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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremextwwlkfablem1 26601 Lemma 1 for extwwlkfab 26617. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
((((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋))

Theoremextwwlkfablem2lem 26602 Lemma for extwwlkfablem2 26605. (Contributed by Alexander van der Vekens, 17-Sep-2018.)
((𝑤 ∈ Word 𝑉 ∧ (#‘𝑤) = 𝑁𝑁 ∈ (ℤ‘2)) → (#‘(𝑤 substr ⟨0, (𝑁 − 2)⟩)) = (𝑁 − 2))

Theoremclwwlkextfrlem1 26603 Lemma for numclwwlk2lem1 26629. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(((𝑋𝑉𝑁 ∈ ℕ ∧ 𝑍𝑉) ∧ (𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ (𝑊‘0) = 𝑋 ∧ ( lastS ‘𝑊) ≠ 𝑋)) → (((𝑊 ++ ⟨“𝑍”⟩)‘0) = 𝑋 ∧ ((𝑊 ++ ⟨“𝑍”⟩)‘𝑁) ≠ 𝑋))

Theoremnumclwwlkfvc 26604* Value of function 𝐶, mapping a nonnegative number n to the closed walks having length n. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       (𝑁 ∈ ℕ0 → (𝐶𝑁) = ((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremextwwlkfablem2 26605* Lemma 2 for extwwlkfab 26617. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       ((((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) ∧ 𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) ∧ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))) → (𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝐶‘(𝑁 − 2)))

Theoremnumclwwlkun 26606* The set of closed walks in an undirected simple graph is the union of the numbers of closed walks starting at each of the vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0) → (𝐶𝑁) = 𝑥𝑉 {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑥})

Theoremnumclwwlkdisj 26607* The sets of closed walks starting at different vertices in an undirected simple graph are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))       Disj 𝑥𝑉 {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑥}

Theoremnumclwwlkovf 26608* Value of operation 𝐹, mapping a vertex v and a nonnegative integer n to the "(For a fixed vertex v, let f(n) be the number of) walks from v to v of length n" according to definition 5 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐹𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ (𝑤‘0) = 𝑋})

Theoremnumclwwlkffin 26609* In a finite graph, the value of operation 𝐹 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((𝑉 ∈ Fin ∧ 𝐸𝑈) ∧ (𝑋𝑉𝑁 ∈ ℕ0)) → (𝑋𝐹𝑁) ∈ Fin)

Theoremnumclwwlkovfel2 26610* Properties of an element of the value of operation 𝐹. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑋𝑉) → (𝐴 ∈ (𝑋𝐹𝑁) ↔ ((𝐴 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝐴) − 1)){(𝐴𝑖), (𝐴‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝐴), (𝐴‘0)} ∈ ran 𝐸) ∧ (#‘𝐴) = 𝑁 ∧ (𝐴‘0) = 𝑋)))

Theoremnumclwwlkovf2 26611* Value of operation 𝐹 for argument 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑋𝑉) → (𝑋𝐹2) = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ ran 𝐸 ∧ (𝑤‘0) = 𝑋)})

Theoremnumclwwlkovf2num 26612* In a k regular graph, therere are k closed walks of length 2 starting at a fixed vertex. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑋𝑉) → (#‘(𝑋𝐹2)) = 𝐾)

Theoremnumclwwlkovf2ex 26613* Extending a closed walk starting at a fixed vertex by an additional edge (forth and back). (Contributed by AV, 22-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) ∧ 𝑄 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ 𝑃 ∈ (𝑋𝐹(𝑁 − 2))) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝐶𝑁))

Theoremnumclwwlkovg 26614* Value of operation 𝐺, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v" according to definition 6 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 14-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) = (𝑤‘0))})

Theoremnumclwwlkovgel 26615* Properties of an element of the value of operation 𝐺. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) ↔ (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∧ (𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0))))

Theoremnumclwwlkovgelim 26616* Properties of an element of the value of operation 𝐺. (Contributed by Alexander van der Vekens, 24-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘2)) → (𝑃 ∈ (𝑋𝐺𝑁) → ((𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ ((𝑃‘0) = 𝑋 ∧ (𝑃‘(𝑁 − 2)) = (𝑃‘0)))))

Theoremextwwlkfab 26617* The set of closed walks (having a fixed length greater than 1 and starting at a fixed vertex) with the last but 2 vertex is identical with the first (and therefore last) vertex can be constructed from the set of closed walks with length smaller by 2 than the fixed length appending a neighbor of the last vertex and afterwards the last vertex (which is the first vertex) itself ("walking forth and back" from the last vertex). 3 ≤ 𝑁 is required since for 𝑁 = 2: (𝑋𝐹(𝑁 − 2)) = (𝑋𝐹0) = ∅, see clwwlkgt0 26299 stating that a walk of length 0 is not represented as word, at least not for an undirected simple graph.) (Contributed by Alexander van der Vekens, 18-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑋𝐺𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤 substr ⟨0, (𝑁 − 2)⟩) ∈ (𝑋𝐹(𝑁 − 2)) ∧ (𝑤‘(𝑁 − 1)) ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋) ∧ (𝑤‘(𝑁 − 2)) = 𝑋)})

Theoremnumclwlk1lem2foa 26618* Going forth and back form the end of a (closed) walk. (Contributed by Alexander van der Vekens, 22-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ((𝑃 ∈ (𝑋𝐹(𝑁 − 2)) ∧ 𝑄 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑋)) → ((𝑃 ++ ⟨“𝑋”⟩) ++ ⟨“𝑄”⟩) ∈ (𝑋𝐺𝑁)))

Theoremnumclwlk1lem2f 26619* T is a function. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)⟶((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2fv 26620* Value of the function T. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → (𝑃 ∈ (𝑋𝐺𝑁) → (𝑇𝑃) = ⟨(𝑃 substr ⟨0, (𝑁 − 2)⟩), (𝑃‘(𝑁 − 1))⟩))

Theoremnumclwlk1lem2f1 26621* T is a 1-1 function. (Contributed by AV, 26-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–1-1→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2fo 26622* T is an onto function. (Contributed by Alexander van der Vekens, 20-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2f1o 26623* T is a 1-1 onto function. (Contributed by Alexander van der Vekens, 26-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑇 = (𝑤 ∈ (𝑋𝐺𝑁) ↦ ⟨(𝑤 substr ⟨0, (𝑁 − 2)⟩), (𝑤‘(𝑁 − 1))⟩)       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → 𝑇:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwlk1lem2 26624* There is a bijection between the set of closed walks (having a fixed length greater than 2 and starting at a fixed vertex) with the last but 2 vertex identical with the first (and therefore last) vertex and the set of closed walks (having a fixed length less by 2 and starting at the same vertex) and the neighbors of this vertex. (Contributed by Alexander van der Vekens, 6-Jul-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       ((𝑉 USGrph 𝐸𝑋𝑉𝑁 ∈ (ℤ‘3)) → ∃𝑓 𝑓:(𝑋𝐺𝑁)–1-1-onto→((𝑋𝐹(𝑁 − 2)) × (⟨𝑉, 𝐸⟩ Neighbors 𝑋)))

Theoremnumclwwlk1 26625* Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since 𝑉, 𝐸 is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})       (((𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) ∧ (𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐺𝑁)) = (𝐾 · (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlkovq 26626* Value of operation Q, mapping a vertex v and a nonnegative integer n to the not closed walks v(0) ... v(n) of length n from a fixed vertex v = v(0). "Not closed" means v(n) =/= v(0). (Contributed by Alexander van der Vekens, 27-Sep-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝑄𝑁) = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)})

Theoremnumclwwlkqhash 26627* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 ∈ Fin) ∧ (𝑋𝑉𝑁 ∈ ℕ)) → (#‘(𝑋𝑄𝑁)) = ((𝐾𝑁) − (#‘(𝑋𝐹𝑁))))

Theoremnumclwwlkovh 26628* Value of operation H, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑋𝑉𝑁 ∈ ℕ0) → (𝑋𝐻𝑁) = {𝑤 ∈ (𝐶𝑁) ∣ ((𝑤‘0) = 𝑋 ∧ (𝑤‘(𝑁 − 2)) ≠ (𝑤‘0))})

Theoremnumclwwlk2lem1 26629* In a friendship graph, for each walk of length n starting with a fixed vertex and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation H. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation H, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem only generally holds for Friendship Graphs, because these guarantee that for the first and last vertex there is a third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝑄𝑁) → ∃!𝑣𝑉 (𝑊 ++ ⟨“𝑣”⟩) ∈ (𝑋𝐻(𝑁 + 2))))

Theoremnumclwlk2lem2f 26630* R is a function. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))⟶(𝑋𝑄𝑁))

Theoremnumclwlk2lem2fv 26631* Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑋𝑉𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋𝐻(𝑁 + 2)) → (𝑅𝑊) = (𝑊 substr ⟨0, (𝑁 + 1)⟩)))

Theoremnumclwlk2lem2f1o 26632* R is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})    &   𝑅 = (𝑥 ∈ (𝑋𝐻(𝑁 + 2)) ↦ (𝑥 substr ⟨0, (𝑁 + 1)⟩))       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → 𝑅:(𝑋𝐻(𝑁 + 2))–1-1-onto→(𝑋𝑄𝑁))

Theoremnumclwwlk2lem3 26633* In a friendship graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex equals the size of the set of all closed walks of length (n+2) starting with this vertex and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Proof shortened by AV, 5-May-2021.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       ((𝑉 FriendGrph 𝐸𝑋𝑉𝑁 ∈ ℕ) → (#‘(𝑋𝑄𝑁)) = (#‘(𝑋𝐻(𝑁 + 2))))

Theoremnumclwwlk2 26634* Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgranumwlkg 26485, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐻𝑁)) = ((𝐾↑(𝑁 − 2)) − (#‘(𝑋𝐹(𝑁 − 2)))))

Theoremnumclwwlk3lem 26635* Lemma for numclwwlk3 26636. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑋𝑉) ∧ 𝑁 ∈ (ℤ‘2)) → (#‘(𝑋𝐹𝑁)) = ((#‘(𝑋𝐻𝑁)) + (#‘(𝑋𝐺𝑁))))

Theoremnumclwwlk3 26636* 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 26625) and with v(n-2) =/= v(n) ( see numclwwlk2 26634): f(n) = kf(n-2) + k^(n-2) - f(n-2) = (k - 1)f(n - 2) + k^(n-2). (Contributed by Alexander van der Vekens, 26-Aug-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})    &   𝐺 = (𝑣𝑉, 𝑛 ∈ (ℤ‘2) ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) = (𝑤‘0))})    &   𝑄 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ ( lastS ‘𝑤) ≠ 𝑣)})    &   𝐻 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ ((𝑤‘0) = 𝑣 ∧ (𝑤‘(𝑛 − 2)) ≠ (𝑤‘0))})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ∈ Fin ∧ 𝑋𝑉𝑁 ∈ (ℤ‘3))) → (#‘(𝑋𝐹𝑁)) = (((𝐾 − 1) · (#‘(𝑋𝐹(𝑁 − 2)))) + (𝐾↑(𝑁 − 2))))

Theoremnumclwwlk4 26637* The total number of closed walks in a finite undirected simple graph is the sum of the numbers of closed walks starting at each of its vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℕ0) → (#‘(𝐶𝑁)) = Σ𝑥𝑉 (#‘(𝑥𝐹𝑁)))

Theoremnumclwwlk5lem 26638* Lemma for numclwwlk5 26639. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 2 ∥ (𝐾 − 1) ∧ 𝑋𝑉) → ((#‘(𝑋𝐹2)) mod 2) = 1)

Theoremnumclwwlk5 26639* Statement 13 in [Huneke] p. 2: "Let p be a prime divisor of k-1; then f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸𝑉 ∈ Fin) ∧ (𝑋𝑉𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝑋𝐹𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk6 26640* For a prime divisor p of k-1, the total number of closed walks of length p in an undirected simple graph with m vertices mod p is equal to the number of vertices mod p. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
𝐶 = (𝑛 ∈ ℕ0 ↦ ((𝑉 ClWWalksN 𝐸)‘𝑛))    &   𝐹 = (𝑣𝑉, 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝐶𝑛) ∣ (𝑤‘0) = 𝑣})       (((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘(𝐶𝑃)) mod 𝑃) = ((#‘𝑉) mod 𝑃))

Theoremnumclwwlk7 26641 Statement 14 in [Huneke] p. 2: "The total number of closed walks of length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)", since the number of vertices in a friendship graph is (k(k-1)+1), see frgregordn0 26597 or frrusgraord 26598, and p divides (k-1), i.e. (k-1) mod p = 0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the empty graph is a friendship graph, see frgra0 26521, as well as k-regular (for any k), see 0vgrargra 26464, but has no closed walk, see clwlk0 26290, this theorem would be false: ((#‘(𝐶𝑃)) mod 𝑃) = 0 ≠ 1, so this case must be excluded. ( (Contributed by Alexander van der Vekens, 1-Sep-2018.)
(((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾𝑉 FriendGrph 𝐸) ∧ (𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) ∧ (𝑃 ∈ ℙ ∧ 𝑃 ∥ (𝐾 − 1))) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 1)

Theoremnumclwwlk8 26642 The size of the set of closed walks of length p, p prime, is divisible by p. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p", see also clwlkndivn 26380. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑃 ∈ ℙ) → ((#‘((𝑉 ClWWalksN 𝐸)‘𝑃)) mod 𝑃) = 0)

Theoremfrgrareggt1 26643 If a finite friendship graph is k-regular with k > 1, then k must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((⟨𝑉, 𝐸⟩ RegUSGrph 𝐾 ∧ 1 < 𝐾) → 𝐾 = 2))

Theoremfrgrareg 26644 If a finite friendship graph is k-regular, then k must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) → ((𝑉 FriendGrph 𝐸 ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → (𝐾 = 0 ∨ 𝐾 = 2)))

Theoremfrgraregord013 26645 If a finite friendship graph is k-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 0 ∨ (#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Theoremfrgraregord13 26646 If a nonempty finite friendship graph is k-regular, then it must have order 1 or 3. Special case of frgraregord013 26645. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
(((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 𝑉 ≠ ∅) ∧ ⟨𝑉, 𝐸⟩ RegUSGrph 𝐾) → ((#‘𝑉) = 1 ∨ (#‘𝑉) = 3))

Theoremfrgraogt3nreg 26647* If a finite friendship graph has an order greater than 3, it cannot be k-regular for any k. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∀𝑘 ∈ ℕ0 ¬ ⟨𝑉, 𝐸⟩ RegUSGrph 𝑘)

Theoremfriendshipgt3 26648* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ∈ Fin ∧ 3 < (#‘𝑉)) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

Theoremfriendship 26649* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
((𝑉 FriendGrph 𝐸𝑉 ≠ ∅ ∧ 𝑉 ∈ Fin) → ∃𝑣𝑉𝑤 ∈ (𝑉 ∖ {𝑣}){𝑣, 𝑤} ∈ ran 𝐸)

PART 18  GUIDES AND MISCELLANEA

18.1  Guides (conventions, explanations, and examples)

18.1.1  Conventions

This section describes the conventions we use. These conventions often refer to existing mathematical practices, which are discussed in more detail in other references. For the general conventions, see conventions 26650, and for conventions related to labels, see conventions-label 26651. 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 26650

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 26651.

• 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 14265) 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 14453). 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 196, df-cleq 2603, df-clel 2606, df-clab 2597) 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 9873, proven by the preceding theorem ax1cn 9849. 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 9873 (not ax1cn 9849) and ax-1ne0 9884 (not ax1ne0 9860), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9849 and ax1ne0 9860 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 8822) 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 7965) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 9140) or axiom of dependent choice (ax-dc 9151) 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 2006 through ax-13 2234, by invoking ax10w 1993 through ax13w 2000. We encourage proving theorems *without* ax-10 2006 through ax-13 2234 and moving them up to the ax-4 1728 through ax-9 1986 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 16864) and connective symbols such as + for some group addition operation. (See prdsplusgval 15956 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 219 or mpbir 220. 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 205 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 957, sylbir 224, or 3imtr4i 280.
• 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 1868 and the related df-sbc 3403 and df-csb 3500.
• 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 3175 and isset 3180. 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 3185 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 4736). 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 5046. This can be used to define a subset, e.g., df-tan 14641 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 14719). 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 5812. 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 6552). For example, the + in (2 + 2); see 2p2e4 11021. 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 384, df-an 385, and df-bi 196 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 1978), equality 𝐴 = 𝐵 (see df-cleq 2603), subset 𝐴𝐵 (see df-ss 3554), and less-than 𝐴 < 𝐵 (see df-lt 9828). For the general definition of a binary relation in the form 𝐴𝑅𝐵, see df-br 4584. For example, 0 < 1 (see 0lt1 10429) 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 10146.
• 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 14634). The function is then defined labeled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 14640). Typically, there are other proofs such as its closure labeled NAMEcl (e.g., coscl 14696), its function application form labeled NAMEval (e.g., cosval 14692), and at least one simple value (e.g., cos0 14719).
• 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 12923 and fac4 12930).
• 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 9822), while "0g" is the group identity element (df-0g 15925), "0." is the poset zero (df-p0 16862), "0𝑝" is the zero polynomial (df-0p 23243), "0vec" is the zero vector in a normed complex vector space (df-0v 26837), and "0" is a class variable for use as a connective symbol (this is used, for example, in p0val 16864). 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 6552, 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 10040). Here are the reasons as discussed in https://groups.google.com/g/metamath/c/2AW7T3d2YiQ/m/iSN7g87t3ikJ:
1. Having the hypotheses immediately shows the intended domain of applicability (is it , *, ω, or something else?), without having to trace back to definitions.
2. Having the hypotheses forces its use in the intended domain, which generally is desirable.
3. The behavior is dependent on accidental behavior of definitions outside of their domains, so the theorems are non-portable and "brittle".
4. Only a few theorems can have their hypotheses removed in this fashion due to happy coincidences for our particular set-theoretical definitions. The poor user (especially a novice learning real number arithmetic) is going to be confused not knowing when hypotheses are needed and when they are not. For someone who hasn't traced back the set-theoretical foundations of the definitions, it is seemingly random and isn't intuitive at all.
5. The consensus of opinion of people on this group seemed to be against doing this.
• Natural numbers. There are different definitions of "natural" numbers in the literature. We use (df-nn 10898) for the set of positive integers starting from 1, and 0 (df-n0 11170) 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 11370, e.g., 4001 (see 4001prm 15690 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 3749). The term "tautology" is also used, especially in propositional calculus. This form was formerly called "theorem form" or "closed theorem form".
• A theorem is in "deduction form" (or is a "deduction") if it has one or more \$e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually 𝜑), and every hypothesis (\$e statement) is either:
1. an implication with the same antecedent as the conclusion, or
2. a definition. A definition can be for a class variable (this is a class variable followed by =, e.g. the definition of 𝐷 in lhop 23583) 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 3751.
• 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 3750).
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 149 is the inference associated with con3 148). 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 81 because it is the inference associated with imim1i 61 which is itself the inference associated with imim1 81.
"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 51). The labels for inferences usually have the suffix "i" (e.g., pm2.43i 50). The labels of theorems in "closed form" would have no special suffix (e.g., pm2.43 54). 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 6851 vs. uniexg 6853.
• 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 4089, which works in certain cases in set theory. We also sometimes use dedhb 3343. 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 26652.
• 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 7355, rec(𝐹, 𝐼) in df-rdg 7393, seq𝜔(𝐹, 𝐼) in df-seqom 7430, and seq𝑀( + , 𝐹) in df-seq 12664. These have characteristic function 𝐹 and initial value 𝐼. (Σg in df-gsum 15926 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7355, but for the "average user" the most useful one is probably df-seq 12664- 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 15697.
• 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 5789 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 6113 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 7955 is the first lemma for sbth 7965. 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 7955 is the first lemma for sbth 7965.
• 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 15224"). 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 1701 (whose description has some important technical details). Similarly, 𝑥𝐴 is read 𝑥 is not free in (class) 𝐴, see df-nfc 2740.
• "\$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 1701) in place of "\$d 𝑥𝜑 \$." when a more general result is desired; ax-5 1827 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 2466 from df-eu 2462.
• "\$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 220 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 2259. 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 196, df-clab 2597, df-cleq 2603, and df-clel 2606); 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 1030 or wsb 1867).
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 1033 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 3174. The definition df-v 3175 V = {𝑥𝑥 = 𝑥} is justified by the justification theorem vjust 3174 {𝑥𝑥 = 𝑥} = {𝑦𝑦 = 𝑦}. Another example of a justification theorem is trujust 1477; the definition df-tru 1478 (⊤ ↔ (∀𝑥𝑥 = 𝑥 → ∀𝑥𝑥 = 𝑥)) is justified by trujust 1477 ((∀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 13974 and absmax 13917.
• 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 26651

The following explains some of the label conventions in use in the Metamath Proof Explorer ("set.mm"). For the general conventions, see conventions 26650.

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 2906"rgen.1 \$e |- ( x e. A -> ph ) \$." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 20231: "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 2551 and stirling 38982.
• 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 41.
• 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 1757, and the restricted quantifier version of Theorem 21 from Section 19 of [Margaris] p. 90 is labeled r19.21 2939.
• 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... 14451. 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 3543, 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 3554. 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 3699. There are many other syntax label fragments, e.g., singleton construct {𝐴} has syntax label fragment "sn" (because it is defined in df-sn 4126), and the pair construct {𝐴, 𝐵} has fragment "pr" ( from df-pr 4128). 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 4260. An "n" is often used for negation (¬), e.g., nan 602.
• 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 9821) and "re" represents real numbers ( definition df-r 9825). The empty set often uses fragment 0, even though it is defined in df-nul 3875. The syntax construct (𝐴 + 𝐵) usually uses the fragment "add" (which is consistent with df-add 9826), 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 11021.
• 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 14719 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 14640) we have value cosval 14692 and closure coscl 14696.
• 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 26652 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 2062 via the use of disjoint variable conditions combined with nfv 1830. If two (or three) such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used, e.g. exlimivv 1847. 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 2466 derived from df-eu 2462. 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 4844. 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 2269 (cbval 2259 in deduction form "d" with a not free variable replaced by a disjoint variable condition "v" with a conjunction as antecedent "a"). Here is a non-exhaustive list of common suffixes:
• a : theorem having a conjunction as antecedent
• b : theorem expressing a logical equivalence
• c : contraction (e.g., sylc 63, syl2anc 691), commutes (e.g., biimpac 502)
• 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 500, rexlimiva 3010
ablAbelian group df-abl 18019 Abel Yes ablgrp 18021, zringabl 19641
absabsorption No ressabs 15766
absabsolute value (of a complex number) df-abs 13824 (abs‘𝐴) Yes absval 13826, absneg 13865, abs1 13885
al"for all" 𝑥𝜑 No alim 1729, alex 1743
ALTalternative/less preferred (suffix) No idALT 23
anand df-an 385 (𝜑𝜓) Yes anor 509, iman 439, imnan 437
assassociative No biass 373, orass 545, mulass 9903
asymasymmetric, antisymmetric No intasym 5430, asymref 5431, posasymb 16775
axaxiom No ax6dgen 1992, ax1cn 9849
bas, base base (set of an extensible structure) df-base 15700 (Base‘𝑆) Yes baseval 15746, ressbas 15757, cnfldbas 19571
b, bibiconditional ("iff", "if and only if") df-bi 196 (𝜑𝜓) Yes impbid 201, sspwb 4844
brbinary relation df-br 4584 𝐴𝑅𝐵 Yes brab1 4630, brun 4633
cbvchange bound variable No cbvalivw 1921, cbvrex 3144
clclosure No ifclda 4070, ovrcl 6584, zaddcl 11294
cncomplex numbers df-c 9821 Yes nnsscn 10902, nncn 10905
cnfldfield of complex numbers df-cnfld 19568 fld Yes cnfldbas 19571, cnfldinv 19596
cntzcentralizer df-cntz 17573 (Cntz‘𝑀) Yes cntzfval 17576, dprdfcntz 18237
cnvconverse df-cnv 5046 𝐴 Yes opelcnvg 5224, f1ocnv 6062
cocomposition df-co 5047 (𝐴𝐵) Yes cnvco 5230, fmptco 6303
comcommutative No orcom 401, bicomi 213, eqcomi 2619
concontradiction, contraposition No condan 831, con2d 128
csbclass substitution df-csb 3500 𝐴 / 𝑥𝐵 Yes csbid 3507, csbie2g 3530
cygcyclic group df-cyg 18103 CycGrp Yes iscyg 18104, zringcyg 19658
ddeduction form (suffix) No idd 24, impbid 201
df(alternate) definition (prefix) No dfrel2 5502, dffn2 5960
di, distrdistributive No andi 907, imdi 377, ordi 904, difindi 3840, ndmovdistr 6721
difclass difference df-dif 3543 (𝐴𝐵) Yes difss 3699, difindi 3840
divdivision df-div 10564 (𝐴 / 𝐵) Yes divcl 10570, divval 10566, divmul 10567
dmdomain df-dm 5048 dom 𝐴 Yes dmmpt 5547, iswrddm0 13184
e, eq, equequals df-cleq 2603 𝐴 = 𝐵 Yes 2p2e4 11021, uneqri 3717, equtr 1935
elelement of 𝐴𝐵 Yes eldif 3550, eldifsn 4260, elssuni 4403
eu"there exists exactly one" df-eu 2462 ∃!𝑥𝜑 Yes euex 2482, euabsn 4205
exexists (i.e. is a set) No brrelex 5080, 0ex 4718
ex"there exists (at least one)" df-ex 1696 𝑥𝜑 Yes exim 1751, alex 1743
expexport No expt 167, expcom 450
f"not free in" (suffix) No equs45f 2338, sbf 2368
ffunction df-f 5808 𝐹:𝐴𝐵 Yes fssxp 5973, opelf 5978
falfalse df-fal 1481 Yes bifal 1488, falantru 1499
fifinite intersection df-fi 8200 (fi‘𝐵) Yes fival 8201, inelfi 8207
fi, finfinite df-fin 7845 Fin Yes isfi 7865, snfi 7923, onfin 8036
fldfield (Note: there is an alternative definition Fld of a field, see df-fld 32961) df-field 18573 Field Yes isfld 18579, fldidom 19126
fnfunction with domain df-fn 5807 𝐴 Fn 𝐵 Yes ffn 5958, fndm 5904
frgpfree group df-frgp 17946 (freeGrp‘𝐼) Yes frgpval 17994, frgpadd 17999
fsuppfinitely supported function df-fsupp 8159 𝑅 finSupp 𝑍 Yes isfsupp 8162, fdmfisuppfi 8167, fsuppco 8190
funfunction df-fun 5806 Fun 𝐹 Yes funrel 5821, ffun 5961
fvfunction value df-fv 5812 (𝐹𝐴) Yes fvres 6117, swrdfv 13276
fzfinite set of sequential integers df-fz 12198 (𝑀...𝑁) Yes fzval 12199, eluzfz 12208
fz0finite set of sequential nonnegative integers (0...𝑁) Yes nn0fz0 12306, fz0tp 12309
fzohalf-open integer range df-fzo 12335 (𝑀..^𝑁) Yes elfzo 12341, elfzofz 12354
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 6853
gragraph No uhgrav 25825, isumgra 25844, usgrares 25898
grpgroup df-grp 17248 Grp Yes isgrp 17251, tgpgrp 21692
gsumgroup sum df-gsum 15926 (𝐺 Σg 𝐹) Yes gsumval 17094, gsumwrev 17619
hashsize (of a set) df-hash 12980 (#‘𝐴) Yes hashgval 12982, hashfz1 12996, hashcl 13009
hbhypothesis builder (prefix) No hbxfrbi 1742, hbald 2028, hbequid 33212
hm(monoid, group, ring) homomorphism No ismhm 17160, isghm 17483, isrhm 18544
iinference (suffix) No eleq1i 2679, tcsni 8502
iimplication (suffix) No brwdomi 8356, infeq5i 8416
ididentity No biid 250
idmidempotent No anidm 674, tpidm13 4235
im, impimplication (label often omitted) df-im 13689 (𝐴𝐵) Yes iman 439, imnan 437, impbidd 199
imaimage df-ima 5051 (𝐴𝐵) Yes resima 5351, imaundi 5464
impimport No biimpa 500, impcom 445
inintersection df-in 3547 (𝐴𝐵) Yes elin 3758, incom 3767
infinfimum df-inf 8232 inf(ℝ+, ℝ*, < ) Yes fiinfcl 8290, infiso 8296
is...is (something a) ...? No isring 18374
jjoining, disjoining No jc 158, jaoi 393
lleft No olcd 407, simpl 472
mapmapping operation or set exponentiation df-map 7746 (𝐴𝑚 𝐵) Yes mapvalg 7754, elmapex 7764
matmatrix df-mat 20033 (𝑁 Mat 𝑅) Yes matval 20036, matring 20068
mdetdeterminant (of a square matrix) df-mdet 20210 (𝑁 maDet 𝑅) Yes mdetleib 20212, mdetrlin 20227
mgmmagma df-mgm 17065 Magma Yes mgmidmo 17082, mgmlrid 17089, ismgm 17066
mgpmultiplicative group df-mgp 18313 (mulGrp‘𝑅) Yes mgpress 18323, ringmgp 18376
mndmonoid df-mnd 17118 Mnd Yes mndass 17125, mndodcong 17784
mo"there exists at most one" df-mo 2463 ∃*𝑥𝜑 Yes eumo 2487, moim 2507
mpmodus ponens ax-mp 5 No mpd 15, mpi 20
mptmodus ponendo tollens No mptnan 1684, mptxor 1685
mptmaps-to notation for a function df-mpt 4645 (𝑥𝐴𝐵) Yes fconstmpt 5085, resmpt 5369
mpt2maps-to notation for an operation df-mpt2 6554 (𝑥𝐴, 𝑦𝐵𝐶) Yes mpt2mpt 6650, resmpt2 6656
mulmultiplication (see "t") df-mul 9827 (𝐴 · 𝐵) Yes mulcl 9899, divmul 10567, mulcom 9901, mulass 9903
n, notnot ¬ 𝜑 Yes nan 602, notnotr 124
nenot equaldf-ne 𝐴𝐵 Yes exmidne 2792, neeqtrd 2851
nelnot element ofdf-nel 𝐴𝐵 Yes neli 2885, nnel 2892
ne0not equal to zero (see n0) ≠ 0 No negne0d 10269, ine0 10344, gt0ne0 10372
nf "not free in" (prefix) No nfnd 1769
ngpnormed group df-ngp 22198 NrmGrp Yes isngp 22210, ngptps 22216
nmnorm (on a group or ring) df-nm 22197 (norm‘𝑊) Yes nmval 22204, subgnm 22247
nnpositive integers df-nn 10898 Yes nnsscn 10902, nncn 10905
nn0nonnegative integers df-n0 11170 0 Yes nnnn0 11176, nn0cn 11179
n0not the empty set (see ne0) ≠ ∅ No n0i 3879, vn0 3883, ssn0 3928
OLDold, obsolete (to be removed soon) No 19.43OLD 1800
opordered pair df-op 4132 𝐴, 𝐵 Yes dfopif 4337, opth 4871
oror df-or 384 (𝜑𝜓) Yes orcom 401, anor 509
otordered triple df-ot 4134 𝐴, 𝐵, 𝐶 Yes euotd 4900, fnotovb 6592
ovoperation value df-ov 6552 (𝐴𝐹𝐵) Yes fnotovb 6592, fnovrn 6707
pplus (see "add"), for all-constant theorems df-add 9826 (3 + 2) = 5 Yes 3p2e5 11037
pfxprefix df-pfx 40245 (𝑊 prefix 𝐿) Yes pfxlen 40254, ccatpfx 40272
pmPrincipia Mathematica No pm2.27 41
pmpartial mapping (operation) df-pm 7747 (𝐴pm 𝐵) Yes elpmi 7762, pmsspw 7778
prpair df-pr 4128 {𝐴, 𝐵} Yes elpr 4146, prcom 4211, prid1g 4239, prnz 4253
prm, primeprime (number) df-prm 15224 Yes 1nprm 15230, dvdsprime 15238
pssproper subset df-pss 3556 𝐴𝐵 Yes pssss 3664, sspsstri 3671
q rational numbers ("quotients") df-q 11665 Yes elq 11666
rright No orcd 406, simprl 790
rabrestricted class abstraction df-rab 2905 {𝑥𝐴𝜑} Yes rabswap 3098, df-oprab 6553
ralrestricted universal quantification df-ral 2901 𝑥𝐴𝜑 Yes ralnex 2975, ralrnmpt2 6673
rclreverse closure No ndmfvrcl 6129, nnarcl 7583
rereal numbers df-r 9825 Yes recn 9905, 0re 9919
relrelation df-rel 5045 Rel 𝐴 Yes brrelex 5080, relmpt2opab 7146
resrestriction df-res 5050 (𝐴𝐵) Yes opelres 5322, f1ores 6064
reurestricted existential uniqueness df-reu 2903 ∃!𝑥𝐴𝜑 Yes nfreud 3091, reurex 3137
rexrestricted existential quantification df-rex 2902 𝑥𝐴𝜑 Yes rexnal 2978, rexrnmpt2 6674
rmorestricted "at most one" df-rmo 2904 ∃*𝑥𝐴𝜑 Yes nfrmod 3092, nrexrmo 3140
rnrange df-rn 5049 ran 𝐴 Yes elrng 5236, rncnvcnv 5270
rng(unital) ring df-ring 18372 Ring Yes ringidval 18326, isring 18374, ringgrp 18375
rotrotation No 3anrot 1036, 3orrot 1037
seliminates need for syllogism (suffix) No ancoms 468
sb(proper) substitution (of a set) df-sb 1868 [𝑦 / 𝑥]𝜑 Yes spsbe 1871, sbimi 1873
sbc(proper) substitution of a class df-sbc 3403 [𝐴 / 𝑥]𝜑 Yes sbc2or 3411, sbcth 3417
scascalar df-sca 15784 (Scalar‘𝐻) Yes resssca 15854, mgpsca 18319
simpsimple, simplification No simpl 472, simp3r3 1164
snsingleton df-sn 4126 {𝐴} Yes eldifsn 4260
spspecialization No spsbe 1871, spei 2249
sssubset df-ss 3554 𝐴𝐵 Yes difss 3699
structstructure df-struct 15697 Struct Yes brstruct 15703, structfn 15708
subsubtract df-sub 10147 (𝐴𝐵) Yes subval 10151, subaddi 10247
supsupremum df-sup 8231 sup(𝐴, 𝐵, < ) Yes fisupcl 8258, supmo 8241
suppsupport (of a function) df-supp 7183 (𝐹 supp 𝑍) Yes ressuppfi 8184, mptsuppd 7205
swapswap (two parts within a theorem) No rabswap 3098, 2reuswap 3377
sylsyllogism syl 17 No 3syl 18
symsymmetric No df-symdif 3806, cnvsym 5429
symgsymmetric group df-symg 17621 (SymGrp‘𝐴) Yes symghash 17628, pgrpsubgsymg 17651
t times (see "mul"), for all-constant theorems df-mul 9827 (3 · 2) = 6 Yes 3t2e6 11056
ththeorem No nfth 1718, sbcth 3417, weth 9200
tptriple df-tp 4130 {𝐴, 𝐵, 𝐶} Yes eltpi 4176, tpeq1 4221
trtransitive No bitrd 267, biantr 968
trutrue df-tru 1478 Yes bitru 1487, truanfal 1498
ununion df-un 3545 (𝐴𝐵) Yes uneqri 3717, uncom 3719
unitunit (in a ring) df-unit 18465 (Unit‘𝑅) Yes isunit 18480, nzrunit 19088
vdisjoint variable conditions used when a not-free hypothesis (suffix) No spimv 2245
vv2 disjoint variables (in a not-free hypothesis) (suffix) No 19.23vv 1890
wweak (version of a theorem) (suffix) No ax11w 1994, spnfw 1915
wrdword df-word 13154 Word 𝑆 Yes iswrdb 13166, wrdfn 13174, ffz0iswrd 13187
xpcross product (Cartesian product) df-xp 5044 (𝐴 × 𝐵) Yes elxp 5055, opelxpi 5072, xpundi 5094
xreXtended reals df-xr 9957 * Yes ressxr 9962, rexr 9964, 0xr 9965
z integers (from German "Zahlen") df-z 11255 Yes elz 11256, zcn 11259
zn ring of integers mod 𝑛 df-zn 19674 (ℤ/nℤ‘𝑁) Yes znval 19702, zncrng 19712, znhash 19726
zringring of integers df-zring 19638 ring Yes zringbas 19643, zringcrng 19639
0, z slashed zero (empty set) (see n0) df-nul 3875 Yes n0i 3879, vn0 3883; snnz 4252, prnz 4253

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

𝜑       𝜑

18.1.2  Natural deduction

Theoremnatded 26652 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 553 jca 553, pm3.2i 470 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 474 simpld 474, adantr 480 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 478 simpr 476, adantl 481 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 449 ex 449 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 699, imp 444 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 407 olc 398, olci 405, olcd 407 Definition I in [Pfenning] p. 18, definition In(1) in [Clemente] p. 12
IR Γ𝜒 => Γ𝜓𝜒 orcd 406 orc 399, orci 404, orcd 406 Definition IR in [Pfenning] p. 18, definition In(2) in [Clemente] p. 12.
E Γ𝜓𝜒 & Γ, 𝜓𝜃 & Γ, 𝜒𝜃 => Γ𝜃 mpjaodan 823 mpjaodan 823, jaodan 822, jaod 394 Definition E in [Pfenning] p. 18, definition Em,n,p in [Clemente] p. 12.
¬I Γ, 𝜓 => Γ¬ 𝜓 inegd 1494 pm2.01d 180
¬I Γ, 𝜓𝜃 & Γ¬ 𝜃 => Γ¬ 𝜓 mtand 689 mtand 689 definition I¬m,n,p in [Clemente] p. 13.
¬I Γ, 𝜓𝜒 & Γ, 𝜓¬ 𝜒 => Γ¬ 𝜓 pm2.65da 598 pm2.65da 598 Contradiction.
¬I Γ, 𝜓¬ 𝜓 => Γ¬ 𝜓 pm2.01da 457 pm2.01d 180, pm2.65da 598, pm2.65d 186 For an alternative falsum-free natural deduction ruleset
¬E Γ𝜓 & Γ¬ 𝜓 => Γ pm2.21fal 1496 pm2.21dd 185
¬E Γ, ¬ 𝜓 => Γ𝜓 pm2.21dd 185 definition E in [Indrzejczak] p. 33.
¬E Γ𝜓 & Γ¬ 𝜓 => Γ𝜃 pm2.21dd 185 pm2.21dd 185, pm2.21d 117, pm2.21 119 For an alternative falsum-free natural deduction ruleset. Definition ¬E in [Pfenning] p. 18.
I Γ a1tru 1491 tru 1479, a1tru 1491, trud 1484 Definition I in [Pfenning] p. 18.
E Γ, ⊥𝜃 falimd 1490 falim 1489 Definition E in [Pfenning] p. 18.
I Γ[𝑎 / 𝑥]𝜓 => Γ𝑥𝜓 alrimiv 1842 alrimiv 1842, ralrimiva 2949 Definition Ia in [Pfenning] p. 18, definition In in [Clemente] p. 32.
E Γ𝑥𝜓 => Γ[𝑡 / 𝑥]𝜓 spsbcd 3416 spcv 3272, rspcv 3278 Definition E in [Pfenning] p. 18, definition En,t in [Clemente] p. 32.
I Γ[𝑡 / 𝑥]𝜓 => Γ𝑥𝜓 spesbcd 3488 spcev 3273, rspcev 3282 Definition I in [Pfenning] p. 18, definition In,t in [Clemente] p. 32.
E Γ𝑥𝜓 & Γ, [𝑎 / 𝑥]𝜓𝜃 => Γ𝜃 exlimddv 1850 exlimddv 1850, exlimdd 2075, exlimdv 1848, rexlimdva 3013 Definition Ea,u in [Pfenning] p. 18, definition Em,n,p,a in [Clemente] p. 32.
C Γ, ¬ 𝜓 => Γ𝜓 efald 1495 efald 1495 Proof by contradiction (classical logic), definition C in [Pfenning] p. 17.
C Γ, ¬ 𝜓𝜓 => Γ𝜓 pm2.18da 458 pm2.18da 458, pm2.18d 123, pm2.18 121 For an alternative falsum-free natural deduction ruleset
¬ ¬C Γ¬ ¬ 𝜓 => Γ𝜓 notnotrd 127 notnotrd 127, notnotr 124 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E¬n in [Clemente] p. 14.
EM Γ𝜓 ∨ ¬ 𝜓 exmidd 431 exmid 430 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
=I Γ𝐴 = 𝐴 eqidd 2611 eqid 2610, eqidd 2611 Introduce equality, definition =I in [Pfenning] p. 127.
=E Γ𝐴 = 𝐵 & Γ[𝐴 / 𝑥]𝜓 => Γ[𝐵 / 𝑥]𝜓 sbceq1dd 3408 sbceq1d 3407, 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 26653, ex-natded5.3 26656, ex-natded5.5 26659, ex-natded5.7 26660, ex-natded5.8 26662, ex-natded5.13 26664, ex-natded9.20 26666, and ex-natded9.26 26668.

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

𝜑       𝜑

18.1.3  Natural deduction examples

These are examples of how natural deduction rules can be applied in Metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 26652 and mmnatded.html 26652. 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 26653 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 553, 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 26654. A proof without context is shown in ex-natded5.2i 26655. (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 26654 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 26653 and ex-natded5.2i 26655. (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 26655 The same as ex-natded5.2 26653 and ex-natded5.2-2 26654 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 26656 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 26657. A proof without context is shown in ex-natded5.3i 26658. 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 480 to move it into the ND hypothesis
25;6 (𝜒𝜃) (𝜑 → (𝜒𝜃)) Given \$e; adantr 480 to move it into the ND hypothesis
31 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 476, to access the new assumption
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15, the MPE equivalent of E, 1.3. adantr 480 was used to transform its dependency (we could also use imp 444 to get this directly from 1)
57 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15, the MPE equivalent of E, 4,6. adantr 480 was used to transform its dependency
68 ... (𝜒𝜃) ((𝜑𝜓) → (𝜒𝜃)) I 4,5 jca 553, the MPE equivalent of I, 4,7
79 (𝜓 → (𝜒𝜃)) (𝜑 → (𝜓 → (𝜒𝜃))) I 3,6 ex 449, 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 26657 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 26656 and ex-natded5.3i 26658. (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 26658 The same as ex-natded5.3 26656 and ex-natded5.3-2 26657 but with no context. Identical to jccir 560, 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 26659 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 480 to move it into the ND hypothesis
25 ¬ 𝜒 (𝜑 → ¬ 𝜒) Given \$e; we'll use adantr 480 to move it into the ND hypothesis
31 ...| 𝜓 (𝜑𝜓) ND hypothesis assumption simpr 476
44 ... 𝜒 ((𝜑𝜓) → 𝜒) E 1,3 mpd 15 1,3
56 ... ¬ 𝜒 ((𝜑𝜓) → ¬ 𝜒) IT 2 adantr 480 5
67 ¬ 𝜓 (𝜑 → ¬ 𝜓) I 3,4,5 pm2.65da 598 4,6

The original used Latin letters; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including 𝜑 and uses the Metamath equivalents of the natural deduction rules. To add an assumption, the antecedent is modified to include it (typically by using adantr 480; simpr 476 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 188; a proof without context is shown in mto 187.

(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 26660 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 26661. 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 480 because we do not move this into an ND hypothesis
21 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption (new scope) simpr 476
32 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) IL 2 orcd 406, the MPE equivalent of IL, 1
43 ...| (𝜒𝜃) ((𝜑 ∧ (𝜒𝜃)) → (𝜒𝜃)) ND hypothesis assumption (new scope) simpr 476
54 ... 𝜒 ((𝜑 ∧ (𝜒𝜃)) → 𝜒) EL 4 simpld 474, the MPE equivalent of EL, 3
66 ... (𝜓𝜒) ((𝜑 ∧ (𝜒𝜃)) → (𝜓𝜒)) IR 5 olcd 407, the MPE equivalent of IR, 4
77 (𝜓𝜒) (𝜑 → (𝜓𝜒)) E 1,3,6 mpjaodan 823, 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 26661 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 26660. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 ∨ (𝜒𝜃)))       (𝜑 → (𝜓𝜒))

Theoremex-natded5.8 26662 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 480 to move it into the ND hypothesis
23;4 (𝜏𝜃) (𝜑 → (𝜏𝜃)) Given \$e; adantr 480 to move it into the ND hypothesis
37;8 𝜒 (𝜑𝜒) Given \$e; adantr 480 to move it into the ND hypothesis
41;2 𝜏 (𝜑𝜏) Given \$e. adantr 480 to move it into the ND hypothesis
56 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND Hypothesis/Assumption simpr 476. New ND hypothesis scope, each reference outside the scope must change antecedent 𝜑 to (𝜑𝜓).
69 ... (𝜓𝜒) ((𝜑𝜓) → (𝜓𝜒)) I 5,3 jca 553 (I), 6,8 (adantr 480 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 598 (¬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 480; simpr 476 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 26663.

(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 26663 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 26662. (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 26664 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 26665. 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 480 to move it into the ND hypothesis
39 𝜏 → ¬ 𝜒) (𝜑 → (¬ 𝜏 → ¬ 𝜒)) Given \$e. ad2antrr 758 to move it into the ND sub-hypothesis
41 ...| 𝜓 ((𝜑𝜓) → 𝜓) ND hypothesis assumption simpr 476
54 ... 𝜃 ((𝜑𝜓) → 𝜃) E 2,4 mpd 15 1,3
65 ... (𝜃𝜏) ((𝜑𝜓) → (𝜃𝜏)) I 5 orcd 406 4
76 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 476
88 ... ...| ¬ 𝜏 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜏) (sub) ND hypothesis assumption simpr 476
911 ... ... ¬ 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → ¬ 𝜒) E 3,8 mpd 15 8,10
107 ... ... 𝜒 (((𝜑𝜒) ∧ ¬ 𝜏) → 𝜒) IT 7 adantr 480 6
1112 ... ¬ ¬ 𝜏 ((𝜑𝜒) → ¬ ¬ 𝜏) ¬I 8,9,10 pm2.65da 598 7,11
1213 ... 𝜏 ((𝜑𝜒) → 𝜏) ¬E 11 notnotrd 127 12
1314 ... (𝜃𝜏) ((𝜑𝜒) → (𝜃𝜏)) I 12 olcd 407 13
1416 (𝜃𝜏) (𝜑 → (𝜃𝜏)) E 1,6,13 mpjaodan 823 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 480; simpr 476 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 26665 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 26664. (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 26666 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 474 1
311 (𝜒𝜃) (𝜑 → (𝜒𝜃)) ER 1 simprd 478 1
44 ...| 𝜒 ((𝜑𝜒) → 𝜒) ND hypothesis assumption simpr 476
55 ... (𝜓𝜒) ((𝜑𝜒) → (𝜓𝜒)) I 2,4 jca 553 3,4
66 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜒) → ((𝜓𝜒) ∨ (𝜓𝜃))) IR 5 orcd 406 5
78 ...| 𝜃 ((𝜑𝜃) → 𝜃) ND hypothesis assumption simpr 476
89 ... (𝜓𝜃) ((𝜑𝜃) → (𝜓𝜃)) I 2,7 jca 553 7,8
910 ... ((𝜓𝜒) ∨ (𝜓𝜃)) ((𝜑𝜃) → ((𝜓𝜒) ∨ (𝜓𝜃))) IL 8 olcd 407 9
1012 ((𝜓𝜒) ∨ (𝜓𝜃)) (𝜑 → ((𝜓𝜒) ∨ (𝜓𝜃))) E 3,6,9 mpjaodan 823 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 480; simpr 476 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 26667. (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 26667 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 26666. (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 26668* 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 476. Later statements will have this scope.
37;5,4 ... 𝜓(𝑥, 𝑦) ((𝜑 ∧ ∀𝑦𝜓) → 𝜓) E 2,y spsbcd 3416 (E), 5,6. To use it we need a1i 11 and vex 3176. This could be immediately done with 19.21bi 2047, 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 3488 (I), 11. To use it we need sylibr 223, which in turn requires sylib 207 and two uses of sbcid 3419. This could be more immediately done using 19.8a 2039, but we want to show the general approach for substitution.
513;1,2 𝑥𝜓(𝑥, 𝑦) (𝜑 → ∃𝑥𝜓) E 1,2,4,a exlimdd 2075 (E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1830 and nfe1 2014 (MPE# 1,2)
614 𝑦𝑥𝜓(𝑥, 𝑦) (𝜑 → ∀𝑦𝑥𝜓) I 5 alrimiv 1842 (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 26669.

(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 26669* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 26668. (Contributed by Mario Carneiro, 9-Feb-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∃𝑥𝑦𝜓)       (𝜑 → ∀𝑦𝑥𝜓)

18.1.4  Definitional examples

Theoremex-or 26670 Example for df-or 384. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 3 ∨ 4 = 4)

Theoremex-an 26671 Example for df-an 385. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
(2 = 2 ∧ 3 = 3)

Theoremex-dif 26672 Example for df-dif 3543. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∖ {1, 8}) = {3}

Theoremex-un 26673 Example for df-un 3545. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∪ {1, 8}) = {1, 3, 8}

Theoremex-in 26674 Example for df-in 3547. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
({1, 3} ∩ {1, 8}) = {1}

Theoremex-uni 26675 Example for df-uni 4373. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
{{1, 3}, {1, 8}} = {1, 3, 8}

Theoremex-ss 26676 Example for df-ss 3554. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊆ {1, 2, 3}

Theoremex-pss 26677 Example for df-pss 3556. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{1, 2} ⊊ {1, 2, 3}

Theoremex-pw 26678 Example for df-pw 4110. 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 26679 Example for df-pr 4128. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐴 ∈ {1, -1} → (𝐴↑2) = 1)

Theoremex-br 26680 Example for df-br 4584. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅 = {⟨2, 6⟩, ⟨3, 9⟩} → 3𝑅9)

Theoremex-opab 26681* Example for df-opab 4644. 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 26682 Example for df-eprel 4949. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
5 E {1, 5}

Theoremex-id 26683 Example for df-id 4953. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(5 I 5 ∧ ¬ 4 I 5)

Theoremex-po 26684 Example for df-po 4959. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
( < Po ℝ ∧ ¬ ≤ Po ℝ)

Theoremex-xp 26685 Example for df-xp 5044. 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 26686 Example for df-cnv 5046. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
{⟨2, 6⟩, ⟨3, 9⟩} = {⟨6, 2⟩, ⟨9, 3⟩}

Theoremex-co 26687 Example for df-co 5047. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((exp ∘ cos)‘0) = e

Theoremex-dm 26688 Example for df-dm 5048. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → dom 𝐹 = {2, 3})

Theoremex-rn 26689 Example for df-rn 5049. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → ran 𝐹 = {6, 9})

Theoremex-res 26690 Example for df-res 5050. 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 26691 Example for df-ima 5051. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
((𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} ∧ 𝐵 = {1, 2}) → (𝐹𝐵) = {6})

Theoremex-fv 26692 Example for df-fv 5812. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
(𝐹 = {⟨2, 6⟩, ⟨3, 9⟩} → (𝐹‘3) = 9)

Theoremex-1st 26693 Example for df-1st 7059. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(1st ‘⟨3, 4⟩) = 3

Theoremex-2nd 26694 Example for df-2nd 7060. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
(2nd ‘⟨3, 4⟩) = 4

Theorem1kp2ke3k 26695 Example for df-dec 11370, 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 11370 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 26696 Example for df-fl 12455. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
((⌊‘(3 / 2)) = 1 ∧ (⌊‘-(3 / 2)) = -2)

Theoremex-ceil 26697 Example for df-ceil 12456. (Contributed by AV, 4-Sep-2021.)
((⌈‘(3 / 2)) = 2 ∧ (⌈‘-(3 / 2)) = -1)

Theoremex-mod 26698 Example for df-mod 12531. (Contributed by AV, 3-Sep-2021.)
((5 mod 3) = 2 ∧ (-7 mod 2) = 1)

Theoremex-exp 26699 Example for df-exp 12723. (Contributed by AV, 4-Sep-2021.)
((5↑2) = 25 ∧ (-3↑-2) = (1 / 9))

Theoremex-fac 26700 Example for df-fac 12923. (Contributed by AV, 4-Sep-2021.)
(!‘5) = 120

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