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Theorem List for Metamath Proof Explorer - 27201-27300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnumclwlk1lem2 27201* There is a bijection between the set of closed walks (having a fixed length greater than 2 and starting at a fixed vertex) with the last but 2 vertex identical with the first (and therefore last) vertex and the set of closed walks (having a fixed length less by 2 and starting at the same vertex) and the neighbors of this vertex. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 29-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝐺 ∈ USGraph ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐶𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (𝐺 NeighbVtx 𝑋)))

Theoremnumclwwlk1 27202* 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 only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 29-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       (((𝑉 ∈ Fin ∧ 𝐺 RegUSGraph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐶𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlkovq 27203* 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). (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 27204* In a 𝐾-regular graph, the size of the set of walks of length 𝑛 starting with a fixed vertex 𝑣 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 starting and ending at this vertex 𝑣. (Contributed by Alexander van der Vekens, 30-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)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})       (((𝐺 RegUSGraph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))

Theoremnumclwwlkovh 27205* Value of operation 𝐻, mapping a vertex 𝑣 and a positive integer 𝑛 to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.) (Revised by AV, 30-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})

Theoremnumclwwlk2lem1 27206* 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.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Theoremnumclwlk2lem2f 27207* 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.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))

Theoremnumclwlk2lem2fv 27208* Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 31-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))

Theoremnumclwlk2lem2f1o 27209* R is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Theoremnumclwwlk2lem3 27210* 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.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝐺 ∈ FriendGraph ∧ 𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2))))

Theoremnumclwwlk2 27211* 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 26853, 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.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((𝐺 RegUSGraph 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlk3lem 27212* Lemma for numclwwlk3 27213. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 1-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       (((𝐺 ∈ FinUSGraph ∧ 𝑋𝑉) ∧ 𝑁 ∈ (ℤ‘2)) → (#‘(𝑋𝐹𝑁)) = ((#‘(𝑋𝐻𝑁)) + (#‘(𝑋𝐶𝑁))))

Theoremnumclwwlk3 27213* 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 27202) and with v(n-2) =/= v(n) ( see numclwwlk2 27211): 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, 1-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝐶 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       (((𝐺 RegUSGraph 𝐾𝐺 ∈ FriendGraph ) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Theoremnumclwwlk4 27214* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})       ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ) → (#‘(𝑁 ClWWalksN 𝐺)) = Σ𝑥𝑉 (#‘(𝑥𝐹𝑁)))

Theoremnumclwwlk5lem 27215* Lemma for numclwwlk5 27216. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 2-Jun-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})       ((𝐺 RegUSGraph 𝐾𝑋𝑉𝐾 ∈ ℕ0) → (2 ∥ (𝐾 − 1) → ((#‘(𝑋𝐹2)) mod 2) = 1))

Theoremnumclwwlk5 27216* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})       (((𝐺 RegUSGraph 𝐾𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑋𝑉𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑋𝐹𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk7lem 27217 Lemma for numclwwlk7 27219, frgrreggt1 27221 and frgrreg 27222: If a finite, non-empty 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 27218 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.)
𝑉 = (Vtx‘𝐺)       (((𝐺 RegUSGraph 𝐾𝐺 ∈ FriendGraph ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = ((#‘𝑉) mod 𝑃))

Theoremnumclwwlk7 27219 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 27178 or frrusgrord 27179, 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 27108, as well as k-regular (for any k), see 0vtxrgr 26453, but has no closed walk, see 0clwlk0 26972, 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 27220 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 26952. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 3-Jun-2021.)
((𝐺 ∈ FinUSGraph ∧ 𝑃 ∈ ℙ) → ((#‘(𝑃 ClWWalksN 𝐺)) mod 𝑃) = 0)

Theoremfrgrreggt1 27221 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 27222 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 27223 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 27224 If a nonempty finite friendship graph is 𝐾-regular, then it must have order 1 or 3. Special case of frgrregord013 27223. (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 27225* 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 27226* 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 27227* 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. For the general conventions, see conventions 27228, and for conventions related to labels, see conventions-label 27229. 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 27228

Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they are identical). For conventions related to labels, see conventions-label 27229.

• 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 14398) 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 14595). 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 197, df-cleq 2613, df-clel 2616, df-clab 2607) that require a more complex metalogical justification by hand.
• Proven axioms. In some cases we wish to treat an expression as an axiom in later theorems, even though it can be proved. For example, we derive the postulates or axioms of complex arithmetic as theorems of ZFC set theory. For convenience, after deriving the postulates, we reintroduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. For more, see mmcomplex.html. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is complex arithmetic axiom ax-1cn 9979, proven by the preceding theorem ax1cn 9955. The metamath.exe program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
• Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
• Statements with hypotheses. Many theorems and some axioms, such as ax-mp 5, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When presented in summarized form such as in the Theorem List (click on "Nearby theorems" on the ax-mp 5 page), the hypotheses are connected with an ampersand and separated from the conclusion with a big arrow, such as in " 𝜑 & (𝜑𝜓) => 𝜓". These symbols are _not_ part of the Metamath language but are just informal notation meaning "and" and "implies".
• Discouraged use and modification. If something should only be used in limited ways, it is marked with "(New usage is discouraged.)". This is used, for example, when something can be constructed in more than one way, and we do not want later theorems to depend on that specific construction. This marking is also used if we want later proofs to use proven axioms. For example, we want later proofs to use ax-1cn 9979 (not ax1cn 9955) and ax-1ne0 9990 (not ax1ne0 9966), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9955 and ax1ne0 9966 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".
• New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use < and for inequality expressions, and use ((sin‘(i · 𝐴)) / i) instead of (sinh‘𝐴) for the hyperbolic sine.
• Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, we prefer proofs that do not use the axiom of choice (df-ac 8924) where such proofs can be found. The axiom of choice is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 8065) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 9242) or axiom of dependent choice (ax-dc 9253) can be used instead. Similarly, any theorem in first order logic (FOL) that contains only set variables that are all mutually distinct, and has no wff variables, can be proved *without* using ax-10 2017 through ax-13 2244, by invoking ax10w 2004 through ax13w 2011. We encourage proving theorems *without* ax-10 2017 through ax-13 2244 and moving them up to the ax-4 1735 through ax-9 1997 section.
• Alternative (ALT) proofs. If a different proof is significantly shorter or clearer but uses more or stronger axioms, we prefer to make that proof an "alternative" proof (marked with an ALT label suffix), even if this alternative proof was formalized first. We then make the proof that requires fewer axioms the main proof. This has the effect of reducing (over time) the number and strength of axioms used by any particular proof. There can be multiple alternatives if it makes sense to do so. Alternative (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternative and main statement comments should use hyperlinks to refer to each other (so that a reader of one will become easily aware of the other).
• Alternative (ALTV) versions. If a theorem or definition is an alternative/variant of an already existing theorem resp. definition, its label should have the same name with suffix ALTV. Such alternatives should be temporary only, until it is decided which alternative should be used in the future. Alternative (*ALTV) theorems or definitions are usually contained in mathboxes. Their comments need not to contain "(Proof modification is discouraged.) (New usage is discouraged.)". Alternative statements should follow the main statement, so that people reading the text in order will see the main statement first.
• Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first. This typically happens when a shorter proof to an existing theorem is found: the existing theorem is kept as an *OLD statement for one year. When a proof is shortened automatically (using Metamath's minimize_with command), then it is not necessary to keep the old proof, nor to add credit for the shortening.
• Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are normally used in this order: 𝜑 = phi, 𝜓 = psi, 𝜒 = chi, 𝜃 = theta, 𝜏 = tau, 𝜂 = eta, 𝜁 = zeta, and 𝜎 = sigma. Individual setvar variables are represented with lowercase Latin letters and are normally used in this order: 𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, and 𝑡. Variables that represent classes are often represented by uppercase Latin letters: 𝐴, 𝐵, 𝐶, 𝐷, 𝐸, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g., 0 for poset zero (see p0val 17022) and connective symbols such as + for some group addition operation. (See prdsplusgval 16114 for an example of the use of +). Class variables are selected in alphabetical order starting from 𝐴 if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.
• Turnstile. "", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff ¬ 𝜑".
• Biconditional (). There are basically two ways to maximize the effectiveness of biconditionals (): you can either have one-directional simplifications of all theorems that produce biconditionals, or you can have one-directional simplifications of theorems that consume biconditionals. Some tools (like Lean) follow the first approach, but set.mm follows the second approach. Practically, this means that in set.mm, for every theorem that uses an implication in the hypothesis, like ax-mp 5, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 220 or mpbir 221. 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 206 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 954, sylbir 225, or 3imtr4i 281.
• Substitution. "[𝑦 / 𝑥]𝜑" should be read "the wff that results from the proper substitution of 𝑦 for 𝑥 in wff 𝜑." See df-sb 1879 and the related df-sbc 3430 and df-csb 3527.
• 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 3197 and isset 3202. 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 3207 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 4800). 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 5112. This can be used to define a subset, e.g., df-tan 14783 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 14861). 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 5884. 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 6638). For example, the + in (2 + 2); see 2p2e4 11129. 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 385, df-an 386, and df-bi 197 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 1989), equality 𝐴 = 𝐵 (see df-cleq 2613), subset 𝐴𝐵 (see df-ss 3581), and less-than 𝐴 < 𝐵 (see df-lt 9934). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 4645. For example, 0 < 1 (see 0lt1 10535) 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 10252.
• 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 14776). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 14782). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 14838), its function application form labeled NAMEval (e.g., cosval 14834), and at least one simple value (e.g., cos0 14861).
• 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 13044 and fac4 13051).
• 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 9928), while "0g" is the group identity element (df-0g 16083), "0." is the poset zero (df-p0 17020), "0𝑝" is the zero polynomial (df-0p 23418), "0vec" is the zero vector in a normed subcomplex vector space (df-0v 27423), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 17022). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including "1", "+", "", and "". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.
• ASCII representation of symbols. We must have an ASCII representation for each symbol. We generally choose short sequences, ideally digraphs, and generally choose sequences that vaguely resemble the mathematical symbol. Here are some of the conventions we use when selecting an ASCII representation.
We generally do not include parentheses inside a symbol because that confuses text editors (such as emacs). Greek letters for wff variables always use the first two letters of their English names, making them easy to type and easy to remember. Symbols that almost look like letters, such as , are often represented by that letter followed by a period. For example, "A." is used to represent , "e." is used to represent , and "E." is used to represent . Single letters are now always variable names, so constants that are often shown as single letters are now typically preceded with "_" in their ASCII representation, for example, "_i" is the ASCII representation for the imaginary unit i. A script font constant is often the letter preceded by "~" meaning "curly", such as "~P" to represent the power class 𝒫.
Originally, all setvar and class variables used only single letters a-z and A-Z, respectively. A big change in recent years was to allow the use of certain symbols as variable names to make formulas more readable, such as a variable representing an additive group operation. The convention is to take the original constant token (in this case "+" which means complex number addition) and put a period in front of it to result in the ASCII representation of the variable ".+", shown as +, that can be used instead of say the letter "P" that had to be used before.
Choosing tokens for more advanced concepts that have no standard symbols but are represented by words in books, is hard. A few are reasonably obvious, like "Grp" for group and "Top" for topology, but often they seem to end up being either too long or too cryptic. It would be nice if the math community came up with standardized short abbreviations for English math terminology, like they have more or less done with symbols, but that probably won't happen any time soon.
Another informal convention that we've somewhat followed, that is also not uncommon in the literature, is to start tokens with a capital letter for collection-like objects and lower case for function-like objects. For example, we have the collections On (ordinal numbers), Fin, Prime, Grp, and we have the functions sin, tan, log, sup. Predicates like Ord and Lim also tend to start with upper case, but in a sense they are really collection-like, e.g. Lim indirectly represents the collection of limit ordinals, but it can't be an actual class since not all limit ordinals are sets. This initial capital vs. lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).
• Typography conventions. Class symbols for functions (e.g., abs, sin) should usually not have leading or trailing blanks in their HTML/Latex representation. This is in contrast to class symbols for operations (e.g., gcd, sadd, eval), which usually do include leading and trailing blanks in their representation. If a class symbol is used for a function as well as an operation (according to the definition df-ov 6638, each operation value can be written as function value of an ordered pair), the convention for its primary usage should be used, e.g. (iEdg‘𝐺) versus (𝑉iEdg𝐸) for the edges of a graph 𝐺 = ⟨𝑉, 𝐸.
• Number construction independence. There are many ways to model complex numbers. After deriving the complex number postulates we reintroduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. This also lets us be independent of the specific construction, which we believe is valuable. See mmcomplex.html for details. Thus, for example, we don't allow the use of ∅ ∉ ℂ, as handy as that would be, because that would be construction-specific. We want proofs about to be independent of whether or not ∅ ∈ ℂ.
• Minimize hypotheses (except for construction independence and number theorem domains). In most cases we try to minimize hypotheses, that is, we eliminate or reduce what must be true to prove something, so that the proof is more general and easier to use. There are exceptions. For example, we intentionally add hypotheses if they help make proofs independent of a particular construction (e.g., the contruction of complex numbers ). We also intentionally add hypotheses for many real and complex number theorems to expressly state their domains even when they aren't strictly needed. For example, we could show that (𝐴 < 𝐵𝐵𝐴) without any other hypotheses, but in practice we also require proving at least some domains (e.g., see ltnei 10146). Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ:
1. Having the hypotheses immediately shows the intended domain of applicability (is it , *, ω, or something else?), without having to trace back to definitions.
2. Having the hypotheses forces its use in the intended domain, which generally is desirable.
3. The behavior is dependent on accidental behavior of definitions outside of their domains, so the theorems are non-portable and "brittle".
4. Only a few theorems can have their hypotheses removed in this fashion due to happy coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who hasn't traced back the set-theoretical foundations of the definitions, it is seemingly random and isn't intuitive at all.
5. The consensus of opinion of people on this group seemed to be against doing this.
• Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 11006) for the set of positive integers starting from 1, and 0 (df-n0 11278) 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 11479, e.g., 4001 (see 4001prm 15833 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 3779). 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 23760) 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 3781. 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 2621, 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 3780).
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 150 is the inference associated with con3 149). 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). When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the closed form with "g" (for "more general") as in uniex 6938 vs. uniexg 6940.
• Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction (its associated inference). See for instance Theorem 3 in [Margaris] p. 56. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see mmdeduction.html. The Deduction Theorem is a metalogical theorem that cannot be applied directly in metamath, and the explosion of steps would be a problem anyway, so alternatives are used. One alternative we use sometimes is the "weak deduction theorem" dedth 4130, which works in certain cases in set theory. We also sometimes use dedhb 3370. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in deduction form (aka "deduction style") as described earlier; the prefixed 𝜑 mimics the context in a deduction proof system. In practice this mechanism works very well. This approach is described in the deduction form and natural deduction page mmnatded.html; a list of translations for common natural deduction rules is given in natded 27230.
• 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 7453, rec(𝐹, 𝐼) in df-rdg 7491, seq𝜔(𝐹, 𝐼) in df-seqom 7528, and seq𝑀( + , 𝐹) in df-seq 12785. These have characteristic function 𝐹 and initial value 𝐼. (Σg in df-gsum 16084 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7453, but for the "average user" the most useful one is probably df-seq 12785- 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 15840.
• 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 5860 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 6188 on the argument being in the domain of the function. It is impossible to derive a contradiction from sound definitions (i.e. that pass the definition check), assuming ZFC is consistent, and he doesn't see the point of all the extra busy work and huge increase in set.mm size that would result from restricting *all* definitions. So instead of implementing a complex system to counter a problem that does not appear to occur in practice, we use a significantly simpler set of approaches.
• Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 8055 is the first lemma for sbth 8065. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
• Limit proof size. It is often preferable to break longer proofs into smaller parts, just as you would do with traditional proofs. One reason is that humans have trouble understanding long proofs. Another reason is that it's generally best to prove reusable results separately, so that others will be able to easily reuse them. Finally, the "minimize" routine can take much longer with very long proofs. We encourage proofs to be no more than 200 essential steps, and generally no more than 500 essential steps, though these are simply guidelines and not hard-and-fast rules. Much smaller proofs are fine! We also acknowledge that some proofs, especially autogenerated ones, should sometimes not be broken up (e.g., because breaking them up might be useless and inefficient due to many interconnections and reused terms within the proof). In Metamath, breaking up longer proofs is done by creating multiple separate proofs of separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 8055 is the first lemma for sbth 8065.
• 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 15367"). When metamath.exe is used to generate HTML it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertion.
• Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g., mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we _require_ page numbers in comments for individual \$a or \$p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the metamath.exe "write bibliography" command.)
• Acceptable shorter proofs Shorter proofs are welcome, and any shorter proof we accept will be acknowledged in the theorem's description. However, in some cases a proof may be "shorter" or not depending on how it is formatted. This section provides general guidelines.

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

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

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

• Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
• Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.
• Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.
• Language and spelling. It is preferred to use American English for comments and symbols, e.g. we use "neighborhood" instead of the British English "neighbourhood". An exception is the word "analog", which can be either a noun or an adjective. Furthermore, "analog" has the confounding meaning "not digital", whereas "analogue" is often used in the sense something that bears analogy to something else also in American English. Therefore, "analogue" is used for the noun and "analogous" for the adjective in set.mm.
• Comments and layout. 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's "WRITE SOURCE" command with the "REWRAP" option. Here are a few other rules, which are not enforced, but that we try follow:
• The file set.mm should have a double blank line before each section header, and at no other places. In particular, there are no triple blank lines. If there is a "@( Begin \$[ ... \$] @)" comment (where "@" is actually "\$") before the section header, then the double blank line should go before that comment.
• 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.
• Header comments are not rewrapped by the Metamath program [as of 24-Oct-2021], but 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.

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 1708 (whose description has some important technical details). Similarly, 𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2751.
• "\$d x y \$." should be read "Assume x and y are distinct variables."
• "\$d x 𝜑 \$." should be read "Assume x does not occur in phi \$." Sometimes a theorem is proved using 𝑥𝜑 (df-nf 1708) in place of "\$d 𝑥𝜑 \$." when a more general result is desired; ax-5 1837 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 2476 from df-eu 2472.
• "\$d x A \$." should be read "Assume x is not a variable occurring in class A."
• "\$d x A \$. \$d x ps \$. \$e |- (𝑥 = 𝐴 → (𝜑𝜓)) \$." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
• " (¬ ∀𝑥𝑥 = 𝑦 → ..." occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the \$d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the \$d eventually becomes a requirement, and after that this device is rarely used.

There is a general technique to replace a \$d x A or \$d x ph condition in a theorem with the corresponding 𝑥𝐴 or 𝑥𝜑; here it is. T[x, A] where , and you wish to prove 𝑥𝐴 T[x, A]. You apply the theorem substituting 𝑦 for 𝑥 and 𝐴 for 𝐴, where 𝑦 is a new dummy variable, so that \$d y A is satisfied. You obtain T[y, A], and apply chvar to obtain T[x, A] (or just use mpbir 221 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 2269. The section mmtheorems32.html#mm3146s also describes the metatheorem that underlies this.

Standard Metamath verifiers do not distinguish between axioms and definitions (both are \$a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions (\$a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 197, df-clab 2607, df-cleq 2613, and df-clel 2616); 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 1035 or wsb 1878).
2. The defined expression must not appear in any statement between its syntax axiom () and its definition, and the defined expression must not be used in its definiens. See df-3an 1038 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 3196. The definition df-v 3197 V = {𝑥𝑥 = 𝑥} is justified by the justification theorem vjust 3196 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}. Another example of a justification theorem is trujust 1483; the definition df-tru 1484 (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1483 ((∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥) ↔ (∀𝑦𝑦 = 𝑦 → ∀𝑦𝑦 = 𝑦)).

• Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
• Maximum text line length is 79 characters. You can fix comment line length by running the commands scripts/rewrap or metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' quit . As a general rule, a math string in a comment should be surrounded by backquotes on the same line, and if it is too long it should be broken into multiple adjacent mathstrings on multiple lines. Those commands don't modify the math content of statements. In statements we try to break before the outermost important connective (not including the typecode and perhaps not the antecedent). For examples, see sqrtmulii 14107 and absmax 14050.
• Discouraged information. A separate file named "discouraged" lists all discouraged statements and uses of them, and this file is checked. If you change the use of discouraged things, you will need to change this file. This makes it obvious when there is a change to anything discouraged (triggering further review).
• LRParser check. Metamath verifiers ensure that \$p statements follow from previous \$a and \$p statements. However, by itself the Metamath language permits certain kinds of syntactic ambiguity that we choose to avoid in this database. Thus, we require that this database unambiguously parse using the "LRParser" check (implemented by at least mmj2). (For details, see mmj2 master file src/mmj/verify/LRParser.java). This check counters, for example, a devious ambiguous construct developed by saueran at oregonstate dot edu posted on Mon, 11 Feb 2019 17:32:32 -0800 (PST) based on creating definitions with mismatched parentheses.
• Proposing specific changes. Please propose specific changes as pull requests (PRs) against the "develop" branch of set.mm, at: https://github.com/metamath/set.mm/tree/develop

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

𝜑       𝜑

Theoremconventions-label 27229

The following explains some of the label conventions in use in the Metamath Proof Explorer ("set.mm"). For the general conventions, see conventions 27228.

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 2919"rgen.1 \$e |- ( x e. A -> ph ) \$." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 20393: "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 2561 and stirling 40069.
• 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 1764, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 2953.
• 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... 14593. Furthermore, the underscore "_" should not be used.
• Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labeled "df-NAME", and normally NAME is the syntax label fragment. For example, the class difference construct (𝐴𝐵) is defined in df-dif 3570, 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 3581. 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 3729. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4169), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4171). 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 4308. An "n" is often used for negation (¬), e.g., nan 603.
• 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 9927) and "re" represents real numbers ( definition df-r 9931). The empty set often uses fragment 0, even though it is defined in df-nul 3908. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 9932), 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 11129.
• 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 14861 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 14782) we have value cosval 14834 and closure coscl 14838.
• 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 27230 for a list), and that is about all you need for most things. As for the rest, you can just assume that if it involves at most three connectives, then it is probably already proved in set.mm, and searching for it will give you the label.
• Suffixes. Suffixes are used to indicate the form of a theorem (see above). Additionally, we sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as 𝑥𝜑 in 19.21 2073 via the use of disjoint variable conditions combined with nfv 1841. If two (or three) such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used, e.g. exlimivv 1858. 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 2476 derived from df-eu 2472. 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 4908. 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 2279 (cbval 2269 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 3251 or rabeqif 3186. 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 692), commutes (e.g., biimpac 503)
• d : theorem in deduction form
• f : theorem with a hypothesis such as 𝑥𝜑
• g : theorem in closed form having an "is a set" antecedent
• i : theorem in inference form
• l : theorem concerning something at the left
• r : theorem concerning something at the right
• r : theorem with something reversed (e.g., a biconditional)
• s : inference that manipulates an antecedent ("s" refers to an application of syl 17 that is eliminated)
• v : theorem with one (main) disjoint variable condition
• vv : theorem with two (main) disjoint variable conditions
• w : weak(er) form of a theorem
• ALT : alternate proof of a theorem
• ALTV : alternate version of a theorem or definition
• OLD : old/obsolete version of a theorem/definition/proof
• Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic, the source theorem or the assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.
AbbreviationMnenomicSource ExpressionSyntax?Example(s)
aand (suffix) No biimpa 501, rexlimiva 3024
ablAbelian group df-abl 18177 Abel Yes ablgrp 18179, zringabl 19803
absabsorption No ressabs 15920
absabsolute value (of a complex number) df-abs 13957 (abs‘𝐴) Yes absval 13959, absneg 13998, abs1 14018
al"for all" 𝑥𝜑 No alim 1736, alex 1751
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 386 (𝜑𝜓) Yes anor 510, iman 440, imnan 438
assassociative No biass 374, orass 546, mulass 10009
asymasymmetric, antisymmetric No intasym 5499, asymref 5500, posasymb 16933
axaxiom No ax6dgen 2003, ax1cn 9955
bas, base base (set of an extensible structure) df-base 15844 (Base‘𝑆) Yes baseval 15899, ressbas 15911, cnfldbas 19731
b, bibiconditional ("iff", "if and only if") df-bi 197 (𝜑𝜓) Yes impbid 202, sspwb 4908
brbinary relation df-br 4645 𝐴𝑅𝐵 Yes brab1 4691, brun 4694
cbvchange bound variable No cbvalivw 1932, cbvrex 3163
clclosure No ifclda 4111, ovrcl 6671, zaddcl 11402
cncomplex numbers df-c 9927 Yes nnsscn 11010, nncn 11013
cnfldfield of complex numbers df-cnfld 19728 fld Yes cnfldbas 19731, cnfldinv 19758
cntzcentralizer df-cntz 17731 (Cntz‘𝑀) Yes cntzfval 17734, dprdfcntz 18395
cnvconverse df-cnv 5112 𝐴 Yes opelcnvg 5291, f1ocnv 6136
cocomposition df-co 5113 (𝐴𝐵) Yes cnvco 5297, fmptco 6382
comcommutative No orcom 402, bicomi 214, eqcomi 2629
concontradiction, contraposition No condan 834, con2d 129
csbclass substitution df-csb 3527 𝐴 / 𝑥𝐵 Yes csbid 3534, csbie2g 3557
cygcyclic group df-cyg 18261 CycGrp Yes iscyg 18262, zringcyg 19820
ddeduction form (suffix) No idd 24, impbid 202
df(alternate) definition (prefix) No dfrel2 5571, dffn2 6034
di, distrdistributive No andi 910, imdi 378, ordi 907, difindi 3873, ndmovdistr 6808
difclass difference df-dif 3570 (𝐴𝐵) Yes difss 3729, difindi 3873
divdivision df-div 10670 (𝐴 / 𝐵) Yes divcl 10676, divval 10672, divmul 10673
dmdomain df-dm 5114 dom 𝐴 Yes dmmpt 5618, iswrddm0 13312
e, eq, equequals df-cleq 2613 𝐴 = 𝐵 Yes 2p2e4 11129, uneqri 3747, equtr 1946
edgedge df-edg 25921 (Edg‘𝐺) Yes edgopval 25925, usgredgppr 26069
elelement of 𝐴𝐵 Yes eldif 3577, eldifsn 4308, elssuni 4458
eu"there exists exactly one" df-eu 2472 ∃!𝑥𝜑 Yes euex 2492, euabsn 4252
exexists (i.e. is a set) No brrelex 5146, 0ex 4781
ex"there exists (at least one)" df-ex 1703 𝑥𝜑 Yes exim 1759, alex 1751
expexport No expt 168, expcom 451
f"not free in" (suffix) No equs45f 2348, sbf 2378
ffunction df-f 5880 𝐹:𝐴𝐵 Yes fssxp 6047, opelf 6052
falfalse df-fal 1487 Yes bifal 1495, falantru 1506
fifinite intersection df-fi 8302 (fi‘𝐵) Yes fival 8303, inelfi 8309
fi, finfinite df-fin 7944 Fin Yes isfi 7964, snfi 8023, onfin 8136
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 33762) df-field 18731 Field Yes isfld 18737, fldidom 19286
fnfunction with domain df-fn 5879 𝐴 Fn 𝐵 Yes ffn 6032, fndm 5978
frgpfree group df-frgp 18104 (freeGrp‘𝐼) Yes frgpval 18152, frgpadd 18157
fsuppfinitely supported function df-fsupp 8261 𝑅 finSupp 𝑍 Yes isfsupp 8264, fdmfisuppfi 8269, fsuppco 8292
funfunction df-fun 5878 Fun 𝐹 Yes funrel 5893, ffun 6035
fvfunction value df-fv 5884 (𝐹𝐴) Yes fvres 6194, swrdfv 13406
fzfinite set of sequential integers df-fz 12312 (𝑀...𝑁) Yes fzval 12313, eluzfz 12322
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 12421, fz0tp 12424
fzohalf-open integer range df-fzo 12450 (𝑀..^𝑁) Yes elfzo 12456, elfzofz 12469
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 6940
grgraph No uhgrf 25938, isumgr 25971, usgrres1 26188
grpgroup df-grp 17406 Grp Yes isgrp 17409, tgpgrp 21863
gsumgroup sum df-gsum 16084 (𝐺 Σg 𝐹) Yes gsumval 17252, gsumwrev 17777
hashsize (of a set) df-hash 13101 (#‘𝐴) Yes hashgval 13103, hashfz1 13117, hashcl 13130
hbhypothesis builder (prefix) No hbxfrbi 1750, hbald 2039, hbequid 34013
hm(monoid, group, ring) homomorphism No ismhm 17318, isghm 17641, isrhm 18702
iinference (suffix) No eleq1i 2690, tcsni 8604
iimplication (suffix) No brwdomi 8458, infeq5i 8518
ididentity No biid 251
iedgindexed edge df-iedg 25858 (iEdg‘𝐺) Yes iedgval0 25913, edgiedgb 25928
idmidempotent No anidm 675, tpidm13 4282
im, impimplication (label often omitted) df-im 13822 (𝐴𝐵) Yes iman 440, imnan 438, impbidd 200
imaimage df-ima 5117 (𝐴𝐵) Yes resima 5419, imaundi 5533
impimport No biimpa 501, impcom 446
inintersection df-in 3574 (𝐴𝐵) Yes elin 3788, incom 3797
infinfimum df-inf 8334 inf(ℝ+, ℝ*, < ) Yes fiinfcl 8392, infiso 8398
is...is (something a) ...? No isring 18532
jjoining, disjoining No jc 159, jaoi 394
lleft No olcd 408, simpl 473
mapmapping operation or set exponentiation df-map 7844 (𝐴𝑚 𝐵) Yes mapvalg 7852, elmapex 7863
matmatrix df-mat 20195 (𝑁 Mat 𝑅) Yes matval 20198, matring 20230
mdetdeterminant (of a square matrix) df-mdet 20372 (𝑁 maDet 𝑅) Yes mdetleib 20374, mdetrlin 20389
mgmmagma df-mgm 17223 Magma Yes mgmidmo 17240, mgmlrid 17247, ismgm 17224
mgpmultiplicative group df-mgp 18471 (mulGrp‘𝑅) Yes mgpress 18481, ringmgp 18534
mndmonoid df-mnd 17276 Mnd Yes mndass 17283, mndodcong 17942
mo"there exists at most one" df-mo 2473 ∃*𝑥𝜑 Yes eumo 2497, moim 2517
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mptmodus ponendo tollens No mptnan 1691, mptxor 1692
mptmaps-to notation for a function df-mpt 4721 (𝑥𝐴𝐵) Yes fconstmpt 5153, resmpt 5437
mpt2maps-to notation for an operation df-mpt2 6640 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpt2mpt 6737, resmpt2 6743
mulmultiplication (see "t") df-mul 9933 (𝐴 · 𝐵) Yes mulcl 10005, divmul 10673, mulcom 10007, mulass 10009
n, notnot ¬ 𝜑 Yes nan 603, notnotr 125
nenot equaldf-ne 𝐴𝐵 Yes exmidne 2801, neeqtrd 2860
nelnot element ofdf-nel 𝐴𝐵 Yes neli 2896, nnel 2903
ne0not equal to zero (see n0) ≠ 0 No negne0d 10375, ine0 10450, gt0ne0 10478
nf "not free in" (prefix) No nfnd 1783
ngpnormed group df-ngp 22369 NrmGrp Yes isngp 22381, ngptps 22387
nmnorm (on a group or ring) df-nm 22368 (norm‘𝑊) Yes nmval 22375, subgnm 22418
nnpositive integers df-nn 11006 Yes nnsscn 11010, nncn 11013
nn0nonnegative integers df-n0 11278 0 Yes nnnn0 11284, nn0cn 11287
n0not the empty set (see ne0) ≠ ∅ No n0i 3912, vn0 3916, ssn0 3967
OLDold, obsolete (to be removed soon) No 19.43OLD 1809
opordered pair df-op 4175 𝐴, 𝐵 Yes dfopif 4390, opth 4935
oror df-or 385 (𝜑𝜓) Yes orcom 402, anor 510
otordered triple df-ot 4177 𝐴, 𝐵, 𝐶 Yes euotd 4965, fnotovb 6679
ovoperation value df-ov 6638 (𝐴𝐹𝐵) Yes fnotovb 6679, fnovrn 6794
pplus (see "add"), for all-constant theorems df-add 9932 (3 + 2) = 5 Yes 3p2e5 11145
pfxprefix df-pfx 41147 (𝑊 prefix 𝐿) Yes pfxlen 41156, ccatpfx 41174
pmPrincipia Mathematica No pm2.27 42
pmpartial mapping (operation) df-pm 7845 (𝐴pm 𝐵) Yes elpmi 7861, pmsspw 7877
prpair df-pr 4171 {𝐴, 𝐵} Yes elpr 4189, prcom 4258, prid1g 4286, prnz 4301
prm, primeprime (number) df-prm 15367 Yes 1nprm 15373, dvdsprime 15381
pssproper subset df-pss 3583 𝐴𝐵 Yes pssss 3694, sspsstri 3701
q rational numbers ("quotients") df-q 11774 Yes elq 11775
rright No orcd 407, simprl 793
rabrestricted class abstraction df-rab 2918 {𝑥𝐴𝜑} Yes rabswap 3116, df-oprab 6639
ralrestricted universal quantification df-ral 2914 𝑥𝐴𝜑 Yes ralnex 2989, ralrnmpt2 6760
rclreverse closure No ndmfvrcl 6206, nnarcl 7681
rereal numbers df-r 9931 Yes recn 10011, 0re 10025
relrelation df-rel 5111 Rel 𝐴 Yes brrelex 5146, relmpt2opab 7244
resrestriction df-res 5116 (𝐴𝐵) Yes opelres 5390, f1ores 6138
reurestricted existential uniqueness df-reu 2916 ∃!𝑥𝐴𝜑 Yes nfreud 3107, reurex 3155
rexrestricted existential quantification df-rex 2915 𝑥𝐴𝜑 Yes rexnal 2992, rexrnmpt2 6761
rmorestricted "at most one" df-rmo 2917 ∃*𝑥𝐴𝜑 Yes nfrmod 3108, nrexrmo 3158
rnrange df-rn 5115 ran 𝐴 Yes elrng 5303, rncnvcnv 5338
rng(unital) ring df-ring 18530 Ring Yes ringidval 18484, isring 18532, ringgrp 18533
rotrotation No 3anrot 1041, 3orrot 1042
seliminates need for syllogism (suffix) No ancoms 469
sb(proper) substitution (of a set) df-sb 1879 [𝑦 / 𝑥]𝜑 Yes spsbe 1882, sbimi 1884
sbc(proper) substitution of a class df-sbc 3430 [𝐴 / 𝑥]𝜑 Yes sbc2or 3438, sbcth 3444
scascalar df-sca 15938 (Scalar‘𝐻) Yes resssca 16012, mgpsca 18477
simpsimple, simplification No simpl 473, simp3r3 1169
snsingleton df-sn 4169 {𝐴} Yes eldifsn 4308
spspecialization No spsbe 1882, spei 2259
sssubset df-ss 3581 𝐴𝐵 Yes difss 3729
structstructure df-struct 15840 Struct Yes brstruct 15847, structfn 15855
subsubtract df-sub 10253 (𝐴𝐵) Yes subval 10257, subaddi 10353
supsupremum df-sup 8333 sup(𝐴, 𝐵, < ) Yes fisupcl 8360, supmo 8343
suppsupport (of a function) df-supp 7281 (𝐹 supp 𝑍) Yes ressuppfi 8286, mptsuppd 7303
swapswap (two parts within a theorem) No rabswap 3116, 2reuswap 3404
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 3836, cnvsym 5498
symgsymmetric group df-symg 17779 (SymGrp‘𝐴) Yes symghash 17786, pgrpsubgsymg 17809
t times (see "mul"), for all-constant theorems df-mul 9933 (3 · 2) = 6 Yes 3t2e6 11164
ththeorem No nfth 1725, sbcth 3444, weth 9302
tptriple df-tp 4173 {𝐴, 𝐵, 𝐶} Yes eltpi 4220, tpeq1 4268
trtransitive No bitrd 268, biantr 971
trutrue df-tru 1484 Yes bitru 1494, truanfal 1505
ununion df-un 3572 (𝐴𝐵) Yes uneqri 3747, uncom 3749
unitunit (in a ring) df-unit 18623 (Unit‘𝑅) Yes isunit 18638, nzrunit 19248
vdisjoint variable conditions used when a not-free hypothesis (suffix) No spimv 2255
vtxvertex df-vtx 25857 (Vtx‘𝐺) Yes vtxval0 25912, opvtxov 25866
vv2 disjoint variables (in a not-free hypothesis) (suffix) No 19.23vv 1901
wweak (version of a theorem) (suffix) No ax11w 2005, spnfw 1926
wrdword df-word 13282 Word 𝑆 Yes iswrdb 13294, wrdfn 13302, ffz0iswrd 13315
xpcross product (Cartesian product) df-xp 5110 (𝐴 × 𝐵) Yes elxp 5121, opelxpi 5138, xpundi 5161
xreXtended reals df-xr 10063 * Yes ressxr 10068, rexr 10070, 0xr 10071
z integers (from German "Zahlen") df-z 11363 Yes elz 11364, zcn 11367
zn ring of integers mod 𝑛 df-zn 19836 (ℤ/nℤ‘𝑁) Yes znval 19864, zncrng 19874, znhash 19888
zringring of integers df-zring 19800 ring Yes zringbas 19805, zringcrng 19801
0, z slashed zero (empty set) (see n0) df-nul 3908 Yes n0i 3912, vn0 3916; snnz 4300, prnz 4301

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

𝜑       𝜑

17.1.2  Natural deduction

Theoremnatded 27230 Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

IT Γ𝜓 => Γ𝜓 idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
I Γ𝜓 & Γ𝜒 => Γ𝜓𝜒 jca 554 jca 554, pm3.2i 471 Definition I in [Pfenning] p. 18, definition Im,n in [Clemente] p. 10, and definition I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
EL Γ𝜓𝜒 => Γ𝜓 simpld 475 simpld 475, adantr 481 Definition EL in [Pfenning] p. 18, definition E(1) in [Clemente] p. 11, and definition E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
ER Γ𝜓𝜒 => Γ𝜒 simprd 479 simpr 477, adantl 482 Definition ER in [Pfenning] p. 18, definition E(2) in [Clemente] p. 11, and definition E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
I Γ, 𝜓𝜒 => Γ𝜓𝜒 ex 450 ex 450 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 701, imp 445 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 408 olc 399, olci 406, olcd 408 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 407 orc 400, orci 405, orcd 407 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 826 mpjaodan 826, jaodan 825, jaod 395 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1501 pm2.01d 181
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 690 mtand 690 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 599 pm2.65da 599 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 458 pm2.01d 181, pm2.65da 599, pm2.65d 187 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1503 pm2.21dd 186
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 186 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 186 pm2.21dd 186, pm2.21d 118, pm2.21 120 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ a1tru 1498 tru 1485, a1tru 1498, trud 1491 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1497 falim 1496 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1853 alrimiv 1853, ralrimiva 2963 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3443 spcv 3294, rspcv 3300 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3515 spcev 3295, rspcev 3304 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1861 exlimddv 1861, exlimdd 2086, exlimdv 1859, rexlimdva 3027 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1502 efald 1502 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 459 pm2.18da 459, pm2.18d 124, pm2.18 122 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 128 notnotrd 128, notnotr 125 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 432 exmid 431 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2621 eqid 2620, eqidd 2621 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3435 sbceq1d 3434, 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 27231, ex-natded5.3 27234, ex-natded5.5 27237, ex-natded5.7 27238, ex-natded5.8 27240, ex-natded5.13 27242, ex-natded9.20 27244, and ex-natded9.26 27246.

(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 27230 and mmnatded.html 27230. 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 27231 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 554, 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 27232. A proof without context is shown in ex-natded5.2i 27233. (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 27232 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 27231 and ex-natded5.2i 27233. (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 27233 The same as ex-natded5.2 27231 and ex-natded5.2-2 27232 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 27234 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 27235. A proof without context is shown in ex-natded5.3i 27236. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e; adantr 481 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given \$e; adantr 481 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 477, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 481 was used to transform its dependency (we could also use imp 445 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 481 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 554, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 450, 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 27235 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 27234 and ex-natded5.3i 27236. (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 27236 The same as ex-natded5.3 27234 and ex-natded5.3-2 27235 but with no context. Identical to jccir 561, 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 27237 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e; adantr 481 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given \$e; we'll use adantr 481 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 477
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 481 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 599 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 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 189; a proof without context is shown in mto 188.

(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 27238 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 27239. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
16 (𝜓 ∨ (𝜒𝜃)) (𝜑 → (𝜓 ∨ (𝜒𝜃))) Given \$e. No need for adantr 481 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 477
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 407, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 477
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 475, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 408, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 826, 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 27239 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 27238. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremex-natded5.8 27240 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11 ((𝜓𝜒) → ¬ 𝜃) (𝜑 → ((𝜓𝜒) → ¬ 𝜃)) Given \$e; adantr 481 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given \$e; adantr 481 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given \$e; adantr 481 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given \$e. adantr 481 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 477. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 554 (I), 6,8 (adantr 481 to bring in scope)
75 ... ¬ 𝜃 ((𝜑𝜓) → ¬ 𝜃) E 1,6 mpd 15 (E), 2,4
812 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 (E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913 ¬ 𝜓 (𝜑 → ¬ 𝜓) ¬I 5,7,8 pm2.65da 599 (¬I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 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 27241.

(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 27241 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 27240. (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 27242 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 27243. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115 (𝜓𝜒) (𝜑 → (𝜓𝜒)) Given \$e.
2;32 (𝜓𝜃) (𝜑 → (𝜓𝜃)) Given \$e. adantr 481 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given \$e. ad2antrr 761 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 477
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 407 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 477
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 477
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 481 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 599 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 128 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 408 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 826 5,14,15

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 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 27243 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 27242. (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 27244 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 475 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 479 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 477
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 554 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 407 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 477
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 554 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 408 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 826 6,10,11

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 481; simpr 477 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 27245. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Theoremex-natded9.20-2 27245 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 27244. (Contributed by David A. Wheeler, 19-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∧ (𝜒𝜃)))       (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃)))

Theoremex-natded9.26 27246* Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that 𝑥 is bound in the conclusion. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13 𝑥𝑦𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝑦𝜓) Given \$e.
26 ...| 𝑦𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∀𝑦𝜓) ND hypothesis assumption simpr 477. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3443 (E), 5,6. To use it we need a1i 11 and vex 3198. This could be immediately done with 19.21bi 2057, but we want to show the general approach for substitution.
412;8,9,10,11 ... 𝑥𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → ∃𝑥𝜓) I 3,a spesbcd 3515 (I), 11. To use it we need sylibr 224, which in turn requires sylib 208 and two uses of sbcid 3446. This could be more immediately done using 19.8a 2050, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2086 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1841 and nfe1 2025 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1853 (I), 13

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps).

Note that in the original proof, 𝜓(𝑥, 𝑦) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 27247.

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

(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)

Theoremex-natded9.26-2 27247* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 27246. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)

17.1.4  Definitional examples

Theoremex-or 27248 Example for df-or 385. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 ∨ 4 = 4)

Theoremex-an 27249 Example for df-an 386. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 ∧ 3 = 3)

Theoremex-dif 27250 Example for df-dif 3570. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∖ {1, 8}) = {3}

Theoremex-un 27251 Example for df-un 3572. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∪ {1, 8}) = {1, 3, 8}

Theoremex-in 27252 Example for df-in 3574. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∩ {1, 8}) = {1}

Theoremex-uni 27253 Example for df-uni 4428. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
{{1, 3}, {1, 8}} = {1, 3, 8}

Theoremex-ss 27254 Example for df-ss 3581. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊆ {1, 2, 3}

Theoremex-pss 27255 Example for df-pss 3583. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊊ {1, 2, 3}

Theoremex-pw 27256 Example for df-pw 4151. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
(𝐴 = {3, 5, 7} → 𝒫 𝐴 = (({∅} ∪ {{3}, {5}, {7}}) ∪ ({{3, 5}, {3, 7}, {5, 7}} ∪ {{3, 5, 7}})))

Theoremex-pr 27257 Example for df-pr 4171. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐴 ∈ {1, -1} → (𝐴↑2) = 1)

Theoremex-br 27258 Example for df-br 4645. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Theoremex-opab 27259* Example for df-opab 4704. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ ∧ (𝑥 + 1) = 𝑦)} → 3𝑅4)

Theoremex-eprel 27260 Example for df-eprel 5019. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
5 E {1, 5}

Theoremex-id 27261 Example for df-id 5014. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(5 I 5 ∧ ¬ 4 I 5)

Theoremex-po 27262 Example for df-po 5025. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
( < Po ℝ ∧ ¬ ≤ Po ℝ)

Theoremex-xp 27263 Example for df-xp 5110. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
({1, 5} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩})

Theoremex-cnv 27264 Example for df-cnv 5112. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}

Theoremex-co 27265 Example for df-co 5113. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((exp ∘ cos)‘0) = e

Theoremex-dm 27266 Example for df-dm 5114. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3})

Theoremex-rn 27267 Example for df-rn 5115. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})

Theoremex-res 27268 Example for df-res 5116. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {⟨2, 6⟩})

Theoremex-ima 27269 Example for df-ima 5117. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6})

Theoremex-fv 27270 Example for df-fv 5884. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9)

Theoremex-1st 27271 Example for df-1st 7153. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(1st ‘⟨3, 4⟩) = 3

Theoremex-2nd 27272 Example for df-2nd 7154. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(2nd ‘⟨3, 4⟩) = 4

Theorem1kp2ke3k 27273 Example for df-dec 11479, 1000 + 2000 = 3000.

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

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

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

This proof heavily relies on the decimal constructor df-dec 11479 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

(1000 + 2000) = 3000

Theoremex-fl 27274 Example for df-fl 12576. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theoremex-ceil 27275 Example for df-ceil 12577. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theoremex-mod 27276 Example for df-mod 12652. (Contributed by AV, 3-Sep-2021.)
((5 mod 3) = 2 ∧ (-7 mod 2) = 1)

Theoremex-exp 27277 Example for df-exp 12844. (Contributed by AV, 4-Sep-2021.)
((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))

Theoremex-fac 27278 Example for df-fac 13044. (Contributed by AV, 4-Sep-2021.)
(!‘5) = 120

Theoremex-bc 27279 Example for df-bc 13073. (Contributed by AV, 4-Sep-2021.)
(5C3) = 10

Theoremex-hash 27280 Example for df-hash 13101. (Contributed by AV, 4-Sep-2021.)
(#‘{0, 1, 2}) = 3

Theoremex-sqrt 27281 Example for df-sqrt 13956. (Contributed by AV, 4-Sep-2021.)
(√‘25) = 5

Theoremex-abs 27282 Example for df-abs 13957. (Contributed by AV, 4-Sep-2021.)
(abs‘-2) = 2

Theoremex-dvds 27283 Example for df-dvds 14965: 3 divides into 6. (Contributed by David A. Wheeler, 19-May-2015.)
3 ∥ 6

Theoremex-gcd 27284 Example for df-gcd 15198. (Contributed by AV, 5-Sep-2021.)
(-6 gcd 9) = 3

Theoremex-lcm 27285 Example for df-lcm 15284. (Contributed by AV, 5-Sep-2021.)
(6 lcm 9) = 18

Theoremex-prmo 27286 Example for df-prmo 15717: (#p‘10) = 2 · 3 · 5 · 7. (Contributed by AV, 6-Sep-2021.)
(#p10) = 210

17.1.5  Other examples

Theoremaevdemo 27287* Proof illustrating the comment of aev2 1984. (Contributed by BJ, 30-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ((∃𝑎𝑏 𝑐 = 𝑑 ∨ ∃𝑒 𝑓 = 𝑔) ∧ ∀(𝑖 = 𝑗𝑘 = 𝑙)))

Theoremex-ind-dvds 27288 Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
(𝑁 ∈ ℕ0 → 3 ∥ ((4↑𝑁) + 2))

17.2  Humor

17.2.1  April Fool's theorem

Theoremavril1 27289 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object 𝑥 equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at mmnotes.txt, under the 1-Apr-2006 entry.

¬ (𝐴𝒫 ℝ(i‘1) ∧ 𝐹∅(0 · 1))

Theorem2bornot2b 27290 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(2 · 𝐵 ∨ ¬ 2 · 𝐵)

Theoremhelloworld 27291 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
¬ ( ∈ (𝐿𝐿0) ∧ 𝑊∅(R1𝑑))

Theorem1p1e2apr1 27292 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 + 1) = 2

Theoremeqid1 27293 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

𝐴 = 𝐴

Theorem1div0apr 27294 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(1 / 0) = ∅

Theoremtopnfbey 27295 Nothing seems to be impossible to Prof. Lirpa. After years of intensive research, he managed to find a proof that when given a chance to reach infinity, one could indeed go beyond, thus giving formal soundness to Buzz Lightyear's motto "To infinity... and beyond!" (Contributed by Prof. Loof Lirpa, 1-Apr-2020.) (Modified by Thierry Arnoux, 2-Aug-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐵 ∈ (0...+∞) → +∞ < 𝐵)

17.3  (Future - to be reviewed and classified)

17.3.1  Planar incidence geometry

Syntaxcplig 27296 Extend class notation with the class of all planar incidence geometries.
class Plig

Definitiondf-plig 27297* Define the class of planar incidence geometries. We use Hilbert's axioms and adapt them to planar geometry. We use for the incidence relation. We could have used a generic binary relation, but using allows us to reuse previous results. Much of what follows is directly borrowed from Aitken, Incidence-Betweenness Geometry, 2008, http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf.

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

Plig = {𝑥 ∣ (∀𝑎 𝑥𝑏 𝑥(𝑎𝑏 → ∃!𝑙𝑥 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝑥𝑎 𝑥𝑏 𝑥(𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎 𝑥𝑏 𝑥𝑐 𝑥𝑙𝑥 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))}

Theoremisplig 27298* The predicate "is a planar incidence geometry" for sets. (Contributed by FL, 2-Aug-2009.)
𝑃 = 𝐺       (𝐺𝐴 → (𝐺 ∈ Plig ↔ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))

Theoremispligb 27299* The predicate "is a planar incidence geometry". (Contributed by BJ, 2-Dec-2021.)
𝑃 = 𝐺       (𝐺 ∈ Plig ↔ (𝐺 ∈ V ∧ (∀𝑎𝑃𝑏𝑃 (𝑎𝑏 → ∃!𝑙𝐺 (𝑎𝑙𝑏𝑙)) ∧ ∀𝑙𝐺𝑎𝑃𝑏𝑃 (𝑎𝑏𝑎𝑙𝑏𝑙) ∧ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))))

Theoremtncp 27300* In any planar incidence geometry, there exist three non-collinear points. (Contributed by FL, 3-Aug-2009.)
𝑃 = 𝐺       (𝐺 ∈ Plig → ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑙𝐺 ¬ (𝑎𝑙𝑏𝑙𝑐𝑙))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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