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

Theoremclwwlknndef 26301 Conditions for ClWWalksN not being defined. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
((𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = ∅)

Theoremclwwlkn0 26302 There is no closed walk of length 0 in an undirected simple graph. (Contributed by Alexander van der Vekens, 15-Sep-2018.)
(𝑉 USGrph 𝐸 → ((𝑉 ClWWalksN 𝐸)‘0) = ∅)

Theoremclwwlkn2 26303 In an undirected simple graph, a closed walk of length 2 represented as word is a word consisting of 2 symbols representing vertices connected by an edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.)
(𝑉 USGrph 𝐸 → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘2) ↔ ((#‘𝑊) = 2 ∧ 𝑊 ∈ Word 𝑉 ∧ {(𝑊‘0), (𝑊‘1)} ∈ ran 𝐸)))

Theoremclwwlknimp 26304* Implications for a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
(𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸))

Theoremclwwlksswrd 26305 Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWWalks 𝐸) ⊆ Word 𝑉)

Theoremclwwlknfi 26306 If there is only a finite number of vertices, the number of closed walk of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉 ∈ Fin ∧ 𝐸𝑋𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) ∈ Fin)

Theoremclwlkisclwwlklem2a1 26307* Lemma 1 for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))

Theoremclwlkisclwwlklem2a2 26308* Lemma 3 for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))

Theoremclwlkisclwwlklem2a3 26309* Lemma 3 for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (𝑃‘(#‘𝐹)) = ( lastS ‘𝑃))

Theoremclwlkisclwwlklem2fv1 26310* Lemma 4a for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       (((#‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((#‘𝑃) − 2))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))

Theoremclwlkisclwwlklem2fv2 26311* Lemma 4b for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       (((#‘𝑃) ∈ ℕ0 ∧ 2 ≤ (#‘𝑃)) → (𝐹‘((#‘𝑃) − 2)) = (𝐸‘{(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)}))

Theoremclwlkisclwwlklem2a4 26312* Lemma 4 for clwlkisclwwlklem2a 26313. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((#‘𝑃) − 1))) → ({(𝑃𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹𝐼)) = {(𝑃𝐼), (𝑃‘(𝐼 + 1))})))

Theoremclwlkisclwwlklem2a 26313* Lemma 2 for clwlkisclwwlklem2 26314. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}), (𝐸‘{(𝑃𝑥), (𝑃‘0)})))       ((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremclwlkisclwwlklem2 26314* Lemma for clwlkisclwwlk 26317. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))))

Theoremclwlkisclwwlklem1 26315* Lemma for clwlkisclwwlk 26317. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
(((𝑉 USGrph 𝐸𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ (∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐹) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸))

Theoremclwlkisclwwlklem0 26316* Lemma for clwlkisclwwlk 26317. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))

Theoremclwlkisclwwlk 26317* A closed walk as word corresponds to a closed walk in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)𝑃 ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (𝑃 substr ⟨0, ((#‘𝑃) − 1)⟩) ∈ (𝑉 ClWWalks 𝐸))))

Theoremclwlkisclwwlk2 26318* A closed walk corresponds to a closed walk as word in an undirected graph. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
((𝑉 USGrph 𝐸𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(𝑉 ClWalks 𝐸)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (𝑉 ClWWalks 𝐸)))

Theoremclwwlkisclwwlkn 26319 A closed walk of a fixed length as word is a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → 𝑃 ∈ (𝑉 ClWWalks 𝐸)))

Theoremclwwlkssclwwlkn 26320 The closed walks of a fixed length as words are closed walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) ⊆ (𝑉 ClWWalks 𝐸))

Theoremclwwlkel 26321* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 20-Oct-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}       (((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) ∧ (𝑃 ∈ Word 𝑉 ∧ (#‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑃), (𝑃‘0)} ∈ ran 𝐸)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ 𝐷)

Theoremclwwlkf 26322* Lemma 1 for clwwlkbij 26327: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwwlkfv 26323* Lemma 2 for clwwlkbij 26327: the value of function F. (Contributed by Alexander van der Vekens, 28-Sep-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       (𝑊𝐷 → (𝐹𝑊) = (𝑊 substr ⟨0, 𝑁⟩))

Theoremclwwlkf1 26324* Lemma 3 for clwwlkbij 26327: F is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Proof shortened by AV, 23-Oct-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwwlkfo 26325* Lemma 4 for clwwlkbij 26327: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwwlkf1o 26326* Lemma 5 for clwwlkbij 26327: F is a 1-1 onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
𝐷 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}    &   𝐹 = (𝑡𝐷 ↦ (𝑡 substr ⟨0, 𝑁⟩))       ((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → 𝐹:𝐷1-1-onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwwlkbij 26327* There is a bijection between the set of closed walks of a fixed length represented by walks (as word) and the set of closed walks (as words) of a fixed length. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)}–1-1-onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwwlknwwlkncl 26328* Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
((𝑁 ∈ ℕ ∧ 𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ ( lastS ‘𝑤) = (𝑤‘0)})

Theoremclwwlkvbij 26329* There is a bijection between the set of closed walks of a fixed length starting at a fixed vertex represented by walks (as word) and the set of closed walks (as words) of a fixed length starting at a fixed vertex. The difference between these two representations is that in the first case the starting vertex is repeated at the end of the word, and in the second case it is not. (Contributed by Alexander van der Vekens, 29-Sep-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑆)}–1-1-onto→{𝑤 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑆})

Theoremclwwlkext2edg 26330 If a word concatenated with a vertex represents a closed walk in (in a graph), there is an edge between this vertex and the last vertex of the word, and between this vertex and the first vertex of the word. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
(((𝑊 ∈ Word 𝑉𝑍𝑉𝑁 ∈ (ℤ‘2)) ∧ (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸))

Theoremwwlkext2clwwlk 26331 If a word represents a walk in (in a graph) and there are edges between the last vertex of the word and another vertex and between this other vertex and the first vertex of the word, then the concatenation of the word representing the walk with this other vertex represents a closed walk. (Contributed by Alexander van der Vekens, 3-Oct-2018.)
((𝑊 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∧ 𝑍𝑉𝑁 ∈ ℕ0) → (({( lastS ‘𝑊), 𝑍} ∈ ran 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ ran 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑉 ClWWalksN 𝐸)‘(𝑁 + 2))))

Theoremwwlksubclwwlk 26332 Any prefix of a word representing a closed walk represents a word. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ‘(𝑀 + 1))) → (𝑋 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → (𝑋 substr ⟨0, 𝑀⟩) ∈ ((𝑉 WWalksN 𝐸)‘(𝑀 − 1))))

Theoremclwwisshclwwlem1 26333* Lemma 1 for clwwisshclwwlem 26334. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
(((𝐿 ∈ (ℤ‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅)

Theoremclwwisshclwwlem 26334* Lemma for clwwisshclww 26335. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 10-Jun-2018.) (Proof shortened by AV, 2-Nov-2018.)
((𝑊 ∈ Word 𝑉𝑁 ∈ (1..^(#‘𝑊))) → ((∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸) → ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ ran 𝐸))

Theoremclwwisshclww 26335 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Mar-2018.) (Revised by Alexander van der Vekens, 10-Jun-2018.)
((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))

Theoremclwwisshclwwn 26336 Cyclically shifting a closed walk as word results in a closed walk as word (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (𝑉 ClWWalks 𝐸))

Theoremclwwnisshclwwn 26337 Cyclically shifting a closed walk as word of fixed length results in a closed walk as word of the same length (in an undirected graph). (Contributed by Alexander van der Vekens, 10-Jun-2018.)
((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑀 ∈ (0...𝑁) → (𝑊 cyclShift 𝑀) ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)))

Theoremerclwwlkrel 26338 is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       Rel

Theoremerclwwlkeq 26339* Two classes are equivalent regarding if both are words and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 ↔ (𝑈 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))

Theoremerclwwlkeqlen 26340* If two classes are equivalent regarding , then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑈𝑋𝑊𝑌) → (𝑈 𝑊 → (#‘𝑈) = (#‘𝑊)))

Theoremerclwwlkref 26341* is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       (𝑥 ∈ (𝑉 ClWWalks 𝐸) ↔ 𝑥 𝑥)

Theoremerclwwlksym 26342* is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       (𝑥 𝑦𝑦 𝑥)

Theoremerclwwlktr 26343* is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}       ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)

Theoremerclwwlk 26344* is an equivalence relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 11-Jun-2018.) (Proof shortened by AV, 1-May-2021.)
= {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (𝑉 ClWWalks 𝐸) ∧ 𝑤 ∈ (𝑉 ClWWalks 𝐸) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}        Er (𝑉 ClWWalks 𝐸)

Theoremeleclclwwlknlem1 26345* Lemma 1 for eleclclwwlkn 26360. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)       ((𝐾 ∈ (0...𝑁) ∧ (𝑋𝑊𝑌𝑊)) → ((𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))

Theoremeleclclwwlknlem2 26346* Lemma 2 for eleclclwwlkn 26360. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)       (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋𝑊𝑥𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))

Theoremclwwlknscsh 26347* The set of cyclical shifts of a word representing a closed walk is the set of closed walks represented by cyclical shifts of a word. (Contributed by Alexander van der Vekens, 15-Jun-2018.)
((𝑁 ∈ ℕ0𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → {𝑦 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word 𝑉 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})

Theoremusg2cwwk2dif 26348 If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2) ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → (𝑊‘1) ≠ (𝑊‘0))

Theoremusg2cwwkdifex 26349* If a word represents a closed walk of length at least 2 in a undirected simple graph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ (ℤ‘2) ∧ 𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁)) → ∃𝑖 ∈ (0..^𝑁)(𝑊𝑖) ≠ (𝑊‘0))

Theoremerclwwlknrel 26350 is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       Rel

Theoremerclwwlkneq 26351* Two classes are equivalent regarding if both are words of the same fixed length and one is the other cyclically shifted. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 ↔ (𝑇𝑊𝑈𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))

Theoremerclwwlkneqlen 26352* If two classes are equivalent regarding , then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑇𝑋𝑈𝑌) → (𝑇 𝑈 → (#‘𝑇) = (#‘𝑈)))

Theoremerclwwlknref 26353* is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝑥𝑊𝑥 𝑥)

Theoremerclwwlknsym 26354* is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝑥 𝑦𝑦 𝑥)

Theoremerclwwlkntr 26355* is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by Alexander van der Vekens, 14-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑥 𝑦𝑦 𝑧) → 𝑥 𝑧)

Theoremerclwwlkn 26356* is an equivalence relation over the set of closed walks (defined as words) with a fixed length. (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Proof shortened by AV, 1-May-2021.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}        Er 𝑊

Theoremqerclwwlknfi 26357* The quotient set of the set of closed walks (defined as words) with a fixed length according to the equivalence relation is finite. (Contributed by Alexander van der Vekens, 10-Apr-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑉 ∈ Fin ∧ 𝐸𝑋𝑁 ∈ ℕ0) → (𝑊 / ) ∈ Fin)

Theoremhashclwwlkn0 26358* The number of closed walks (defined as words) with a fixed length is the sum of the sizes of all equivalence classes according to . (Contributed by Alexander van der Vekens, 10-Apr-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑉 ∈ Fin ∧ 𝐸𝑋𝑁 ∈ ℕ0) → (#‘𝑊) = Σ𝑥 ∈ (𝑊 / )(#‘𝑥))

Theoremeclclwwlkn1 26359* An equivalence class according to . (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by Alexander van der Vekens, 15-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       (𝐵𝑋 → (𝐵 ∈ (𝑊 / ) ↔ ∃𝑥𝑊 𝐵 = {𝑦𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))

Theoremeleclclwwlkn 26360* A member of an equivalence class according to . (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by Alexander van der Vekens, 15-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝐵 ∈ (𝑊 / ) ∧ 𝑋𝐵) → (𝑌𝐵 ↔ (𝑌𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))

Theoremhashecclwwlkn1 26361* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number is 1 or equals this length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / )) → ((#‘𝑈) = 1 ∨ (#‘𝑈) = 𝑁))

Theoremusghashecclwwlk 26362* The size of every equivalence class of the equivalence relation over the set of closed walks (defined as words) with a fixed length which is a prime number equals this length (in an undirected simple graph). (Contributed by Alexander van der Vekens, 17-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ) → (#‘𝑈) = 𝑁))

Theoremhashclwwlkn 26363* The size of the set of closed walks (defined as words) with a fixed length which is a prime number is the product of the number of equivalence classes for over the set of closed walks and the fixed length. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
𝑊 = ((𝑉 ClWWalksN 𝐸)‘𝑁)    &    = {⟨𝑡, 𝑢⟩ ∣ (𝑡𝑊𝑢𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}       ((𝑉 ∈ Fin ∧ 𝑉 USGrph 𝐸𝑁 ∈ ℙ) → (#‘𝑊) = ((#‘(𝑊 / )) · 𝑁))

Theoremclwwlkndivn 26364 The size of the set of closed walks (defined as words) of length n is divisible by n. (Contributed by Alexander van der Vekens, 17-Jun-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘((𝑉 ClWWalksN 𝐸)‘𝑁)))

Theoremwlklenvp1 26365 The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (#‘𝑃) = ((#‘𝐹) + 1))

Theoremwlklenvclwlk 26366 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.)
((𝑊 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑊)) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (𝑉 Walks 𝐸) → (#‘𝐹) = (#‘𝑊)))

Theoremclwlkfclwwlk2wrd 26367 The second component of a closed walk is a word over the "vertices". (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶𝐵 ∈ Word 𝑉)

Theoremclwlkfclwwlk1hashn 26368* The size of the first component of a closed walk. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (#‘(1st𝑊)) = 𝑁)

Theoremclwlkfclwwlk1hash 26369* The size of the first component of a closed walk is an integer in the range between 0 and the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) ∈ (0...(#‘𝐵)))

Theoremclwlkfclwwlk2sswd 26370* The size of a subword of the second component of a closed walk with length of the size of the second component. (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑐𝐶 → (#‘𝐴) = (#‘(𝐵 substr ⟨0, (#‘𝐴)⟩)))

Theoremclwlkfclwwlk 26371* There is a function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 25-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶⟶((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkfoclwwlk 26372* There is an onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 30-Jun-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkf1clwwlklem1 26373* Lemma 1 for clwlkf1clwwlklem 26376. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶𝑁 ≤ (#‘(2nd𝑊)))

Theoremclwlkf1clwwlklem2 26374* Lemma 2 for clwlkf1clwwlklem 26376. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → ((2nd𝑊)‘0) = ((2nd𝑊)‘𝑁))

Theoremclwlkf1clwwlklem3 26375* Lemma 3 for clwlkf1clwwlklem 26376. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       (𝑊𝐶 → (2nd𝑊) ∈ Word 𝑉)

Theoremclwlkf1clwwlklem 26376* Lemma for clwlkf1clwwlk 26377. (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑁 ∈ ℕ ∧ 𝑈𝐶𝑊𝐶) → (((2nd𝑈) substr ⟨0, (#‘(1st𝑈))⟩) = ((2nd𝑊) substr ⟨0, (#‘(1st𝑊))⟩) → ∀𝑦 ∈ (0...𝑁)((2nd𝑈)‘𝑦) = ((2nd𝑊)‘𝑦)))

Theoremclwlkf1clwwlk 26377* There is a one-to-one function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlkf1oclwwlk 26378* There is a one-to-one onto function between the set of closed walks (defined as words) of length n and the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 5-Jul-2018.)
𝐴 = (1st𝑐)    &   𝐵 = (2nd𝑐)    &   𝐶 = {𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘𝐴) = 𝑁}    &   𝐹 = (𝑐𝐶 ↦ (𝐵 substr ⟨0, (#‘𝐴)⟩))       ((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝐹:𝐶1-1-onto→((𝑉 ClWWalksN 𝐸)‘𝑁))

Theoremclwlksizeeq 26379* The size of the set of closed walks (defined as words) of length n corresponds to the size of the set of closed walks of length n (in an undirected simple graph). (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Proof shortened by AV, 4-May-2021.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → (#‘((𝑉 ClWWalksN 𝐸)‘𝑁)) = (#‘{𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))

Theoremclwlkndivn 26380* The size of the set of closed walks of length n is divisible by n. 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". (Contributed by Alexander van der Vekens, 6-Jul-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (#‘{𝑐 ∈ (𝑉 ClWalks 𝐸) ∣ (#‘(1st𝑐)) = 𝑁}))

17.1.5.7  Walks/paths of length 2 as ordered triples

Syntaxc2wlkot 26381 Extend class notation with walks (of a graph) of length 2 as ordered triple.
class 2WalksOt

Syntaxc2wlkonot 26382 Extend class notation with walks between two vertices (within a graph) of length 2 as ordered triple.
class 2WalksOnOt

Syntaxc2spthot 26383 Extend class notation with paths (of a graph) of length 2 as ordered triple.
class 2SPathsOt

Syntaxc2pthonot 26384 Extend class notation with simple paths between two vertices (within a graph) of length 2 as ordered triple.
class 2SPathOnOt

Definitiondf-2wlkonot 26385* Define the collection of walks of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
2WalksOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Definitiondf-2wlksot 26386* Define the collection of all walks of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
2WalksOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2WalksOnOt 𝑒)𝑏)})

Definitiondf-2spthonot 26387* Define the collection of simple paths of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
2SPathOnOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑣 SPathOn 𝑒)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Definitiondf-2spthsot 26388* Define the collection of all simple paths of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
2SPathsOt = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑡 ∈ ((𝑣 × 𝑣) × 𝑣) ∣ ∃𝑎𝑣𝑏𝑣 𝑡 ∈ (𝑎(𝑣 2SPathOnOt 𝑒)𝑏)})

Theoremel2wlkonotlem 26389 Lemma for el2wlkonot 26396. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2) → (𝑝‘1) ∈ 𝑉)

Theoremis2wlkonot 26390* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 WalkOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Theoremis2spthonot 26391* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2SPathOnOt 𝐸) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝑎(𝑉 SPathOn 𝐸)𝑏)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝑎 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝑏))}))

Theorem2wlkonot 26392* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2WalksOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})

Theorem2spthonot 26393* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 2SPathOnOt 𝐸)𝐵) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑓𝑝(𝑓(𝐴(𝑉 SPathOn 𝐸)𝐵)𝑝 ∧ (#‘𝑓) = 2 ∧ ((1st ‘(1st𝑡)) = 𝐴 ∧ (2nd ‘(1st𝑡)) = (𝑝‘1) ∧ (2nd𝑡) = 𝐵))})

Theorem2wlksot 26394* The set of walks of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2WalksOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2WalksOnOt 𝐸)𝑏)})

Theorem2spthsot 26395* The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 2SPathsOt 𝐸) = {𝑡 ∈ ((𝑉 × 𝑉) × 𝑉) ∣ ∃𝑎𝑉𝑏𝑉 𝑡 ∈ (𝑎(𝑉 2SPathOnOt 𝐸)𝑏)})

Theoremel2wlkonot 26396* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2spthonot 26397* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ∃𝑓𝑝(𝑓(𝑉 SPaths 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2spthonot0 26398* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (𝑇 ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶) ↔ ∃𝑏𝑉 (𝑇 = ⟨𝐴, 𝑏, 𝐶⟩ ∧ ⟨𝐴, 𝑏, 𝐶⟩ ∈ (𝐴(𝑉 2SPathOnOt 𝐸)𝐶))))

Theoremel2wlkonotot0 26399* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝑅𝑉𝑆𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝑅(𝑉 2WalksOnOt 𝐸)𝑆) ↔ (𝐴 = 𝑅𝐶 = 𝑆 ∧ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))))

Theoremel2wlkonotot 26400* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐶𝑉)) → (⟨𝐴, 𝐵, 𝐶⟩ ∈ (𝐴(𝑉 2WalksOnOt 𝐸)𝐶) ↔ ∃𝑓𝑝(𝑓(𝑉 Walks 𝐸)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2)))))

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