.71r 558, caovdir 7602
r	right			No	orcd 874, simprl 771
rab	restricted class abstraction	df-rab 3402	{𝑥 ∈ 𝐴 ∣ 𝜑}	Yes	rabswap 3410, df-oprab 7372
ral	restricted universal quantification	df-ral 3053	∀𝑥 ∈ 𝐴𝜑	Yes	ralnex 3064, ralrnmpo 7507
rcl	reverse closure			No	ndmfvrcl 6875, nnarcl 8554
re	real numbers	df-r 11048	ℝ	Yes	recn 11128, 0re 11146
rel	relation	df-rel 5639	Rel 𝐴	Yes	brrelex1 5685, relmpoopab 8046
res	restriction	df-res 5644	(𝐴 ↾ 𝐵)	Yes	opelres 5952, f1ores 6796
reu	restricted existential uniqueness	df-reu 3353	∃!𝑥 ∈ 𝐴𝜑	Yes	nfreud 3398, reurex 3356
rex	restricted existential quantification	df-rex 3063	∃𝑥 ∈ 𝐴𝜑	Yes	rexnal 3090, rexrnmpo 7508
rmo	restricted "at most one"	df-rmo 3352	∃*𝑥 ∈ 𝐴𝜑	Yes	nfrmod 3397, nrexrmo 3371
rn	range	df-rn 5643	ran 𝐴	Yes	elrng 5848, rncnvcnv 5891
ring	(unital) ring	df-ring 20182	Ring	Yes	ringidval 20130, isring 20184, ringgrp 20185
rng	non-unital ring	df-rng 20100	Rng	Yes	isrng 20101, rngabl 20102, rnglz 20112
rot	rotation			No	3anrot 1100, 3orrot 1092
s	eliminates need for syllogism (suffix)			No	ancoms 458
sb	(proper) substitution (of a set)	df-sb 2069	[𝑦 / 𝑥]𝜑	Yes	spsbe 2088, sbimi 2080
sbc	(proper) substitution of a class	df-sbc 3743	[𝐴 / 𝑥]𝜑	Yes	sbc2or 3751, sbcth 3757
sca	scalar	df-sca 17205	(Scalar‘𝐻)	Yes	resssca 17275, mgpsca 20093
simp	simple, simplification			No	simpl 482, simp3r3 1285
sn	singleton	df-sn 4583	{𝐴}	Yes	eldifsn 4744
sp	specialization			No	spsbe 2088, spei 2399
ss	subset	df-ss 3920	𝐴 ⊆ 𝐵	Yes	difss 4090
struct	structure	df-struct 17086	Struct	Yes	brstruct 17087, structfn 17095
sub	subtract	df-sub 11378	(𝐴 − 𝐵)	Yes	subval 11383, subaddi 11480
sup	supremum	df-sup 9357	sup(𝐴, 𝐵, < )	Yes	fisupcl 9385, supmo 9367
supp	support (of a function)	df-supp 8113	(𝐹 supp 𝑍)	Yes	ressuppfi 9310, mptsuppd 8139
swap	swap (two parts within a theorem)			No	rabswap 3410, 2reuswap 3706
syl	syllogism	syl 17		No	3syl 18
sym	symmetric			No	df-symdif 4207, cnvsym 6079
symg	symmetric group	df-symg 19311	(SymGrp‘𝐴)	Yes	symghash 19319, pgrpsubgsymg 19350
t	times (see "mul"), for all-constant theorems	df-mul 11050	(3 · 2) = 6	Yes	3t2e6 12318
th, t	theorem			No	nfth 1803, sbcth 3757, weth 10417, ancomst 464
tp	triple	df-tp 4587	{𝐴, 𝐵, 𝐶}	Yes	eltpi 4647, tpeq1 4701
tr	transitive			No	bitrd 279, biantr 806
tru, t	true, truth	df-tru 1545	⊤	Yes	bitru 1551, truanfal 1576, biimt 360
un	union	df-un 3908	(𝐴 ∪ 𝐵)	Yes	uneqri 4110, uncom 4112
unit	unit (in a ring)	df-unit 20306	(Unit‘𝑅)	Yes	isunit 20321, nzrunit 20469
v	setvar (especially for specializations of theorems when a class is replaced by a setvar variable)		x	Yes	cv 1541, vex 3446, velpw 4561, vtoclf 3523
v	disjoint variable condition used in place of nonfreeness hypothesis (suffix)			No	spimv 2395
vtx	vertex	df-vtx 29083	(Vtx‘𝐺)	Yes	vtxval0 29124, opvtxov 29090
vv	two disjoint variable conditions used in place of nonfreeness hypotheses (suffix)			No	19.23vv 1945
w	weak (version of a theorem) (suffix)			No	ax11w 2136, spnfw 1981
wrd	word	df-word 14449	Word 𝑆	Yes	iswrdb 14455, wrdfn 14463, ffz0iswrd 14476
xp	cross product (Cartesian product)	df-xp 5638	(𝐴 × 𝐵)	Yes	elxp 5655, opelxpi 5669, xpundi 5701
xr	eXtended reals	df-xr 11182	ℝ^*	Yes	ressxr 11188, rexr 11190, 0xr 11191
z	integers (from German "Zahlen")	df-z 12501	ℤ	Yes	elz 12502, zcn 12505
zn	ring of integers mod 𝑁	df-zn 21473	(ℤ/nℤ‘𝑁)	Yes	znval 21502, zncrng 21511, znhash 21525
zring	ring of integers	df-zring 21414	ℤ_ring	Yes	zringbas 21420, zringcrng 21415
0, z	slashed zero (empty set)	df-nul 4288	∅	Yes	n0i 4294, vn0 4299; snnz 4735, prnz 4736

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

𝜑 ⇒ 𝜑

Theorem

conventions-comments 30489

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding comments, and more generally nonmathematical conventions. For other conventions, see conventions 30487 and links therein.

Input format.
The input format is ASCII. Tab characters are not allowed. If non-ASCII characters have to be displayed in comments, use embedded mathematical symbols when they have been defined (e.g., "` -> `" for " →") or HTML entities (e.g., "é" for "é"). Default indentation is by two spaces. Lines are hard-wrapped to be at most 79-character long, excluding the newline character (this can be achieved, except currently for section comments, by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command or by running the script scripts/rewrap). The file ends with an empty line. There are no trailing spaces. As for line wrapping in statements, we try to break lines before the most important token.
Language and spelling.
The MPE uses American English, e.g., we write "neighborhood" instead of the British English "neighbourhood". An exception is the word "analog", which can be either a noun or an adjective (furthermore, "analog" has the confounding meaning "not digital"); therefore, "analogue" is used for the noun and "analogous" for the adjective. We favor regular plurals, e.g., "formulas" instead of "formulae", "lemmas" instead of "lemmata". We use the serial comma (Oxford comma) in enumerations. We use commas after "i.e." and "e.g.".

We avoid beginning a sentence with a symbol (for instance, by writing "The function F is ..." instead of "F is...").

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

When compound words have several variants, we prefer the concatenated variant (e.g., nonempty, nontrivial, nonpositive, nonzero, nonincreasing, nondegenerate...).
Quotation style.
We use the "logical quotation style", which means that when a quoted text is followed by punctuation not pertaining to the quote, then the quotation mark precedes the punctuation (like at the beginning of this sentence). We use the double quote as default quotation mark (since the single quote also serves as apostrophe), and the single quote in the case of a nested quotation.
Sectioning and section headers.
The database set.mm has a sectioning system with four levels of titles, signaled by "decoration lines" which are 79-character long repetitions of ####, #*#*, =-=-, and -.-. (in descending order of sectioning level). Sections of any level are separated by two blank lines (if there is a "@( Begin $[ ... $] @)" comment (where "@" is actually "$") before a section header, then the double blank line should go before that comment, which is considered as belonging to that section). The format of section headers is best seen in the source file (set.mm); it is as follows:
- a line with "@(" (with the "@" replaced by "$");
- a decoration line;
- section title indented with two spaces;
- a (matching) decoration line;
- [blank line; header comment indented with two spaces; blank line;]
- a line with "@)" (with the "@" replaced by "$");
- one blank line.
As everywhere else, lines are hard-wrapped to be 79-character long. It is expected that in a future version, the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command will reformat section headers to automatically conform with this format.
Comments.
As for formatting of the file set.mm, and in particular formatting and layout of the comments, the foremost rule is consistency. The first sections of set.mm, in particular Part 1 "Classical first-order logic with equality" can serve as a model for contributors. Some formatting rules are enforced when using the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command. Here are a few other rules, which are not enforced, but that we try to follow:
- A math string in a comment should be surrounded by space-separated backquotes on the same line, and if it is too long it should be broken into multiple adjacent math strings on multiple lines.
- The file set.mm should have a double blank line between sections, and at no other places. In particular, there are no triple blank lines.
- The header comments should be spaced as those of Part 1, namely, with a blank line before and after the comment, and an indentation of two spaces.
- As of 20-Sep-2022, section comments are not rewrapped by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command, though this is expected in a future version. Similar spacing and wrapping should be used as for other comments: double spaces after a period ending a sentence, line wrapping with line width of 79, and no trailing spaces at the end of lines.
Contributors.
Each assertion (theorem, definition or axiom) has a contribution tag of the form "(Contributed by xxx, dd-Mmm-yyyy.)" (see Metamath Book, p. 142). The date cannot serve as a proof of anteriority since there is currently no formal guarantee that the date is correct (a claim of anteriority can be backed, for instance, by the uploading of a result to a public repository with verifiable date). The contributor is the first person who proved (or stated, in the case of a definition or axiom) the statement. The list of contributors appears at the beginning of set.mm.

An exception should be made if a theorem is essentially an extract or a variant of an already existing theorem, in which case the contributor should be that of the statement from which it is derived, with the modification signaled by a "(Revised by xxx, dd-Mmm-yyyy.)" tag.
Usage of parentheticals.
Usually, the comment of a theorem should contain at most one of the "Revised by" and "Proof shortened by" parentheticals, see Metamath Book, pp. 142-143 (there must always be a "Contributed by" parenthetical for every theorem). Exceptions for "Proof shortened by" parentheticals are essential additional shortenings by a different person. If a proof is shortened by the same person, the date within the "Proof shortened by" parenthetical should be updated only. This also holds for "Revised by" parentheticals, except that also more than one of such parentheticals for the same person are acceptable (if there are good reasons for this). A revision tag is optionally preceded by a short description of the revision. Since this is somewhat subjective, judgment and intellectual honesty should be applied, with collegial settlement in case of dispute.
Explaining new labels.
A comment should explain the first use of an abbreviation within a label. This is often in a definition (e.g., Definition df-an 396 introduces the abbreviation "an" for conjunction ("and")), but not always (e.g., Theorem alim 1812 introduces the abbreviation "al" for the universal quantifier ("for all")). See conventions-labels 30488 for a table of abbreviations.

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

𝜑 ⇒ 𝜑

18.1.2 Natural deduction

Theorem

natded 30490

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

Name	Natural Deduction Rule	Translation	Recommendation	Comments
IT	Γ⊢ 𝜓 => Γ⊢ 𝜓	idi 1	nothing	Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
∧I	Γ⊢ 𝜓 & Γ⊢ 𝜒 => Γ⊢ 𝜓 ∧ 𝜒	jca 511	jca 511, pm3.2i 470	Definition ∧I in [Pfenning] p. 18, definition I∧m,n in [Clemente] p. 10, and definition ∧I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
∧E_L	Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜓	simpld 494	simpld 494, adantr 480	Definition ∧E_L 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)
∧E_R	Γ⊢ 𝜓 ∧ 𝜒 => Γ⊢ 𝜒	simprd 495	simpr 484, adantl 481	Definition ∧E_R 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, 𝜓⊢ 𝜒 => Γ⊢ 𝜓 → 𝜒	ex 412	ex 412	Definition →I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition →I in [Indrzejczak] p. 33.
→E	Γ⊢ 𝜓 → 𝜒 & Γ⊢ 𝜓 => Γ⊢ 𝜒	mpd 15	ax-mp 5, mpd 15, mpdan 688, imp 406	Definition →E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition →E in [Indrzejczak] p. 33.
∨I_L	Γ⊢ 𝜓 => Γ⊢ 𝜓 ∨ 𝜒	olcd 875	olc 869, olci 867, olcd 875	Definition ∨I in [Pfenning] p. 18, definition I∨n(1) in [Clemente] p. 12
∨I_R	Γ⊢ 𝜒 => Γ⊢ 𝜓 ∨ 𝜒	orcd 874	orc 868, orci 866, orcd 874	Definition ∨I_R in [Pfenning] p. 18, definition I∨n(2) in [Clemente] p. 12.
∨E	Γ⊢ 𝜓 ∨ 𝜒 & Γ, 𝜓⊢ 𝜃 & Γ, 𝜒⊢ 𝜃 => Γ⊢ 𝜃	mpjaodan 961	mpjaodan 961, jaodan 960, jaod 860	Definition ∨E in [Pfenning] p. 18, definition E∨m,n,p in [Clemente] p. 12.
¬I	Γ, 𝜓⊢ ⊥ => Γ⊢ ¬ 𝜓	inegd 1562	pm2.01d 190
¬I	Γ, 𝜓⊢ 𝜃 & Γ⊢ ¬ 𝜃 => Γ⊢ ¬ 𝜓	mtand 816	mtand 816	definition I¬m,n,p in [Clemente] p. 13.
¬I	Γ, 𝜓⊢ 𝜒 & Γ, 𝜓⊢ ¬ 𝜒 => Γ⊢ ¬ 𝜓	pm2.65da 817	pm2.65da 817	Contradiction.
¬I	Γ, 𝜓⊢ ¬ 𝜓 => Γ⊢ ¬ 𝜓	pm2.01da 799	pm2.01d 190, pm2.65da 817, pm2.65d 196	For an alternative falsum-free natural deduction ruleset
¬E	Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ ⊥	pm2.21fal 1564	pm2.21dd 195
¬E	Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓		pm2.21dd 195	definition →E in [Indrzejczak] p. 33.
¬E	Γ⊢ 𝜓 & Γ⊢ ¬ 𝜓 => Γ⊢ 𝜃	pm2.21dd 195	pm2.21dd 195, pm2.21d 121, pm2.21 123	For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
⊤I	Γ⊢ ⊤	trud 1552	tru 1546, trud 1552, mptru 1549	Definition ⊤I in [Pfenning] p. 18.
⊥E	Γ, ⊥⊢ 𝜃	falimd 1560	falim 1559	Definition ⊥E in [Pfenning] p. 18.
∀I	Γ⊢ [𝑎 / 𝑥]𝜓 => Γ⊢ ∀𝑥𝜓	alrimiv 1929	alrimiv 1929, ralrimiva 3130	Definition ∀I^a in [Pfenning] p. 18, definition I∀n in [Clemente] p. 32.
∀E	Γ⊢ ∀𝑥𝜓 => Γ⊢ [𝑡 / 𝑥]𝜓	spsbcd 3756	spcv 3561, rspcv 3574	Definition ∀E in [Pfenning] p. 18, definition E∀n,t in [Clemente] p. 32.
∃I	Γ⊢ [𝑡 / 𝑥]𝜓 => Γ⊢ ∃𝑥𝜓	spesbcd 3835	spcev 3562, rspcev 3578	Definition ∃I in [Pfenning] p. 18, definition I∃n,t in [Clemente] p. 32.
∃E	Γ⊢ ∃𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓⊢ 𝜃 => Γ⊢ 𝜃	exlimddv 1937	exlimddv 1937, exlimdd 2228, exlimdv 1935, rexlimdva 3139	Definition ∃E^a,u in [Pfenning] p. 18, definition E∃m,n,p,a in [Clemente] p. 32.
⊥C	Γ, ¬ 𝜓⊢ ⊥ => Γ⊢ 𝜓	efald 1563	efald 1563	Proof by contradiction (classical logic), definition ⊥C in [Pfenning] p. 17.
⊥C	Γ, ¬ 𝜓⊢ 𝜓 => Γ⊢ 𝜓	pm2.18da 800	pm2.18da 800, pm2.18d 127, pm2.18 128	For an alternative falsum-free natural deduction ruleset
¬ ¬C	Γ⊢ ¬ ¬ 𝜓 => Γ⊢ 𝜓	notnotrd 133	notnotrd 133, notnotr 130	Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM	Γ⊢ 𝜓 ∨ ¬ 𝜓	exmidd 896	exmid 895	Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I	Γ⊢ 𝐴 = 𝐴	eqidd 2738	eqid 2737, eqidd 2738	Introduce equality, definition =I in [Pfenning] p. 127.
=E	Γ⊢ 𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ⊢ [𝐵 / 𝑥]𝜓	sbceq1dd 3748	sbceq1d 3747, 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 antecedent 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 30491, ex-natded5.3 30494, ex-natded5.5 30497, ex-natded5.7 30498, ex-natded5.8 30500, ex-natded5.13 30502, ex-natded9.20 30504, and ex-natded9.26 30506.

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

𝜑 ⇒ 𝜑

18.1.3 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 30490 and mmnatded.html 30490. Since these examples should not be used within proofs of other theorems, especially in mathboxes, they are marked with "(New usage is discouraged.)".

Theorem

ex-natded5.2 30491

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 Translation	ND Rationale	MPE Rationale
1	5	((𝜓 ∧ 𝜒) → 𝜃)	(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃))	Given	$e.
2	2	(𝜒 → 𝜓)	(𝜑 → (𝜒 → 𝜓))	Given	$e.
3	1	𝜒	(𝜑 → 𝜒)	Given	$e.
4	3	𝜓	(𝜑 → 𝜓)	→E 2,3	mpd 15, the MPE equivalent of →E, 1,2
5	4	(𝜓 ∧ 𝜒)	(𝜑 → (𝜓 ∧ 𝜒))	∧I 4,3	jca 511, the MPE equivalent of ∧I, 3,1
6	6	𝜃	(𝜑 → 𝜃)	→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 30492. A proof without context is shown in ex-natded5.2i 30493. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & (𝜑 → (𝜒 → 𝜓)) & (𝜑 → 𝜒) ⇒ (𝜑 → 𝜃)

Theorem ex-natded5.2-2 30492 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 30491 and ex-natded5.2i 30493. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → ((𝜓 ∧ 𝜒) → 𝜃)) & (𝜑 → (𝜒 → 𝜓)) & (𝜑 → 𝜒) ⇒ (𝜑 → 𝜃)

Theorem ex-natded5.2i 30493 The same as ex-natded5.2 30491 and ex-natded5.2-2 30492 but with no context. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

((𝜓 ∧ 𝜒) → 𝜃) & (𝜒 → 𝜓) & 𝜒 ⇒ 𝜃

Theorem

ex-natded5.3 30494

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 30495. A proof without context is shown in ex-natded5.3i 30496. 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 Translation	ND Rationale	MPE Rationale
1	2;3	(𝜓 → 𝜒)	(𝜑 → (𝜓 → 𝜒))	Given	$e; adantr 480 to move it into the ND hypothesis
2	5;6	(𝜒 → 𝜃)	(𝜑 → (𝜒 → 𝜃))	Given	$e; adantr 480 to move it into the ND hypothesis
3	1	...\| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption	simpr 484, to access the new assumption
4	4	... 𝜒	((𝜑 ∧ 𝜓) → 𝜒)	→E 1,3	mpd 15, the MPE equivalent of →E, 1.3. adantr 480 was used to transform its dependency (we could also use imp 406 to get this directly from 1)
5	7	... 𝜃	((𝜑 ∧ 𝜓) → 𝜃)	→E 2,4	mpd 15, the MPE equivalent of →E, 4,6. adantr 480 was used to transform its dependency
6	8	... (𝜒 ∧ 𝜃)	((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃))	∧I 4,5	jca 511, the MPE equivalent of ∧I, 4,7
7	9	(𝜓 → (𝜒 ∧ 𝜃))	(𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))	→I 3,6	ex 412, the MPE equivalent of →I, 8

(𝜑 → (𝜓 → 𝜒)) & (𝜑 → (𝜒 → 𝜃)) ⇒ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))

Theorem ex-natded5.3-2 30495 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 30494 and ex-natded5.3i 30496. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 → 𝜒)) & (𝜑 → (𝜒 → 𝜃)) ⇒ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃)))

Theorem ex-natded5.3i 30496 The same as ex-natded5.3 30494 and ex-natded5.3-2 30495 but with no context. Identical to jccir 521, which should be used instead. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜓 → 𝜒) & (𝜒 → 𝜃) ⇒ (𝜓 → (𝜒 ∧ 𝜃))

Theorem

ex-natded5.5 30497

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 Translation	ND Rationale	MPE Rationale
1	2;3	(𝜓 → 𝜒)	(𝜑 → (𝜓 → 𝜒))	Given	$e; adantr 480 to move it into the ND hypothesis
2	5	¬ 𝜒	(𝜑 → ¬ 𝜒)	Given	$e; we'll use adantr 480 to move it into the ND hypothesis
3	1	...\| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption	simpr 484
4	4	... 𝜒	((𝜑 ∧ 𝜓) → 𝜒)	→E 1,3	mpd 15 1,3
5	6	... ¬ 𝜒	((𝜑 ∧ 𝜓) → ¬ 𝜒)	IT 2	adantr 480 5
6	7	¬ 𝜓	(𝜑 → ¬ 𝜓)	∧I 3,4,5	pm2.65da 817 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 480; simpr 484 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 198; a proof without context is shown in mto 197.

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

(𝜑 → (𝜓 → 𝜒)) & (𝜑 → ¬ 𝜒) ⇒ (𝜑 → ¬ 𝜓)

Theorem

ex-natded5.7 30498

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 30499. 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 Translation	ND Rationale	MPE Rationale
1	6	(𝜓 ∨ (𝜒 ∧ 𝜃))	(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃)))	Given	$e. No need for adantr 480 because we do not move this into an ND hypothesis
2	1	...\| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND hypothesis assumption (new scope)	simpr 484
3	2	... (𝜓 ∨ 𝜒)	((𝜑 ∧ 𝜓) → (𝜓 ∨ 𝜒))	∨I_L 2	orcd 874, the MPE equivalent of ∨I_L, 1
4	3	...\| (𝜒 ∧ 𝜃)	((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜒 ∧ 𝜃))	ND hypothesis assumption (new scope)	simpr 484
5	4	... 𝜒	((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜒)	∧E_L 4	simpld 494, the MPE equivalent of ∧E_L, 3
6	6	... (𝜓 ∨ 𝜒)	((𝜑 ∧ (𝜒 ∧ 𝜃)) → (𝜓 ∨ 𝜒))	∨I_R 5	olcd 875, the MPE equivalent of ∨I_R, 4
7	7	(𝜓 ∨ 𝜒)	(𝜑 → (𝜓 ∨ 𝜒))	∨E 1,3,6	mpjaodan 961, the MPE equivalent of ∨E, 2,5,6

(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ (𝜑 → (𝜓 ∨ 𝜒))

Theorem ex-natded5.7-2 30499 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 30498. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 ∨ (𝜒 ∧ 𝜃))) ⇒ (𝜑 → (𝜓 ∨ 𝜒))

Theorem

ex-natded5.8 30500

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 Translation	ND Rationale	MPE Rationale
1	10;11	((𝜓 ∧ 𝜒) → ¬ 𝜃)	(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃))	Given	$e; adantr 480 to move it into the ND hypothesis
2	3;4	(𝜏 → 𝜃)	(𝜑 → (𝜏 → 𝜃))	Given	$e; adantr 480 to move it into the ND hypothesis
3	7;8	𝜒	(𝜑 → 𝜒)	Given	$e; adantr 480 to move it into the ND hypothesis
4	1;2	𝜏	(𝜑 → 𝜏)	Given	$e. adantr 480 to move it into the ND hypothesis
5	6	...\| 𝜓	((𝜑 ∧ 𝜓) → 𝜓)	ND Hypothesis/Assumption	simpr 484. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑 ∧ 𝜓).
6	9	... (𝜓 ∧ 𝜒)	((𝜑 ∧ 𝜓) → (𝜓 ∧ 𝜒))	∧I 5,3	jca 511 (∧I), 6,8 (adantr 480 to bring in scope)
7	5	... ¬ 𝜃	((𝜑 ∧ 𝜓) → ¬ 𝜃)	→E 1,6	mpd 15 (→E), 2,4
8	12	... 𝜃	((𝜑 ∧ 𝜓) → 𝜃)	→E 2,4	mpd 15 (→E), 9,11; note the contradiction with ND line 7 (MPE line 5)
9	13	¬ 𝜓	(𝜑 → ¬ 𝜓)	¬I 5,7,8	pm2.65da 817 (¬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

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 30501.

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

(𝜑 → ((𝜓 ∧ 𝜒) → ¬ 𝜃)) & (𝜑 → (𝜏 → 𝜃)) & (𝜑 → 𝜒) & (𝜑 → 𝜏) ⇒ (𝜑 → ¬ 𝜓)

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