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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremfrgrwopreg1 28101* 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 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (♯‘𝐴) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrwopreg2 28102* 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 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (♯‘𝐵) = 1) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)

Theoremfrgrwopreglem5lem 28103* Lemma for frgrwopreglem5 28104. (Contributed by AV, 5-Feb-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       (((𝑎𝐴𝑥𝐴) ∧ (𝑏𝐵𝑦𝐵)) → ((𝐷𝑎) = (𝐷𝑥) ∧ (𝐷𝑎) ≠ (𝐷𝑏) ∧ (𝐷𝑥) ≠ (𝐷𝑦)))

Theoremfrgrwopreglem5 28104* Lemma 5 for frgrwopreg 28106. 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 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Theoremfrgrwopreglem5ALT 28105* Alternate direct proof of frgrwopreglem5 28104, not using frgrwopreglem5a 28094. This proof would be even a little bit shorter than the proof of frgrwopreglem5 28104 without using frgrwopreglem5lem 28103. (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.)
𝑉 = (Vtx‘𝐺)    &   𝐷 = (VtxDeg‘𝐺)    &   𝐴 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 1 < (♯‘𝐴) ∧ 1 < (♯‘𝐵)) → ∃𝑎𝐴𝑥𝐴𝑏𝐵𝑦𝐵 ((𝑎𝑥𝑏𝑦) ∧ ({𝑎, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝑥} ∈ 𝐸) ∧ ({𝑥, 𝑦} ∈ 𝐸 ∧ {𝑦, 𝑎} ∈ 𝐸)))

Theoremfrgrwopreg 28106* 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 = {𝑥𝑉 ∣ (𝐷𝑥) = 𝐾}    &   𝐵 = (𝑉𝐴)       (𝐺 ∈ FriendGraph → (((♯‘𝐴) = 1 ∨ 𝐴 = ∅) ∨ ((♯‘𝐵) = 1 ∨ 𝐵 = ∅)))

Theoremfrgrregorufr0 28107* 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 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∀𝑣𝑉 (𝐷𝑣) ≠ 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸))

Theoremfrgrregorufr 28108* 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 → (∃𝑎𝑉 (𝐷𝑎) = 𝐾 → (∀𝑣𝑉 (𝐷𝑣) = 𝐾 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theoremfrgrregorufrg 28109* 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 28108 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 (∃𝑎𝑉 ((VtxDeg‘𝐺)‘𝑎) = 𝑘 → (𝐺 RegUSGraph 𝑘 ∨ ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ 𝐸)))

Theoremfrgr2wwlkeu 28110* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → ∃!𝑐𝑉 ⟨“𝐴𝑐𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵))

Theoremfrgr2wwlkn0 28111 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 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (𝐴(2 WWalksNOn 𝐺)𝐵) ≠ ∅)

Theoremfrgr2wwlk1 28112 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 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (♯‘(𝐴(2 WWalksNOn 𝐺)𝐵)) = 1)

Theoremfrgr2wsp1 28113 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 ∧ (𝐴𝑉𝐵𝑉) ∧ 𝐴𝐵) → (♯‘(𝐴(2 WSPathsNOn 𝐺)𝐵)) = 1)

Theoremfrgr2wwlkeqm 28114 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 ∧ 𝐴𝐵 ∧ (𝑃𝑋𝑄𝑌)) → ((⟨“𝐴𝑃𝐵”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐵) ∧ ⟨“𝐵𝑄𝐴”⟩ ∈ (𝐵(2 WWalksNOn 𝐺)𝐴)) → 𝑄 = 𝑃))

Theoremfrgrhash2wsp 28115 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)))

Theoremfusgreg2wsplem 28116* Lemma for fusgreg2wsp 28119 and related theorems. (Contributed by AV, 8-Jan-2022.)
𝑉 = (Vtx‘𝐺)    &   𝑀 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝑁𝑉 → (𝑝 ∈ (𝑀𝑁) ↔ (𝑝 ∈ (2 WSPathsN 𝐺) ∧ (𝑝‘1) = 𝑁)))

Theoremfusgr2wsp2nb 28117* 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 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       ((𝐺 ∈ FinUSGraph ∧ 𝑁𝑉) → (𝑀𝑁) = 𝑥 ∈ (𝐺 NeighbVtx 𝑁) 𝑦 ∈ ((𝐺 NeighbVtx 𝑁) ∖ {𝑥}){⟨“𝑥𝑁𝑦”⟩})

Theoremfusgreghash2wspv 28118* 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 28115. (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 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝐺 ∈ FinUSGraph → ∀𝑣𝑉 (((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(𝑀𝑣)) = (𝐾 · (𝐾 − 1))))

Theoremfusgreg2wsp 28119* 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 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       (𝐺 ∈ FinUSGraph → (2 WSPathsN 𝐺) = 𝑥𝑉 (𝑀𝑥))

Theorem2wspmdisj 28120* 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 = (𝑎𝑉 ↦ {𝑤 ∈ (2 WSPathsN 𝐺) ∣ (𝑤‘1) = 𝑎})       Disj 𝑥𝑉 (𝑀𝑥)

Theoremfusgreghash2wsp 28121* 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 ≠ ∅) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘(2 WSPathsN 𝐺)) = ((♯‘𝑉) · (𝐾 · (𝐾 − 1)))))

Theoremfrrusgrord0lem 28122* Lemma for frrusgrord0 28123. (Contributed by AV, 12-Jan-2022.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾) → (𝐾 ∈ ℂ ∧ (♯‘𝑉) ∈ ℂ ∧ (♯‘𝑉) ≠ 0))

Theoremfrrusgrord0 28123* 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 ≠ ∅) → (∀𝑣𝑉 ((VtxDeg‘𝐺)‘𝑣) = 𝐾 → (♯‘𝑉) = ((𝐾 · (𝐾 − 1)) + 1)))

Theoremfrrusgrord 28124 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 28123, using the definition RegUSGraph (df-rusgr 27346). (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)))

Theoremnumclwwlk2lem1lem 28125 Lemma for numclwwlk2lem1 28159. (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)))

Theorem2clwwlklem 28126 Lemma for clwwnonrepclwwnon 28128 and extwwlkfab 28135. (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))

Theoremclwwnrepclwwn 28127 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 27811. (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))

Theoremclwwnonrepclwwnon 28128 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 28127. (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)))

Theorem2clwwlk2clwwlklem 28129 Lemma for 2clwwlk2clwwlk 28133. (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))

Theorem2clwwlk 28130* 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 28133. (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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋})

Theorem2clwwlk2 28131* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})       (𝑋𝑉 → (𝑋𝐶2) = (𝑋(ClWWalksNOn‘𝐺)2))

Theorem2clwwlkel 28132* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∧ (𝑊‘(𝑁 − 2)) = 𝑋)))

Theorem2clwwlk2clwwlk 28133* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ ∃𝑎 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))∃𝑏 ∈ (𝑋(ClWWalksNOn‘𝐺)2)𝑊 = (𝑎 ++ 𝑏)))

Theoremnumclwwlk1lem2foalem 28134 Lemma for numclwwlk1lem2foa 28137. (Contributed by AV, 29-May-2021.) (Revised by AV, 1-Nov-2022.)
(((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 − 2)) ∧ (𝑋𝑉𝑌𝑉) ∧ 𝑁 ∈ (ℤ‘3)) → ((((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) prefix (𝑁 − 2)) = 𝑊 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 1)) = 𝑌 ∧ (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑁 − 2)) = 𝑋))

Theoremextwwlkfab 28135* 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 27875 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 28131. (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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))       ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑤‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)})

Theoremextwwlkfabel 28136* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))       ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑊 ∈ (𝑋𝐶𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ ((𝑊 prefix (𝑁 − 2)) ∈ 𝐹 ∧ (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx 𝑋) ∧ (𝑊‘(𝑁 − 2)) = 𝑋))))

Theoremnumclwwlk1lem2foa 28137* 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), pn = 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))       ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑊𝐹𝑌 ∈ (𝐺 NeighbVtx 𝑋)) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋𝐶𝑁)))

Theoremnumclwwlk1lem2f 28138* 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 = (𝑣𝑉, 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 ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐶𝑁)⟶(𝐹 × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1lem2fv 28139* 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 = (𝑣𝑉, 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) → (𝑇𝑊) = ⟨(𝑊 prefix (𝑁 − 2)), (𝑊‘(𝑁 − 1))⟩)

Theoremnumclwwlk1lem2f1 28140* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (𝑣𝑉, 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 ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐶𝑁)–1-1→(𝐹 × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1lem2fo 28141* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (𝑣𝑉, 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 ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐶𝑁)–onto→(𝐹 × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1lem2f1o 28142* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = (𝑣𝑉, 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 ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐶𝑁)–1-1-onto→(𝐹 × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1lem2 28143* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))       ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐶𝑁) ≈ (𝐹 × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1 28144* 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 28131, 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 28152. If the general representation of (closed) walk is used, Huneke's statement can be proven even for n = 2, see numclwlk1 28154. 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐹 = (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))       (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐶𝑁)) = (𝐾 · (♯‘𝐹)))

Theoremclwwlknonclwlknonf1o 28145* 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) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)}    &   𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))       ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝐹:𝑊1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))

Theoremclwwlknonclwlknonen 28146* 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) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋)} ≈ (𝑋(ClWWalksNOn‘𝐺)𝑁))

Theoremdlwwlknondlwlknonf1olem1 28147 Lemma 1 for dlwwlknondlwlknonf1o 28148. (Contributed by AV, 29-May-2022.) (Revised by AV, 1-Nov-2022.)
(((♯‘(1st𝑐)) = 𝑁𝑐 ∈ (ClWalks‘𝐺) ∧ 𝑁 ∈ (ℤ‘2)) → (((2nd𝑐) prefix (♯‘(1st𝑐)))‘(𝑁 − 2)) = ((2nd𝑐)‘(𝑁 − 2)))

Theoremdlwwlknondlwlknonf1o 28148* 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) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}    &   𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}    &   𝐹 = (𝑐𝑊 ↦ ((2nd𝑐) prefix (♯‘(1st𝑐))))       ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → 𝐹:𝑊1-1-onto𝐷)

Theoremdlwwlknondlwlknonen 28149* 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) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}    &   𝐷 = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) = 𝑋}       ((𝐺 ∈ USPGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘2)) → 𝑊𝐷)

Theoremwlkl0 28150* There is exactly one walk of length 0 on each vertex 𝑋. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtx‘𝐺)       (𝑋𝑉 → {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 0 ∧ ((2nd𝑤)‘0) = 𝑋)} = {⟨∅, {⟨0, 𝑋⟩}⟩})

Theoremclwlknon2num 28151* There are k walks of length 2 on each vertex 𝑋 in a k-regular simple graph. Variant of clwwlknon2num 27888, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾𝑋𝑉) → (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 2 ∧ ((2nd𝑤)‘0) = 𝑋)}) = 𝐾)

Theoremnumclwlk1lem1 28152* Lemma 1 for numclwlk1 28154 (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) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}    &   𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}       (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 = 2)) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Theoremnumclwlk1lem2 28153* Lemma 2 for numclwlk1 28154 (Statement 9 in [Huneke] p. 2 for n>2). This theorem corresponds to numclwwlk1 28144, using the general definition of walks instead of walks as words. (Contributed by AV, 4-Jun-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}    &   𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}       (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Theoremnumclwlk1 28154* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = 𝑁 ∧ ((2nd𝑤)‘0) = 𝑋 ∧ ((2nd𝑤)‘(𝑁 − 2)) = 𝑋)}    &   𝐹 = {𝑤 ∈ (ClWalks‘𝐺) ∣ ((♯‘(1st𝑤)) = (𝑁 − 2) ∧ ((2nd𝑤)‘0) = 𝑋)}       (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘2))) → (♯‘𝐶) = (𝐾 · (♯‘𝐹)))

Theoremnumclwwlkovh0 28155* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ∣ (𝑤‘(𝑁 − 2)) ≠ 𝑋})

Theoremnumclwwlkovh 28156* 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})

Theoremnumclwwlkovq 28157* 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 28158 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝑄𝑁) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (lastS‘𝑤) ≠ 𝑋)})

Theoremnumclwwlkqhash 28158* 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})       (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (♯‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁))))

Theoremnumclwwlk2lem1 28159* 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Theoremnumclwlk2lem2f 28160* 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))

Theoremnumclwlk2lem2fv 28161* 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 prefix (𝑁 + 1))))

Theoremnumclwlk2lem2f1o 28162* 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.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 prefix (𝑁 + 1)))       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Theoremnumclwwlk2lem3 28163* 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (♯‘(𝑋𝑄𝑁)) = (♯‘(𝑋𝐻(𝑁 + 2))))

Theoremnumclwwlk2 28164* 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 27761, 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 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (lastS‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       (((𝐺 RegUSGraph 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))))

Theoremnumclwwlk3lem1 28165 Lemma 2 for numclwwlk3 28168. (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))))

Theoremnumclwwlk3lem2lem 28166* Lemma for numclwwlk3lem2 28167: 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ((𝑋𝐻𝑁) ∪ (𝑋𝐶𝑁)))

Theoremnumclwwlk3lem2 28167* Lemma 1 for numclwwlk3 28168: 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 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑣(ClWWalksNOn‘𝐺)𝑛) ∣ (𝑤‘(𝑛 − 2)) ≠ 𝑣})       (((𝐺 ∈ FinUSGraph ∧ 𝑋𝑉) ∧ 𝑁 ∈ (ℤ‘2)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = ((♯‘(𝑋𝐻𝑁)) + (♯‘(𝑋𝐶𝑁))))

Theoremnumclwwlk3 28168 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 28144) and with v(n-2) =/= v(n) (see numclwwlk2 28164): 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 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (♯‘(𝑋(ClWWalksNOn‘𝐺)𝑁)) = (((𝐾 − 1) · (♯‘(𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Theoremnumclwwlk4 28169* 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)) = Σ𝑥𝑉 (♯‘(𝑥(ClWWalksNOn‘𝐺)𝑁)))

Theoremnumclwwlk5lem 28170 Lemma for numclwwlk5 28171. (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 𝐾𝑋𝑉𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)2)) mod 2) = 1))

Theoremnumclwwlk5 28171 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 𝐾𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋𝑉𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑋(ClWWalksNOn‘𝐺)𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk7lem 28172 Lemma for numclwwlk7 28174, frgrreggt1 28176 and frgrreg 28177: 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 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin)) → 𝐾 ∈ ℕ0)

Theoremnumclwwlk6 28173 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 𝐾𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((♯‘𝑉) mod 𝑃))

Theoremnumclwwlk7 28174 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 28123 or frrusgrord 28124, 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 28048, as well as k-regular (for any k), see 0vtxrgr 27364, but has no closed walk, see 0clwlk0 27915, 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 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((♯‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 1)

Theoremnumclwwlk8 28175 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 27869. (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)

Theoremfrgrreggt1 28176 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))

Theoremfrgrreg 28177 If a finite nonempty friendship graph is 𝐾-regular, then 𝐾 must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 3-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝐺 ∈ FriendGraph ∧ 𝐺 RegUSGraph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Theoremfrgrregord013 28178 If a finite friendship graph is 𝐾-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 0 ∨ (♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Theoremfrgrregord13 28179 If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 28178. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ 𝐺 RegUSGraph 𝐾) → ((♯‘𝑉) = 1 ∨ (♯‘𝑉) = 3))

Theoremfrgrogt3nreg 28180* If a finite friendship graph has an order greater than 3, it cannot be 𝑘-regular for any 𝑘. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ 𝐺 RegUSGraph 𝑘)

Theoremfriendshipgt3 28181* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.) (Revised by AV, 4-Jun-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin ∧ 3 < (♯‘𝑉)) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

Theoremfriendship 28182* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
𝑉 = (Vtx‘𝐺)       ((𝐺 ∈ FriendGraph ∧ 𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ (Edg‘𝐺))

PART 17  GUIDES AND MISCELLANEA

17.1  Guides (conventions, explanations, and examples)

17.1.1  Conventions

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

Logic and set theory provide a foundation for all of mathematics. To learn about them, you should study one or more of the references listed below. We indicate references using square brackets. The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

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

Theoremconventions 28183

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

• Notation. Where possible, the notation attempts to conform to modern conventions, with variations due to our choice of the axiom system or to make proofs shorter. However, our notation is strictly sequential (left-to-right). For example, summation is written in the form Σ𝑘𝐴𝐵 (df-sum 15034) 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 15229). The notation is usually explained in more detail when first introduced.

• Axiomatic assertions (\$a). All axiomatic assertions (\$a statements) starting with " " have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats \$a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound as discussed below, except for 4 definitions (df-bi 210, df-cleq 2815, df-clel 2894, df-clab 2801) 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 2801. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is the complex arithmetic axiom ax-1cn 10584, proven by the preceding theorem ax1cn 10560. The Metamath program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.

• Definitions (df-...). We encourage definitions to include hypertext links to proven examples.

• Statements with hypotheses. Many theorems and some axioms, such as ax-mp 5, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When displayed in summarized form such as in the "Theorem List" page (to get to it, click on "Nearby theorems" on the ax-mp 5 page), the hypotheses are connected with an ampersand and separated from the conclusion with a double right arrow, such as in " 𝜑 & (𝜑𝜓) ⇒ 𝜓". These symbols are not part of the Metamath language but are just informal notation meaning "and" and "implies".

• Discouraged use and modification. If something should only be used in limited ways, it is marked with "(New usage is discouraged.)". This is used, for example, when something can be constructed in more than one way, and we do not want later theorems to depend on that specific construction. This marking is also used if we want later proofs to use proven axioms. For example, we want later proofs to use ax-1cn 10584 (not ax1cn 10560) and ax-1ne0 10595 (not ax1ne0 10571), as these are proven axioms for complex arithmetic. Thus, both ax1cn 10560 and ax1ne0 10571 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".

• New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use < and for inequality expressions, and use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴) for the hyperbolic sine.

• Minimizing axiom dependencies. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, because of the non-constructive nature of the axiom of choice df-ac 9531, we prefer proofs that do not use it, or use weaker versions like countable choice ax-cc 9846 or dependent choice ax-dc 9857. An example is our proof of the Schroeder-Bernstein Theorem sbth 8625, 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 2145 through ax-13 2391, by using ax10w 2133 through ax13w 2140 instead.

We do not try to similarly reduce dependencies on definitions, since definitions are conservative (they do not increase the proving power of a deductive system), and are introduced in order to be used to increase readability). An exception is made for the definitions df-clab 2801, df-cleq 2815, df-clel 2894, since they can be considered as axioms under some definitions of what a definition is exactly (see their comments).

• Alternate proofs (ALT). If a different proof is shorter or clearer but uses more or stronger axioms, we make that proof an "alternate" proof (marked with an ALT label suffix), even if this alternate proof was formalized first. We then make the proof that requires fewer axioms the main proof. Alternate proofs can also occur in other cases when an alternate proof gives some particular insight. Their comment should begin with "Alternate proof of ~ xxx " followed by a description of the specificity of that alternate proof. There can be multiple alternates. Alternate (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternate and main statement comments should use hyperlinks to refer to each other.

• Alternate versions (ALTV). The suffix ALTV is reserved for theorems (or definitions) which are alternate versions, or variants, of an existing theorem. This is reserved to statements in mathboxes and is typically used temporarily, when it is not clear yet which variant to use. If it is decided that both variants should be kept and moved to the main part of set.mm, then a label for the variant should be found with a more explicit suffix indicating how it is a variant (e.g., commutation of some subformula, antecedent replaced with hypothesis, (un)curried variant, biconditional instead of implication, etc.). There is no requirement to add discouragement tags, but their comment should have a link to the main version of the statement and describe how it is a variant of it.

• Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-Mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first. This typically happens when a shorter proof to an existing theorem is found: the existing theorem is kept as an *OLD statement for one year. When a proof is shortened automatically (using the Metamath program "MM-PA> MINIMIZE__WITH *" command), then it is not necessary to keep the old proof, nor to add credit for the shortening.

• Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are generally used in this order: 𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta, 𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma. Individual setvar variables are represented with lowercase Latin letters and are generally used in this order: 𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡. Variables that represent classes are often represented by uppercase Latin letters: 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g., 0 for a zero element (e.g., fsuppcor 8855) and connective symbols such as + for some group addition operation (e.g., grprinvd ). Class variables are selected in alphabetical order starting from 𝐴 if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.

• Turnstile. "", meaning "It is provable that", is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff ¬ 𝜑".

• Biconditional (). There are basically two ways to maximize the effectiveness of biconditionals (): you can either have one-directional simplifications of all theorems that produce biconditionals, or you can have one-directional simplifications of theorems that consume biconditionals. Some tools (like Lean) follow the first approach, but set.mm follows the second approach. Practically, this means that in set.mm, for every theorem that uses an implication in the hypothesis, like ax-mp 5, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 233 or mpbir 234. 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 219 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 710, sylbir 238, or 3imtr4i 295.

• Quantifiers. The quantifiers are named as follows:

• : universal quantifier (wal 1536);
• : existential quantifier (df-ex 1782);
• ∃*: at-most-one quantifier (df-mo 2622);
• ∃!: unique existential quantifier (df-eu 2653).

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 2070 and the related df-sbc 3748 and df-csb 3856.

• 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 3471 and isset 3481. However, instead of using 𝐼 ∈ V in the antecedent of a theorem for some variable 𝐼, we now prefer to use 𝐼𝑉 (or another variable if 𝑉 is not available) to make it more general. That way we can often avoid needing extra uses of elex 3487 and syl 17 in the common case where 𝐼 is already a member of something. For hypotheses (\$e statement) of theorems (mostly in inference form), however, 𝐴 ∈ V is used rather than 𝐴𝑉 (e.g., difexi 5208). This is because 𝐴 ∈ V is almost always satisfied using an existence theorem stating "... ∈ V", and a hard-coded V in the \$e statement saves a couple of syntax building steps that substitute V into 𝑉. Notice that this does not hold for hypotheses of theorems in deduction form: Here still (𝜑𝐴𝑉) should be used rather than (𝜑𝐴 ∈ V).

• Converse. "𝑅" should be read "converse of (relation) 𝑅" and is the same as the more standard notation R^{-1} (the standard notation is ambiguous). See df-cnv 5540. This can be used to define a subset, e.g., df-tan 15416 notates "the set of values whose cosine is a nonzero complex number" as (cos “ (ℂ ∖ {0})).

• Function application. "(𝐹𝑥)" should be read "the value of function 𝐹 at 𝑥" and has the same meaning as the more familiar but ambiguous notation F(x). For example, (cos‘0) = 1 (see cos0 15494). 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 6342. 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 7143). For example, the + in (2 + 2); see 2p2e4 11760. Function application is itself an example of this. Similarly, predicate expressions in infix form that take two or three wffs and produce a wff are also always surrounded by parentheses, such as (𝜑𝜓), (𝜑𝜓), (𝜑𝜓), and (𝜑𝜓) (see wi 4, df-or 845, df-an 400, and df-bi 210 respectively). In contrast, a binary relation (which compares two _classes_ and produces a _wff_) applied in an infix expression is _not_ surrounded by parentheses. This includes set membership 𝐴𝐵 (see wel 2115), equality 𝐴 = 𝐵 (see df-cleq 2815), subset 𝐴𝐵 (see df-ss 3925), and less-than 𝐴 < 𝐵 (see df-lt 10539). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 5043. For example, 0 < 1 (see 0lt1 11151) does not use parentheses.

• Unary minus. The symbol - is used to indicate a unary minus, e.g., -1. It is specially defined because it is so commonly used. See cneg 10860.

• Function definition. Functions are typically defined by first defining the constant symbol (using \$c) and declaring that its symbol is a class with the label cNAME (e.g., ccos 15409). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 15415). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 15471), its function application form labeled NAMEval (e.g., cosval 15467), and at least one simple value (e.g., cos0 15494). 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 45013 (which is based on the sequence builder seq, see df-seq 13365).

• 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 13630 and fac4 13637).

• Unambiguous symbols. A given symbol has a single unambiguous meaning in general. Thus, where the literature might use the same symbol with different meanings, here we use different (variant) symbols for different meanings. These variant symbols often have suffixes, subscripts, or underlines to distinguish them. For example, here "0" always means the value zero (df-0 10533), while "0g" is the group identity element (df-0g 16706), "0." is the poset zero (df-p0 17640), "0𝑝" is the zero polynomial (df-0p 24272), "0vec" is the zero vector in a normed subcomplex vector space (df-0v 28379), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 17642). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including "1", "+", "", and "". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.

• ASCII representation of symbols. We must have an ASCII representation for each symbol. We generally choose short sequences, ideally digraphs, and generally choose sequences that vaguely resemble the mathematical symbol. Here are some of the conventions we use when selecting an ASCII representation.

We generally do not include parentheses inside a symbol because that confuses text editors (such as emacs). Greek letters for wff variables always use the first two letters of their English names, making them easy to type and easy to remember. Symbols that almost look like letters, such as , are often represented by that letter followed by a period. For example, "A." is used to represent , "e." is used to represent , and "E." is used to represent . Single letters are now always variable names, so constants that are often shown as single letters are now typically preceded with "_" in their ASCII representation, for example, "_i" is the ASCII representation for the imaginary unit i. A script font constant is often the letter preceded by "~" meaning "curly", such as "~P" to represent the power class 𝒫.

Originally, all setvar and class variables used only single letters a-z and A-Z, respectively. A big change in recent years was to allow the use of certain symbols as variable names to make formulas more readable, such as a variable representing an additive group operation. The convention is to take the original constant token (in this case "+" which means complex number addition) and put a period in front of it to result in the ASCII representation of the variable ".+", shown as +, that can be used instead of say the letter "P" that had to be used before.

Choosing tokens for more advanced concepts that have no standard symbols but are represented by words in books, is hard. A few are reasonably obvious, like "Grp" for group and "Top" for topology, but often they seem to end up being either too long or too cryptic. It would be nice if the math community came up with standardized short abbreviations for English math terminology, like they have more or less done with symbols, but that probably won't happen any time soon.

Another informal convention that we've somewhat followed, that is also not uncommon in the literature, is to start tokens with a capital letter for collection-like objects and lower case for function-like objects. For example, we have the collections On (ordinal numbers), Fin, Prime, Grp, and we have the functions sin, tan, log, sup. Predicates like Ord and Lim also tend to start with upper case, but in a sense they are really collection-like, e.g. Lim indirectly represents the collection of limit ordinals, but it can't be an actual class since not all limit ordinals are sets. This initial upper versus lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).

• Typography conventions. Class symbols for functions (e.g., abs, sin) should usually not have leading or trailing blanks in their HTML representation. This is in contrast to class symbols for operations (e.g., gcd, sadd, eval), which usually do include leading and trailing blanks in their representation. If a class symbol is used for a function as well as an operation (according to the definition df-ov 7143, each operation value can be written as function value of an ordered pair), the convention for its primary usage should be used, e.g. (iEdg‘𝐺) versus (𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸.

• LaTeX definitions. Each token has a "LaTeX definition" which is used by the Metamath program to output tex files. When writing LaTeX definitions, contributors should favor simplicity over perfection of the display, and should only use core LaTeX symbols or symbols from standard packages; if packages other than amssymb, amsmath, mathtools, mathrsfs, phonetic, graphicx are needed, this should be discussed. A useful resource is The Comprehensive LaTeX Symbol List.

• Number construction independence. There are many ways to model complex numbers. After deriving the complex number postulates we reintroduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. This also lets us be independent of the specific construction, which we believe is valuable. See mmcomplex.html 7143 for details. Thus, for example, we don't allow the use of ∅ ∉ ℂ, as handy as that would be, because that would be construction-specific. We want proofs about to be independent of whether or not ∅ ∈ ℂ.

• Minimize hypotheses. In most cases we try to minimize hypotheses, so that the statement be more general and easier to use. There are exceptions. For example, we intentionally add hypotheses if they help make proofs independent of a particular construction (e.g., the contruction of the complex numbers ). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they are not needed. For example, we could show that (𝐴 < 𝐵𝐵𝐴) without any hypotheses, but we require that theorems using this result prove that 𝐴 and 𝐵 are real numbers, so that the statement we use is ltnei 10753. Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ 10753:

1. Having the hypotheses immediately shows the intended domain of applicability (is it , *, ω, or something else?), without having to trace back to definitions.
2. Having the hypotheses forces the intended use of the statement, which generally is desirable.
3. Many out-of-domain values are dependent on contingent details of definitions, so hypothesis-free theorems would be non-portable and "brittle".
4. Only a few theorems can have their hypotheses removed in this fashion, due to coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning, e.g., real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who has not traced back the set-theoretical foundations of the definitions, it is seemingly random and is not intuitive at all.
5. Ultimately, this is a matter of consensus, and the consensus in the group was in favor of keeping sometimes redundant hypotheses.
• Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 11626) for the set of positive integers starting from 1, and 0 (df-n0 11886) 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 12087, e.g., 4001 (see 4001prm 16469 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 4135). 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 24617) 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 4137. 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 2823, 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 4136).

Any theorem whose conclusion is an implication has an associated inference, whose hypotheses are the hypotheses of that theorem together with the antecedent of its conclusion, and whose conclusion is the consequent of that conclusion. When both theorems are in set.mm, then the associated inference is often labeled by adding the suffix "i" to the label of the original theorem (for instance, con3i 157 is the inference associated with con3 156). The inference associated with a theorem is easily derivable from that theorem by a simple use of ax-mp 5. The other direction is the subject of the Deduction Theorem discussed below. We may also use the term "associated inference" when the above process is iterated. For instance, syl 17 is an inference associated with imim1 83 because it is the inference associated with imim1i 63 which is itself the inference associated with imim1 83.

"Deduction form" is the preferred form for theorems because this form allows us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that would end up requiring us to use the deduction theorem.

Deductions have a label suffix of "d", especially if there are other forms of the same theorem (e.g., pm2.43d 53). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 52). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 56) or, if the non-suffixed label is already used, then we add the suffix "t" (for "theorem" or "tautology", e.g., ancomst 468 or nfimt 1896). When an inference with an "is a set" hypothesis (e.g., 𝐴 ∈ V) is converted to a theorem (in closed form) by replacing the hypothesis with an antecedent of the form (𝐴𝑉, we sometimes suffix the closed form with "g" (for "more general") as in uniex 7452 versus uniexg 7451. 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 5838. Its inference form which was the original "brres", now is brresi 5840. 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 5840. 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 4495, which works in certain cases in set theory. We also sometimes use dedhb 3670. 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 3670; a list of translations for common natural deduction rules is given in natded 28186.

• Recursion. We define recursive functions using various "recursion constructors". These allow us to define, with compact direct definitions, functions that are usually defined in textbooks with indirect self-referencing recursive definitions. This produces compact definition and much simpler proofs, and greatly reduces the risk of creating unsound definitions. Examples of recursion constructors include recs(𝐹) in df-recs 7995, rec(𝐹, 𝐼) in df-rdg 8033, seqω(𝐹, 𝐼) in df-seqom 8071, and seq𝑀( + , 𝐹) in df-seq 13365. These have characteristic function 𝐹 and initial value 𝐼. (Σg in df-gsum 16707 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7995, but for the "average user" the most useful one is probably df-seq 13365- 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 16476.

• Undefined results and "junk theorems". Some expressions are only expected to be meaningful in certain contexts. For example, consider Russell's definition description binder iota, where (℩𝑥𝜑) is meant to be "the 𝑥 such that 𝜑" (where 𝜑 typically depends on x). What should that expression produce when there is no such 𝑥? In set.mm we primarily use one of two approaches. One approach is to make the expression evaluate to the empty set whenever the expression is being used outside of its expected context. While not perfect, it makes it a bit more clear when something is undefined, and it has the advantage that it makes more things equal outside their domain which can remove hypotheses when you feel like exploiting these so-called junk theorems. Note that Quine does this with iota (his definition of iota evaluates to the empty set when there is no unique value of 𝑥). Quine has no problem with that and we don't see why we should, so we define iota exactly the same way that Quine does. The main place where you see this being systematically exploited is in "reverse closure" theorems like 𝐴 ∈ (𝐹𝐵) → 𝐵 ∈ dom 𝐹, which is useful when 𝐹 is a family of sets. (by this we mean it's a set set even in a type theoretic interpretation.)

The second approach uses "(New usage is discouraged.)" to prevent unintentional uses of certain properties. For example, you could define some construct df-NAME whose usage is discouraged, and prove only the specific properties you wish to use (and add those proofs to the list of permitted uses of "discouraged" information). From then on, you can only use those specific properties without a warning. Other approaches often have hidden problems. For example, you could try to "not define undefined terms" by creating definitions like \${ \$d 𝑦𝑥 \$. \$d 𝑦𝜑 \$. df-iota \$a (∃!𝑥𝜑 → (℩𝑥𝜑) = {𝑥𝜑}) \$. \$}. This will be rejected by the definition checker, but the bigger theoretical reason to reject this axiom is that it breaks equality - the metatheorem (𝑥 = 𝑦 P(x) = P(y) ) fails to hold if definitions don't unfold without some assumptions. (That is, iotabidv 6318 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 6665 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 8615 is the first lemma for sbth 8625. 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 8615 is the first lemma for sbth 8625.

• 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 16005"). 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.

• Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g., mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we _require_ page numbers in comments for individual \$a or \$p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the Metamath program "MM> WRITE BIBLIOGRAPHY" command.)

• Acceptable shorter proofs. Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem description. However, in some cases a proof may be "shorter" or not depending on how it is formatted. This section provides general guidelines.

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

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

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

• Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see mmdefinitions.html 23. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.

• Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.

The challenge of varying mathematical conventions

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

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

Distinctness or freeness

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

• 𝑥𝜑 is read " 𝑥 is not free in (wff) 𝜑"; see df-nf 1786 (whose description has some important technical details). Similarly, 𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2962.
• "\$d 𝑥𝑦 \$." should be read "Assume 𝑥 and 𝑦 are distinct variables."
• "\$d 𝜑𝑥 \$." should be read "Assume 𝑥 does not occur in ϕ." Sometimes a theorem is proved using 𝑥𝜑 (df-nf 1786) in place of "\$d 𝜑𝑥 \$." when a more general result is desired; ax-5 1911 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 2660 from eu6 2658.
• "\$d 𝐴𝑥 \$." should be read "Assume 𝑥 is not a variable occurring in class 𝐴."
• "\$d 𝐴𝑥 \$. \$d 𝜓𝑥 \$. \$e |- (𝑥 = 𝐴 → (𝜑𝜓)) \$." is an idiom often used instead of explicit substitution, meaning "Assume ψ results from the proper substitution of 𝐴 for 𝑥 in ϕ." Therefore, we often use the term "implicit substitution" for such a hypothesis.
• Class and wff variables should appear at the beginning of distinct variable conditions, and setvars should be in alphabetical order. E.g., "\$d 𝑍𝑥𝑦 \$.", "\$d 𝜓𝑎𝑥 \$.". This convention should be applied for new theorems (formerly, the class and wff variables mostly appear at the end) and will be assured by a formatter in the future.
• " (¬ ∀𝑥𝑥 = 𝑦 → ..." occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the \$d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the \$d eventually becomes a requirement, and after that this device is rarely used.

There is a general technique to replace a \$d x A or \$d x ph condition in a theorem with the corresponding 𝑥𝐴 or 𝑥𝜑; here it is. T[x, A] where \$d 𝑥𝐴, and you wish to prove 𝑥𝐴 T[x, A]. You apply the theorem substituting 𝑦 for 𝑥 and 𝐴 for 𝐴, where 𝑦 is a new dummy variable, so that \$d y A is satisfied. You obtain T[y, A], and apply chvar to obtain T[x, A] (or just use mpbir 234 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 2417. The section mmtheorems32.html#mm3146s 2417 also describes the metatheorem that underlies this.

Standard Metamath verifiers do not distinguish between axioms and definitions (both are \$a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions (\$a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 210, df-clab 2801, df-cleq 2815, and df-clel 2894); see those definitions for more information. These additional rules for definitions are checked by at least mmj2's definition check (see mmj2 master file mmj2jar/macros/definitionCheck.js). This definition check relies on the database being very much like set.mm, down to the names of certain constants and types, so it cannot apply to all Metamath databases... but it is useful in set.mm. In this definition check, a \$a-statement with a given label and typecode passes the test if and only if it respects the following rules (these rules require that we have an unambiguous tree parse, which is checked separately):

1. The expression must be a biconditional or an equality (i.e. its root-symbol must be or =). If the proposed definition passes this first rule, we then define its definiendum as its left hand side (LHS) and its definiens as its right hand side (RHS). We define the *defined symbol* as the root-symbol of the LHS. We define a *dummy variable* as a variable occurring in the RHS but not in the LHS. Note that the "root-symbol" is the root of the considered tree; it need not correspond to a single token in the database (e.g., see w3o 1083 or wsb 2069).

2. The defined expression must not appear in any statement between its syntax axiom (\$a wff ) and its definition, and the defined expression must not be used in its definiens. See df-3an 1086 for an example where the same symbol is used in different ways (this is allowed).

3. No two variables occurring in the LHS may share a disjoint variable (DV) condition.

4. All dummy variables are required to be disjoint from any other (dummy or not) variable occurring in this labeled expression.

5. Either
(a) there must be no non-setvar dummy variables, or
(b) there must be a justification theorem.

The justification theorem must be of form ( definiens root-symbol definiens' ) where definiens' is definiens but the dummy variables are all replaced with other unused dummy variables of the same type. Note that root-symbol is or =, and that setvar variables are simply variables with the setvar typecode.

6. One of the following must be true:
(a) there must be no setvar dummy variables,
(b) there must be a justification theorem as described in rule 5, or
(c) if there are setvar dummy variables, every one must not be free.

That is, it must be true that (𝜑 → ∀𝑥𝜑) for each setvar dummy variable 𝑥 where 𝜑 is the definiens. We use two different tests for non-freeness; one must succeed for each setvar dummy variable 𝑥. The first test requires that the setvar dummy variable 𝑥 be syntactically bound (this is sometimes called the "fast" test, and this implies that we must track binding operators). The second test requires a successful search for the directly-stated proof of (𝜑 → ∀𝑥𝜑) Part c of this rule is how most setvar dummy variables are handled.

Rule 3 may seem unnecessary, but it is needed. Without this rule, you can define something like

```       cbar \$a wff Foo x y \$.
\${ \$d x y \$. df-foo \$a |- ( Foo x y <-> x = y ) \$. \$}```
and now "Foo x x" is not eliminable; there is no way to prove that it means anything in particular, because the definitional theorem that is supposed to be responsible for connecting it to the original language wants nothing to do with this expression, even though it is well formed.

A justification theorem for a definition (if used this way) must be proven before the definition that depends on it. One example of a justification theorem is vjust 3470. The definition df-v 3471 V = {𝑥𝑥 = 𝑥} is justified by the justification theorem vjust 3470 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}. Another example of a justification theorem is trujust 1540; the definition df-tru 1541 (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1540 ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).

• Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.

• 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 1540.

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

𝜑       𝜑

Theoremconventions-labels 28184

The following gives conventions used in the Metamath Proof Explorer (MPE, set.mm) regarding labels. For other conventions, see conventions 28183 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 3140"rgen.1 \$e |- ( x e. A -> ph ) \$." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 21209: "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 2749 and stirling 42670.
• 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 1840, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 3204.
• 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... 15228. Furthermore, the underscore "_" should not be used. Finally, only lower case characters should be used (except the special suffixes OLD, ALT, and ALTV mentioned in bullet point "Suffixes"), at least in main set.mm (exceptions are tolerated in mathboxes).
• Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct (𝐴𝐵) is defined in df-dif 3911, and thus its syntax label fragment is "dif". Similarly, the subclass relation 𝐴𝐵 has syntax label fragment "ss" because it is defined in df-ss 3925. Most theorem names follow from these fragments, for example, the theorem proving (𝐴𝐵) ⊆ 𝐴 involves a class difference ("dif") of a subset ("ss"), and thus is labeled difss 4083. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4540), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4542). Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
• Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct 𝐴𝐵 does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with (𝐴 ∈ (𝐵 ∖ {𝐶}) uses is-element-of ("el") of a class difference ("dif") of a singleton ("sn"), it is labeled eldifsn 4693. An "n" is often used for negation (¬), e.g., nan 828.
• Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The definition should note this exception as part of its definition. In addition, the table below attempts to list all such cases and marks them in bold. For example, the label fragment "cn" represents complex numbers (even though its definition is in df-c 10532) and "re" represents real numbers ( definition df-r 10536). The empty set often uses fragment 0, even though it is defined in df-nul 4266. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 10537), but "p" is used as the fragment for constant theorems. Equality (𝐴 = 𝐵) often uses "e" as the fragment. As a result, "two plus two equals four" is labeled 2p2e4 11760.
• 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 15494 is the theorem that provides the value for the cosine of 0; we would need to look at the theorem itself to see what that value is. The label "cos0" is concise and we use it instead of "cos0eq1". There is no need to add the "eq1", because there will never be a case where we have to disambiguate between different values produced by the cosine of zero, and we generally prefer shorter labels if they are unambiguous.
• Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value labeled "NAMEval" and of its closure labeld "NAMEcl". E.g., for cosine (df-cos 15415) we have value cosval 15467 and closure coscl 15471.
• 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 28186 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 1940 versus 19.21 2208. This often permits to prove the result using fewer axioms, and/or to eliminate a nonfreeness hypothesis (such as 𝑥𝜑 in 19.21 2208). If no constraint is put on axiom use, then the v-version can be proved from the original theorem using nfv 1915. If two (resp. three) such disjoint variable conditions are added, then the suffix "vv" (resp. "vvv") is used, e.g., exlimivv 1933. 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 2660 derived from eu6 2658. 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 5319. 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 2431 (cbval 2417 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 3530. 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 587), commutes (e.g., biimpac 482)
• d : theorem in deduction form
• f : theorem with a hypothesis such as 𝑥𝜑
• g : theorem in closed form having an "is a set" antecedent
• i : theorem in inference form
• l : theorem concerning something at the left
• r : theorem concerning something at the right
• r : theorem with something reversed (e.g., a biconditional)
• s : inference that manipulates an antecedent ("s" refers to an application of syl 17 that is eliminated)
• t : theorem in closed form (not having an "is a set" antecedent)
• v : theorem with one (main) disjoint variable condition
• vv : theorem with two (main) disjoint variable conditions
• w : weak(er) form of a theorem
• ALT : alternate proof of a theorem
• ALTV : alternate version of a theorem or definition (mathbox only)
• OLD : old/obsolete version of a theorem (or proof) or definition
• Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

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

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
aand (suffix) No biimpa 480, rexlimiva 3267
ablAbelian group df-abl 18900 Abel Yes ablgrp 18902, zringabl 20165
absabsorption No ressabs 16554
absabsolute value (of a complex number) df-abs 14586 (abs‘𝐴) Yes absval 14588, absneg 14628, abs1 14648
al"for all" 𝑥𝜑 No alim 1812, alex 1827
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 400 (𝜑𝜓) Yes anor 980, iman 405, imnan 403
assassociative No biass 389, orass 919, mulass 10614
asymasymmetric, antisymmetric No intasym 5953, asymref 5954, posasymb 17553
axaxiom No ax6dgen 2132, ax1cn 10560
bas, base base (set of an extensible structure) df-base 16480 (Base‘𝑆) Yes baseval 16533, ressbas 16545, cnfldbas 20093
b, bibiconditional ("iff", "if and only if") df-bi 210 (𝜑𝜓) Yes impbid 215, sspwb 5319
brbinary relation df-br 5043 𝐴𝑅𝐵 Yes brab1 5090, brun 5093
cbvchange bound variable No cbvalivw 2014, cbvrex 3421
clclosure No ifclda 4473, ovrcl 7181, zaddcl 12010
cncomplex numbers df-c 10532 Yes nnsscn 11630, nncn 11633
cnfldfield of complex numbers df-cnfld 20090 fld Yes cnfldbas 20093, cnfldinv 20120
cntzcentralizer df-cntz 18438 (Cntz‘𝑀) Yes cntzfval 18441, dprdfcntz 19128
cnvconverse df-cnv 5540 𝐴 Yes opelcnvg 5728, f1ocnv 6609
cocomposition df-co 5541 (𝐴𝐵) Yes cnvco 5733, fmptco 6873
comcommutative No orcom 867, bicomi 227, eqcomi 2831
concontradiction, contraposition No condan 817, con2d 136
csbclass substitution df-csb 3856 𝐴 / 𝑥𝐵 Yes csbid 3868, csbie2g 3895
cygcyclic group df-cyg 18988 CycGrp Yes iscyg 18989, zringcyg 20182
ddeduction form (suffix) No idd 24, impbid 215
df(alternate) definition (prefix) No dfrel2 6024, dffn2 6496
di, distrdistributive No andi 1005, imdi 394, ordi 1003, difindi 4232, ndmovdistr 7322
difclass difference df-dif 3911 (𝐴𝐵) Yes difss 4083, difindi 4232
divdivision df-div 11287 (𝐴 / 𝐵) Yes divcl 11293, divval 11289, divmul 11290
dmdomain df-dm 5542 dom 𝐴 Yes dmmpt 6072, iswrddm0 13881
e, eq, equequals (equ for setvars, eq for classes) df-cleq 2815 𝐴 = 𝐵 Yes 2p2e4 11760, uneqri 4102, equtr 2028
edgedge df-edg 26839 (Edg‘𝐺) Yes edgopval 26842, usgredgppr 26984
elelement of 𝐴𝐵 Yes eldif 3918, eldifsn 4693, elssuni 4843
enequinumerous df-en 𝐴𝐵 Yes domen 8509, enfi 8722
eu"there exists exactly one" eu6 2658 ∃!𝑥𝜑 Yes euex 2661, euabsn 4636
exexists (i.e. is a set) ∈ V No brrelex1 5582, 0ex 5187
ex, e"there exists (at least one)" df-ex 1782 𝑥𝜑 Yes exim 1835, alex 1827
expexport No expt 180, expcom 417
f"not free in" (suffix) No equs45f 2483, sbf 2272
ffunction df-f 6338 𝐹:𝐴𝐵 Yes fssxp 6515, opelf 6520
falfalse df-fal 1551 Yes bifal 1554, falantru 1573
fifinite intersection df-fi 8863 (fi‘𝐵) Yes fival 8864, inelfi 8870
fi, finfinite df-fin 8500 Fin Yes isfi 8520, snfi 8581, onfin 8698
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 35388) df-field 19496 Field Yes isfld 19502, fldidom 20069
fnfunction with domain df-fn 6337 𝐴 Fn 𝐵 Yes ffn 6494, fndm 6434
frgpfree group df-frgp 18827 (freeGrp‘𝐼) Yes frgpval 18875, frgpadd 18880
fsuppfinitely supported function df-fsupp 8822 𝑅 finSupp 𝑍 Yes isfsupp 8825, fdmfisuppfi 8830, fsuppco 8853
funfunction df-fun 6336 Fun 𝐹 Yes funrel 6351, ffun 6497
fvfunction value df-fv 6342 (𝐹𝐴) Yes fvres 6671, swrdfv 14001
fzfinite set of sequential integers df-fz 12886 (𝑀...𝑁) Yes fzval 12887, eluzfz 12897
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 13000, fz0tp 13003
fzohalf-open integer range df-fzo 13029 (𝑀..^𝑁) Yes elfzo 13035, elfzofz 13048
gmore general (suffix); eliminates "is a set" hypotheses No uniexg 7451
grgraph No uhgrf 26853, isumgr 26886, usgrres1 27103
grpgroup df-grp 18097 Grp Yes isgrp 18100, tgpgrp 22681
gsumgroup sum df-gsum 16707 (𝐺 Σg 𝐹) Yes gsumval 17878, gsumwrev 18485
hashsize (of a set) df-hash 13687 (♯‘𝐴) Yes hashgval 13689, hashfz1 13702, hashcl 13713
hbhypothesis builder (prefix) No hbxfrbi 1826, hbald 2175, hbequid 36163
hm(monoid, group, ring) homomorphism No ismhm 17949, isghm 18349, isrhm 19467
iinference (suffix) No eleq1i 2904, tcsni 9173
iimplication (suffix) No brwdomi 9020, infeq5i 9087
ididentity No biid 264
iedgindexed edge df-iedg 26790 (iEdg‘𝐺) Yes iedgval0 26831, edgiedgb 26845
idmidempotent No anidm 568, tpidm13 4666
im, impimplication (label often omitted) df-im 14451 (𝐴𝐵) Yes iman 405, imnan 403, impbidd 213
imaimage df-ima 5545 (𝐴𝐵) Yes resima 5865, imaundi 5986
impimport No biimpa 480, impcom 411
inintersection df-in 3915 (𝐴𝐵) Yes elin 3924, incom 4152
infinfimum df-inf 8895 inf(ℝ+, ℝ*, < ) Yes fiinfcl 8953, infiso 8960
is...is (something a) ...? No isring 19292
jjoining, disjoining No jc 164, jaoi 854
lleft No olcd 871, simpl 486
mapmapping operation or set exponentiation df-map 8395 (𝐴m 𝐵) Yes mapvalg 8403, elmapex 8414
matmatrix df-mat 21011 (𝑁 Mat 𝑅) Yes matval 21014, matring 21046
mdetdeterminant (of a square matrix) df-mdet 21188 (𝑁 maDet 𝑅) Yes mdetleib 21190, mdetrlin 21205
mgmmagma df-mgm 17843 Magma Yes mgmidmo 17861, mgmlrid 17868, ismgm 17844
mgpmultiplicative group df-mgp 19231 (mulGrp‘𝑅) Yes mgpress 19241, ringmgp 19294
mndmonoid df-mnd 17903 Mnd Yes mndass 17911, mndodcong 18661
mo"there exists at most one" df-mo 2622 ∃*𝑥𝜑 Yes eumo 2662, moim 2626
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mpomaps-to notation for an operation df-mpo 7145 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpompt 7250, resmpo 7256
mptmodus ponendo tollens No mptnan 1770, mptxor 1771
mptmaps-to notation for a function df-mpt 5123 (𝑥𝐴𝐵) Yes fconstmpt 5591, resmpt 5883
mpt2maps-to notation for an operation (deprecated). We are in the process of replacing mpt2 with mpo in labels. df-mpo 7145 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpompt 7250, resmpo 7256
mulmultiplication (see "t") df-mul 10538 (𝐴 · 𝐵) Yes mulcl 10610, divmul 11290, mulcom 10612, mulass 10614
n, notnot ¬ 𝜑 Yes nan 828, notnotr 132
nenot equaldf-ne 𝐴𝐵 Yes exmidne 3021, neeqtrd 3080
nelnot element ofdf-nel 𝐴𝐵 Yes neli 3117, nnel 3124
ne0not equal to zero (see n0) ≠ 0 No negne0d 10984, ine0 11064, gt0ne0 11094
nf "not free in" (prefix) No nfnd 1859
ngpnormed group df-ngp 23188 NrmGrp Yes isngp 23200, ngptps 23206
nmnorm (on a group or ring) df-nm 23187 (norm‘𝑊) Yes nmval 23194, subgnm 23237
nnpositive integers df-nn 11626 Yes nnsscn 11630, nncn 11633
nn0nonnegative integers df-n0 11886 0 Yes nnnn0 11892, nn0cn 11895
n0not the empty set (see ne0) ≠ ∅ No n0i 4271, vn0 4276, ssn0 4326
OLDold, obsolete (to be removed soon) No 19.43OLD 1884
onordinal number df-on 6173 𝐴 ∈ On Yes elon 6178, 1on 8096 onelon 6194
opordered pair df-op 4546 𝐴, 𝐵 Yes dfopif 4773, opth 5345
oror df-or 845 (𝜑𝜓) Yes orcom 867, anor 980
otordered triple df-ot 4548 𝐴, 𝐵, 𝐶 Yes euotd 5380, fnotovb 7190
ovoperation value df-ov 7143 (𝐴𝐹𝐵) Yes fnotovb 7190, fnovrn 7308
pplus (see "add"), for all-constant theorems df-add 10537 (3 + 2) = 5 Yes 3p2e5 11776
pfxprefix df-pfx 14024 (𝑊 prefix 𝐿) Yes pfxlen 14036, ccatpfx 14054
pmPrincipia Mathematica No pm2.27 42
pmpartial mapping (operation) df-pm 8396 (𝐴pm 𝐵) Yes elpmi 8412, pmsspw 8428
prpair df-pr 4542 {𝐴, 𝐵} Yes elpr 4562, prcom 4642, prid1g 4670, prnz 4686
prm, primeprime (number) df-prm 16005 Yes 1nprm 16012, dvdsprime 16020
pssproper subset df-pss 3927 𝐴𝐵 Yes pssss 4047, sspsstri 4054
q rational numbers ("quotients") df-q 12337 Yes elq 12338
rright No orcd 870, simprl 770
rabrestricted class abstraction df-rab 3139 {𝑥𝐴𝜑} Yes rabswap 3464, df-oprab 7144
ralrestricted universal quantification df-ral 3135 𝑥𝐴𝜑 Yes ralnex 3224, ralrnmpo 7273
rclreverse closure No ndmfvrcl 6683, nnarcl 8229
rereal numbers df-r 10536 Yes recn 10616, 0re 10632
relrelation df-rel 5539 Rel 𝐴 Yes brrelex1 5582, relmpoopab 7776
resrestriction df-res 5544 (𝐴𝐵) Yes opelres 5837, f1ores 6611
reurestricted existential uniqueness df-reu 3137 ∃!𝑥𝐴𝜑 Yes nfreud 3353, reurex 3404
rexrestricted existential quantification df-rex 3136 𝑥𝐴𝜑 Yes rexnal 3226, rexrnmpo 7274
rmorestricted "at most one" df-rmo 3138 ∃*𝑥𝐴𝜑 Yes nfrmod 3354, nrexrmo 3408
rnrange df-rn 5543 ran 𝐴 Yes elrng 5739, rncnvcnv 5781
rng(unital) ring df-ring 19290 Ring Yes ringidval 19244, isring 19292, ringgrp 19293
rotrotation No 3anrot 1097, 3orrot 1089
seliminates need for syllogism (suffix) No ancoms 462
sb(proper) substitution (of a set) df-sb 2070 [𝑦 / 𝑥]𝜑 Yes spsbe 2088, sbimi 2079
sbc(proper) substitution of a class df-sbc 3748 [𝐴 / 𝑥]𝜑 Yes sbc2or 3756, sbcth 3762
scascalar df-sca 16572 (Scalar‘𝐻) Yes resssca 16641, mgpsca 19237
simpsimple, simplification No simpl 486, simp3r3 1280
snsingleton df-sn 4540 {𝐴} Yes eldifsn 4693
spspecialization No spsbe 2088, spei 2413
sssubset df-ss 3925 𝐴𝐵 Yes difss 4083
structstructure df-struct 16476 Struct Yes brstruct 16483, structfn 16491
subsubtract df-sub 10861 (𝐴𝐵) Yes subval 10866, subaddi 10962
supsupremum df-sup 8894 sup(𝐴, 𝐵, < ) Yes fisupcl 8921, supmo 8904
suppsupport (of a function) df-supp 7818 (𝐹 supp 𝑍) Yes ressuppfi 8847, mptsuppd 7840
swapswap (two parts within a theorem) No rabswap 3464, 2reuswap 3712
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 4193, cnvsym 5952
symgsymmetric group df-symg 18487 (SymGrp‘𝐴) Yes symghash 18497, pgrpsubgsymg 18528
t times (see "mul"), for all-constant theorems df-mul 10538 (3 · 2) = 6 Yes 3t2e6 11791
th, t theorem No nfth 1803, sbcth 3762, weth 9906, ancomst 468
tptriple df-tp 4544 {𝐴, 𝐵, 𝐶} Yes eltpi 4599, tpeq1 4652
trtransitive No bitrd 282, biantr 805
tru, t true, truth df-tru 1541 Yes bitru 1547, truanfal 1572, biimt 364
ununion df-un 3913 (𝐴𝐵) Yes uneqri 4102, uncom 4104
unitunit (in a ring) df-unit 19386 (Unit‘𝑅) Yes isunit 19401, nzrunit 20031
v setvar (especially for specializations of theorems when a class is replaced by a setvar variable) x Yes cv 1537, vex 3472, velpw 4516, vtoclf 3533
v disjoint variable condition used in place of nonfreeness hypothesis (suffix) No spimv 2409
vtx vertex df-vtx 26789 (Vtx‘𝐺) Yes vtxval0 26830, opvtxov 26796
vv two disjoint variable conditions used in place of nonfreeness hypotheses (suffix) No 19.23vv 1944
wweak (version of a theorem) (suffix) No ax11w 2134, spnfw 1984
wrdword df-word 13858 Word 𝑆 Yes iswrdb 13863, wrdfn 13871, ffz0iswrd 13884
xpcross product (Cartesian product) df-xp 5538 (𝐴 × 𝐵) Yes elxp 5555, opelxpi 5569, xpundi 5597
xreXtended reals df-xr 10668 * Yes ressxr 10674, rexr 10676, 0xr 10677
z integers (from German "Zahlen") df-z 11970 Yes elz 11971, zcn 11974
zn ring of integers mod 𝑁 df-zn 20198 (ℤ/nℤ‘𝑁) Yes znval 20225, zncrng 20234, znhash 20248
zringring of integers df-zring 20162 ring Yes zringbas 20167, zringcrng 20163
0, z slashed zero (empty set) df-nul 4266 Yes n0i 4271, vn0 4276; snnz 4685, prnz 4686

(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

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 28183 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., "&eacute;" for "é"). Default indentation is by two spaces. Lines are hard-wrapped to be at most 79-character long, excluding the newline character (this can be achieved, except currently for section comments, by the Metamath program "MM> WRITE SOURCE set.mm / REWRAP" command or by running the script scripts/rewrap). The file ends with an empty line. There are no trailing spaces. As for line wrapping in statements, we try to break lines before the most important token.

• Language and spelling.

The MPE uses American English, e.g., we write "neighborhood" instead of the British English "neighbourhood". An exception is the word "analog", which can be either a noun or an adjective (furthermore, "analog" has the confounding meaning "not digital"); therefore, "analogue" is used for the noun and "analogous" for the adjective. We favor regular plurals, e.g., "formulas" instead of "formulae", "lemmas" instead of "lemmata".

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

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

• Quotation style.

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

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.

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

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

• Contributors.

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

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

• Usage of parentheticals.

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

• Explaining new labels.

A comment should explain the first use of an abbreviation within a label. This is often in a definition (e.g., the definition df-an 400 introduces the abbreviation "an" for conjunction ("and")), but not always (e.g., the theorem alim 1812 introduces the abbreviation "al" for the universal quantifier ("for all")). See conventions-labels 28184 for a table of abbreviations.

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

𝜑       𝜑

17.1.2  Natural deduction

Theoremnatded 28186 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.

IT Γ𝜓 => Γ𝜓 idi 1 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
I Γ𝜓 & Γ𝜒 => Γ𝜓𝜒 jca 515 jca 515, pm3.2i 474 Definition I in [Pfenning] p. 18, definition Im,n in [Clemente] p. 10, and definition I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
EL Γ𝜓𝜒 => Γ𝜓 simpld 498 simpld 498, adantr 484 Definition EL in [Pfenning] p. 18, definition E(1) in [Clemente] p. 11, and definition E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
ER Γ𝜓𝜒 => Γ𝜒 simprd 499 simpr 488, adantl 485 Definition ER in [Pfenning] p. 18, definition E(2) in [Clemente] p. 11, and definition E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
I Γ, 𝜓𝜒 => Γ𝜓𝜒 ex 416 ex 416 Definition I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition I in [Indrzejczak] p. 33.
E Γ𝜓𝜒 & Γ𝜓 => Γ𝜒 mpd 15 ax-mp 5, mpd 15, mpdan 686, imp 410 Definition E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition E in [Indrzejczak] p. 33.
IL Γ𝜓 => Γ𝜓𝜒 olcd 871 olc 865, olci 863, olcd 871 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 870 orc 864, orci 862, orcd 870 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 956 mpjaodan 956, jaodan 955, jaod 856 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1558 pm2.01d 193
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 815 mtand 815 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 816 pm2.65da 816 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 798 pm2.01d 193, pm2.65da 816, pm2.65d 199 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1560 pm2.21dd 198
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 198 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 198 pm2.21dd 198, pm2.21d 121, pm2.21 123 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ trud 1548 tru 1542, trud 1548, mptru 1545 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1556 falim 1555 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1928 alrimiv 1928, ralrimiva 3174 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3761 spcv 3581, rspcv 3593 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3839 spcev 3582, rspcev 3598 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1936 exlimddv 1936, exlimdd 2221, exlimdv 1934, rexlimdva 3270 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1559 efald 1559 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 799 pm2.18da 799, pm2.18d 127, pm2.18 128 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 135 notnotrd 135, notnotr 132 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 893 exmid 892 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2823 eqid 2822, eqidd 2823 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3753 sbceq1d 3752, equality theorems Eliminate equality, definition =E in [Pfenning] p. 127. (Both E1 and E2.)

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

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

For annotated examples where some traditional ND rules are directly applied in MPE, see ex-natded5.2 28187, ex-natded5.3 28190, ex-natded5.5 28193, ex-natded5.7 28194, ex-natded5.8 28196, ex-natded5.13 28198, ex-natded9.20 28200, and ex-natded9.26 28202.

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

𝜑       𝜑

17.1.3  Natural deduction examples

These are examples of how natural deduction rules can be applied in Metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 28186 and mmnatded.html 28186. Since these examples should not be used within proofs of other theorems, especially in mathboxes, they are marked with "(New usage is discouraged.)".

Theoremex-natded5.2 28187 Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15 ((𝜓𝜒) → 𝜃) (𝜑 → ((𝜓𝜒) → 𝜃)) Given \$e.
22 (𝜒𝜓) (𝜑 → (𝜒𝜓)) Given \$e.
31 𝜒 (𝜑𝜒) Given \$e.
43 𝜓 (𝜑𝜓) E 2,3 mpd 15, the MPE equivalent of E, 1,2
54 (𝜓𝜒) (𝜑 → (𝜓𝜒)) I 4,3 jca 515, the MPE equivalent of I, 3,1
66 𝜃 (𝜑𝜃) E 1,5 mpd 15, the MPE equivalent of E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. Below is the final Metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 28188. A proof without context is shown in ex-natded5.2i 28189. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → ((𝜓𝜒) → 𝜃))    &   (𝜑 → (𝜒𝜓))    &   (𝜑𝜒)       (𝜑𝜃)

Theoremex-natded5.2-2 28188 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 28187 and ex-natded5.2i 28189. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → 𝜃))    &   (𝜑 → (𝜒𝜓))    &   (𝜑𝜒)       (𝜑𝜃)

Theoremex-natded5.2i 28189 The same as ex-natded5.2 28187 and ex-natded5.2-2 28188 but with no context. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝜓𝜒) → 𝜃)    &   (𝜒𝜓)    &   𝜒       𝜃

Theoremex-natded5.3 28190 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 28191. A proof without context is shown in ex-natded5.3i 28192. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e; adantr 484 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given \$e; adantr 484 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 488, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 484 was used to transform its dependency (we could also use imp 410 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 484 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 515, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 416, the MPE equivalent of I, 8

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

(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremex-natded5.3-2 28191 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 28190 and ex-natded5.3i 28192. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜒𝜃))       (𝜑 → (𝜓 → (𝜒𝜃)))

Theoremex-natded5.3i 28192 The same as ex-natded5.3 28190 and ex-natded5.3-2 28191 but with no context. Identical to jccir 525, which should be used instead. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜓𝜒)    &   (𝜒𝜃)       (𝜓 → (𝜒𝜃))

Theoremex-natded5.5 28193 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e; adantr 484 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given \$e; we'll use adantr 484 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 488
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 484 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 816 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 484; simpr 488 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 201; a proof without context is shown in mto 200.

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

(𝜑 → (𝜓𝜒))    &   (𝜑 → ¬ 𝜒)       (𝜑 → ¬ 𝜓)

Theoremex-natded5.7 28194 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 28195. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16 (𝜓 ∨ (𝜒𝜃)) (𝜑 → (𝜓 ∨ (𝜒𝜃))) Given \$e. No need for adantr 484 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 488
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 870, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 488
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 498, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 871, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 956, the MPE equivalent of E, 2,5,6

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

(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremex-natded5.7-2 28195 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 28194. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremex-natded5.8 28196 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11 ((𝜓𝜒) → ¬ 𝜃) (𝜑 → ((𝜓𝜒) → ¬ 𝜃)) Given \$e; adantr 484 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given \$e; adantr 484 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given \$e; adantr 484 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given \$e. adantr 484 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 488. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 515 (I), 6,8 (adantr 484 to bring in scope)
75 ... ¬ 𝜃 ((𝜑𝜓) → ¬ 𝜃) E 1,6 mpd 15 (E), 2,4
812 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 ¬ 𝜓 (𝜑 → ¬ 𝜓) ¬I 5,7,8 pm2.65da 816 (¬I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 484; simpr 488 is useful when you want to depend directly on the new assumption). Below is the final Metamath proof (which reorders some steps).

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.8-2 28197.

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

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

Theoremex-natded5.8-2 28197 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 28196. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ((𝜓𝜒) → ¬ 𝜃))    &   (𝜑 → (𝜏𝜃))    &   (𝜑𝜒)    &   (𝜑𝜏)       (𝜑 → ¬ 𝜓)

Theoremex-natded5.13 28198 Theorem 5.13 of [Clemente] p. 20, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 28199. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given \$e. adantr 484 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given \$e. ad2antrr 725 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 488
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 870 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 488
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 488
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 484 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 816 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 135 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 871 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 956 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 484; simpr 488 is useful when you want to depend directly on the new assumption). (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))

Theoremex-natded5.13-2 28199 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 28198. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓𝜒))    &   (𝜑 → (𝜓𝜃))    &   (𝜑 → (¬ 𝜏 → ¬ 𝜒))       (𝜑 → (𝜃𝜏))

Theoremex-natded9.20 28200 Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11 (𝜓 ∧ (𝜒𝜃)) (𝜑 → (𝜓 ∧ (𝜒𝜃))) Given \$e
22 𝜓 (𝜑𝜓) EL 1 simpld 498 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 499 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 488
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 515 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 870 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 488
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 515 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 871 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 956 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 484; simpr 488 is useful when you want to depend directly on the new assumption). Below is the final Metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 28201. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))

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