P. 284 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 28301-28400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	isclwwlknx 28301*	Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁)))
Theorem	clwwlknp 28302*	Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸))
Theorem	clwwlknwwlksn 28303	A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 − 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if ⟨“𝑎𝑏𝑐”⟩ ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then ⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if 𝑎 ≠ 𝑐) of length 2, and ⟨“𝑎𝑏𝑐𝑎”⟩ ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) (Revised by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺))
Theorem	clwwlknlbonbgr1 28304	The last but one vertex in a closed walk is a neighbor of the first vertex of the closed walk. (Contributed by AV, 17-Feb-2022.)
⊢ ((𝐺 ∈ USGraph ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘(𝑁 − 1)) ∈ (𝐺 NeighbVtx (𝑊‘0)))
Theorem	clwwlkinwwlk 28305	If the initial vertex of a walk occurs another time in the walk, the walk starts with a closed walk. Since the walk is expressed as a word over vertices, the closed walk can be expressed as a subword of this word. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 23-Jan-2022.) (Revised by AV, 30-Oct-2022.)
⊢ (((𝑁 ∈ ℕ ∧ 𝑀 ∈ (ℤ≥‘𝑁)) ∧ 𝑊 ∈ (𝑀 WWalksN 𝐺) ∧ (𝑊‘𝑁) = (𝑊‘0)) → (𝑊 prefix 𝑁) ∈ (𝑁 ClWWalksN 𝐺))
Theorem	clwwlkn1 28306	A closed walk of length 1 represented as word is a word consisting of 1 symbol representing a vertex connected to itself by (at least) one edge, that is, a loop. (Contributed by AV, 24-Apr-2021.) (Revised by AV, 11-Feb-2022.)
⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 1 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0)} ∈ (Edg‘𝐺)))
Theorem	loopclwwlkn1b 28307	The singleton word consisting of a vertex 𝑉 represents a closed walk of length 1 iff there is a loop at vertex 𝑉. (Contributed by AV, 11-Feb-2022.)
⊢ (𝑉 ∈ (Vtx‘𝐺) → ({𝑉} ∈ (Edg‘𝐺) ↔ ⟨“𝑉”⟩ ∈ (1 ClWWalksN 𝐺)))
Theorem	clwwlkn1loopb 28308*	A word represents a closed walk of length 1 iff this word is a singleton word consisting of a vertex with an attached loop. (Contributed by AV, 11-Feb-2022.)
⊢ (𝑊 ∈ (1 ClWWalksN 𝐺) ↔ ∃𝑣 ∈ (Vtx‘𝐺)(𝑊 = ⟨“𝑣”⟩ ∧ {𝑣} ∈ (Edg‘𝐺)))
Theorem	clwwlkn2 28309	A closed walk of length 2 represented as word is a word consisting of 2 symbols representing (not necessarily different) vertices connected by (at least) one edge. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Apr-2021.)
⊢ (𝑊 ∈ (2 ClWWalksN 𝐺) ↔ ((♯‘𝑊) = 2 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ {(𝑊‘0), (𝑊‘1)} ∈ (Edg‘𝐺)))
Theorem	clwwlknfi 28310	If there is only a finite number of vertices, the number of closed walks of fixed length (as words) is also finite. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.) (Proof shortened by JJ, 18-Nov-2022.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 ClWWalksN 𝐺) ∈ Fin)
Theorem	clwwlkel 28311*	Obtaining a closed walk (as word) by appending the first symbol to the word representing a walk. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 25-Apr-2021.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ (𝑃 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑃) = 𝑁) ∧ (∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ (Edg‘𝐺) ∧ {(lastS‘𝑃), (𝑃‘0)} ∈ (Edg‘𝐺))) → (𝑃 ++ ⟨“(𝑃‘0)”⟩) ∈ 𝐷)
Theorem	clwwlkf 28312*	Lemma 1 for clwwlkf1o 28316: F is a function. (Contributed by Alexander van der Vekens, 27-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷⟶(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkfv 28313*	Lemma 2 for clwwlkf1o 28316: the value of function 𝐹. (Contributed by Alexander van der Vekens, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑊 ∈ 𝐷 → (𝐹‘𝑊) = (𝑊 prefix 𝑁))
Theorem	clwwlkf1 28314*	Lemma 3 for clwwlkf1o 28316: 𝐹 is a 1-1 function. (Contributed by AV, 28-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkfo 28315*	Lemma 4 for clwwlkf1o 28316: F is an onto function. (Contributed by Alexander van der Vekens, 29-Sep-2018.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlkf1o 28316*	F is a 1-1 onto function, that means that there is a bijection between the set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the 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.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐷 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (𝑡 prefix 𝑁))    ⇒   ⊢ (𝑁 ∈ ℕ → 𝐹:𝐷–1-1-onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwwlken 28317*	The set of closed walks of a fixed length represented by walks (as words) and the set of closed walks (as words) of the fixed length are equinumerous. (Contributed by AV, 5-Jun-2022.) (Proof shortened by AV, 2-Nov-2022.)
⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)} ≈ (𝑁 ClWWalksN 𝐺))
Theorem	clwwlknwwlkncl 28318*	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.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (lastS‘𝑤) = (𝑤‘0)})
Theorem	clwwlkwwlksb 28319	A nonempty word over vertices represents a closed walk iff the word concatenated with its first symbol represents a walk. (Contributed by AV, 4-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) → (𝑊 ∈ (ClWWalks‘𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (WWalks‘𝐺)))
Theorem	clwwlknwwlksnb 28320	A word over vertices represents a closed walk of a fixed length 𝑁 greater than zero iff the word concatenated with its first symbol represents a walk of length 𝑁. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, because (𝑊 ++ ⟨“(𝑊‘0)”⟩) = ⟨“∅”⟩ ∈ (0 WWalksN 𝐺) could be true, but not 𝑊 ∈ (0 ClWWalksN 𝐺) ↔ ∅ ∈ ∅. (Contributed by AV, 4-Mar-2022.) (Proof shortened by AV, 22-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ++ ⟨“(𝑊‘0)”⟩) ∈ (𝑁 WWalksN 𝐺)))
Theorem	clwwlkext2edg 28321	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.) (Revised by AV, 27-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑊 ∈ Word 𝑉 ∧ 𝑍 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘2)) ∧ (𝑊 ++ ⟨“𝑍”⟩) ∈ (𝑁 ClWWalksN 𝐺)) → ({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸))
Theorem	wwlksext2clwwlk 28322	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.) (Revised by AV, 27-Apr-2021.) (Revised by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑍 ∈ 𝑉) → (({(lastS‘𝑊), 𝑍} ∈ 𝐸 ∧ {𝑍, (𝑊‘0)} ∈ 𝐸) → (𝑊 ++ ⟨“𝑍”⟩) ∈ ((𝑁 + 2) ClWWalksN 𝐺)))
Theorem	wwlksubclwwlk 28323	Any prefix of a word representing a closed walk represents a walk. (Contributed by Alexander van der Vekens, 5-Oct-2018.) (Revised by AV, 28-Apr-2021.) (Revised by AV, 1-Nov-2022.)
⊢ ((𝑀 ∈ ℕ ∧ 𝑁 ∈ (ℤ≥‘(𝑀 + 1))) → (𝑋 ∈ (𝑁 ClWWalksN 𝐺) → (𝑋 prefix 𝑀) ∈ ((𝑀 − 1) WWalksN 𝐺)))
Theorem	clwwnisshclwwsn 28324	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.) (Revised by AV, 29-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ ((𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ 𝑀 ∈ (0...𝑁)) → (𝑊 cyclShift 𝑀) ∈ (𝑁 ClWWalksN 𝐺))
Theorem	eleclclwwlknlem1 28325*	Lemma 1 for eleclclwwlkn 28341. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    ⇒   ⊢ ((𝐾 ∈ (0...𝑁) ∧ (𝑋 ∈ 𝑊 ∧ 𝑌 ∈ 𝑊)) → ((𝑋 = (𝑌 cyclShift 𝐾) ∧ ∃𝑚 ∈ (0...𝑁)𝑍 = (𝑌 cyclShift 𝑚)) → ∃𝑛 ∈ (0...𝑁)𝑍 = (𝑋 cyclShift 𝑛)))
Theorem	eleclclwwlknlem2 28326*	Lemma 2 for eleclclwwlkn 28341. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    ⇒   ⊢ (((𝑘 ∈ (0...𝑁) ∧ 𝑋 = (𝑥 cyclShift 𝑘)) ∧ (𝑋 ∈ 𝑊 ∧ 𝑥 ∈ 𝑊)) → (∃𝑚 ∈ (0...𝑁)𝑌 = (𝑥 cyclShift 𝑚) ↔ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛)))
Theorem	clwwlknscsh 28327*	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.) (Revised by AV, 30-Apr-2021.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → {𝑦 ∈ (𝑁 ClWWalksN 𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)} = {𝑦 ∈ Word (Vtx‘𝐺) ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑊 cyclShift 𝑛)})
Theorem	clwwlknccat 28328	The concatenation of two words representing closed walks anchored at the same vertex represents a closed walk with a length which is the sum of the lengths of the two walks. The resulting walk is a "double loop", starting at the common vertex, coming back to the common vertex by the first walk, following the second walk and finally coming back to the common vertex again. (Contributed by AV, 24-Apr-2022.)
⊢ ((𝐴 ∈ (𝑀 ClWWalksN 𝐺) ∧ 𝐵 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝐴‘0) = (𝐵‘0)) → (𝐴 ++ 𝐵) ∈ ((𝑀 + 𝑁) ClWWalksN 𝐺))
Theorem	umgr2cwwk2dif 28329	If a word represents a closed walk of length at least 2 in a multigraph, the first two symbols of the word must be different. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 30-Apr-2021.)
⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → (𝑊‘1) ≠ (𝑊‘0))
Theorem	umgr2cwwkdifex 28330*	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.) (Revised by AV, 30-Apr-2021.)
⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ (ℤ≥‘2) ∧ 𝑊 ∈ (𝑁 ClWWalksN 𝐺)) → ∃𝑖 ∈ (0..^𝑁)(𝑊‘𝑖) ≠ (𝑊‘0))
Theorem	erclwwlknrel 28331	∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ Rel ∼
Theorem	erclwwlkneq 28332*	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 AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 ↔ (𝑇 ∈ 𝑊 ∧ 𝑈 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑇 = (𝑈 cyclShift 𝑛))))
Theorem	erclwwlkneqlen 28333*	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 AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑇 ∈ 𝑋 ∧ 𝑈 ∈ 𝑌) → (𝑇 ∼ 𝑈 → (♯‘𝑇) = (♯‘𝑈)))
Theorem	erclwwlknref 28334*	∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 26-Mar-2018.) (Revised by AV, 30-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∈ 𝑊 ↔ 𝑥 ∼ 𝑥)
Theorem	erclwwlknsym 28335*	∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)
Theorem	erclwwlkntr 28336*	∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)
Theorem	erclwwlkn 28337*	∼ 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.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ∼ Er 𝑊
Theorem	qerclwwlknfi 28338*	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.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (𝑊 / ∼ ) ∈ Fin)
Theorem	hashclwwlkn0 28339*	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.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘𝑊) = Σ𝑥 ∈ (𝑊 / ∼ )(♯‘𝑥))
Theorem	eclclwwlkn1 28340*	An equivalence class according to ∼. (Contributed by Alexander van der Vekens, 12-Apr-2018.) (Revised by AV, 30-Apr-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ (𝐵 ∈ 𝑋 → (𝐵 ∈ (𝑊 / ∼ ) ↔ ∃𝑥 ∈ 𝑊 𝐵 = {𝑦 ∈ 𝑊 ∣ ∃𝑛 ∈ (0...𝑁)𝑦 = (𝑥 cyclShift 𝑛)}))
Theorem	eleclclwwlkn 28341*	A member of an equivalence class according to ∼. (Contributed by Alexander van der Vekens, 11-May-2018.) (Revised by AV, 1-May-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐵 ∈ (𝑊 / ∼ ) ∧ 𝑋 ∈ 𝐵) → (𝑌 ∈ 𝐵 ↔ (𝑌 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑌 = (𝑋 cyclShift 𝑛))))
Theorem	hashecclwwlkn1 28342*	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.) (Revised by AV, 1-May-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝑁 ∈ ℙ ∧ 𝑈 ∈ (𝑊 / ∼ )) → ((♯‘𝑈) = 1 ∨ (♯‘𝑈) = 𝑁))
Theorem	umgrhashecclwwlk 28343*	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.) (Revised by AV, 1-May-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ 𝑁 ∈ ℙ) → (𝑈 ∈ (𝑊 / ∼ ) → (♯‘𝑈) = 𝑁))
Theorem	fusgrhashclwwlkn 28344*	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.) (Revised by AV, 1-May-2021.)
⊢ 𝑊 = (𝑁 ClWWalksN 𝐺)    &   ⊢ ∼ = {⟨𝑡, 𝑢⟩ ∣ (𝑡 ∈ 𝑊 ∧ 𝑢 ∈ 𝑊 ∧ ∃𝑛 ∈ (0...𝑁)𝑡 = (𝑢 cyclShift 𝑛))}    ⇒   ⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → (♯‘𝑊) = ((♯‘(𝑊 / ∼ )) · 𝑁))
Theorem	clwwlkndivn 28345	The size of the set of closed walks (defined as words) of length 𝑁 is divisible by 𝑁 if 𝑁 is a prime number. (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 2-May-2021.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (♯‘(𝑁 ClWWalksN 𝐺)))
Theorem	clwlknf1oclwwlknlem1 28346	Lemma 1 for clwlknf1oclwwlkn 28349. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
⊢ ((𝐶 ∈ (ClWalks‘𝐺) ∧ 1 ≤ (♯‘(1st ‘𝐶))) → (♯‘((2nd ‘𝐶) prefix ((♯‘(2nd ‘𝐶)) − 1))) = (♯‘(1st ‘𝐶)))
Theorem	clwlknf1oclwwlknlem2 28347*	Lemma 2 for clwlknf1oclwwlkn 28349: The closed walks of a positive length are nonempty closed walks of this length. (Contributed by AV, 26-May-2022.)
⊢ (𝑁 ∈ ℕ → {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁} = {𝑐 ∈ (ClWalks‘𝐺) ∣ (1 ≤ (♯‘(1st ‘𝑐)) ∧ (♯‘(1st ‘𝑐)) = 𝑁)})
Theorem	clwlknf1oclwwlknlem3 28348*	Lemma 3 for clwlknf1oclwwlkn 28349: The bijective function of clwlknf1oclwwlkn 28349 is the bijective function of clwlkclwwlkf1o 28276 restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐴 = (1st ‘𝑐)    &   ⊢ 𝐵 = (2nd ‘𝑐)    &   ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹 = ((𝑐 ∈ {𝑤 ∈ (ClWalks‘𝐺) ∣ 1 ≤ (♯‘(1st ‘𝑤))} ↦ (𝐵 prefix (♯‘𝐴))) ↾ 𝐶))
Theorem	clwlknf1oclwwlkn 28349*	There is a one-to-one onto function between the set of closed walks as words of length 𝑁 and the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 5-Jul-2018.) (Revised by AV, 3-May-2021.) (Revised by AV, 1-Nov-2022.)
⊢ 𝐴 = (1st ‘𝑐)    &   ⊢ 𝐵 = (2nd ‘𝑐)    &   ⊢ 𝐶 = {𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}    &   ⊢ 𝐹 = (𝑐 ∈ 𝐶 ↦ (𝐵 prefix (♯‘𝐴)))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → 𝐹:𝐶–1-1-onto→(𝑁 ClWWalksN 𝐺))
Theorem	clwlkssizeeq 28350*	The size of the set of closed walks as words of length 𝑁 corresponds to the size of the set of closed walks of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 6-Jul-2018.) (Revised by AV, 4-May-2021.) (Revised by AV, 26-May-2022.) (Proof shortened by AV, 3-Nov-2022.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ) → (♯‘(𝑁 ClWWalksN 𝐺)) = (♯‘{𝑤 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑤)) = 𝑁}))
Theorem	clwlksndivn 28351*	The size of the set of closed walks of prime 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.) (Revised by AV, 4-May-2021.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℙ) → 𝑁 ∥ (♯‘{𝑐 ∈ (ClWalks‘𝐺) ∣ (♯‘(1st ‘𝑐)) = 𝑁}))
16.3.10.3  Closed walks on a vertex of a fixed length as words
Syntax	cclwwlknon 28352	Extend class notation with closed walks (in an undirected graph) anchored at a fixed vertex and of a fixed length as word over the set of vertices.
class ClWWalksNOn
Definition	df-clwwlknon 28353*	Define the set of all closed walks a graph 𝑔, anchored at a fixed vertex 𝑣 (i.e., a walk starting and ending at the fixed vertex 𝑣, also called "a closed walk on vertex 𝑣") and having a fixed length 𝑛 as words over the set of vertices. Such a word corresponds to the sequence v=p(0) p(1) ... p(n-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0)=v as defined in df-clwlks 28040. The set ((𝑣(ClWWalksNOn‘𝑔)𝑛) corresponds to the set of "walks from v to v of length n" in a statement of [Huneke] p. 2. (Contributed by AV, 24-Feb-2022.)
⊢ ClWWalksNOn = (𝑔 ∈ V ↦ (𝑣 ∈ (Vtx‘𝑔), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝑔) ∣ (𝑤‘0) = 𝑣}))
Theorem	clwwlknonmpo 28354*	(ClWWalksNOn‘𝐺) is an operator mapping a vertex 𝑣 and a nonnegative integer 𝑛 to the set of closed walks on 𝑣 of length 𝑛 as words over the set of vertices in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Proof shortened by AV, 2-Mar-2024.)
⊢ (ClWWalksNOn‘𝐺) = (𝑣 ∈ (Vtx‘𝐺), 𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑛 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑣})
Theorem	clwwlknon 28355*	The set of closed walks on vertex 𝑋 of length 𝑁 in a graph 𝐺 as words over the set of vertices. (Contributed by Alexander van der Vekens, 14-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)
⊢ (𝑋(ClWWalksNOn‘𝐺)𝑁) = {𝑤 ∈ (𝑁 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
Theorem	isclwwlknon 28356	A word over the set of vertices representing a closed walk on vertex 𝑋 of length 𝑁 in a graph 𝐺. (Contributed by AV, 25-Feb-2022.) (Revised by AV, 24-Mar-2022.)
⊢ (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ∧ (𝑊‘0) = 𝑋))
Theorem	clwwlk0on0 28357	There is no word over the set of vertices representing a closed walk on vertex 𝑋 of length 0 in a graph 𝐺. (Contributed by AV, 17-Feb-2022.) (Revised by AV, 25-Feb-2022.)
⊢ (𝑋(ClWWalksNOn‘𝐺)0) = ∅
Theorem	clwwlknon0 28358	Sufficient conditions for ClWWalksNOn to be empty. (Contributed by AV, 25-Mar-2022.)
⊢ (¬ (𝑋 ∈ (Vtx‘𝐺) ∧ 𝑁 ∈ ℕ) → (𝑋(ClWWalksNOn‘𝐺)𝑁) = ∅)
Theorem	clwwlknonfin 28359	In a finite graph 𝐺, the set of closed walks on vertex 𝑋 of length 𝑁 is also finite. (Contributed by Alexander van der Vekens, 26-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑉 ∈ Fin → (𝑋(ClWWalksNOn‘𝐺)𝑁) ∈ Fin)
Theorem	clwwlknonel 28360*	Characterization of a word over the set of vertices representing a closed walk on vertex 𝑋 of (nonzero) length 𝑁 in a graph 𝐺. This theorem would not hold for 𝑁 = 0 if 𝑊 = 𝑋 = ∅. (Contributed by Alexander van der Vekens, 20-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 24-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ≠ 0 → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = 𝑁 ∧ (𝑊‘0) = 𝑋)))
Theorem	clwwlknonccat 28361	The concatenation of two words representing closed walks on a vertex 𝑋 represents a closed walk on vertex 𝑋. The resulting walk is a "double loop", starting at vertex 𝑋, coming back to 𝑋 by the first walk, following the second walk and finally coming back to 𝑋 again. (Contributed by AV, 24-Apr-2022.)
⊢ ((𝐴 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑀) ∧ 𝐵 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁)) → (𝐴 ++ 𝐵) ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑀 + 𝑁)))
Theorem	clwwlknon1 28362*	The set of closed walks on vertex 𝑋 of length 1 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 24-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → (𝑋𝐶1) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 = ⟨“𝑋”⟩ ∧ {𝑋} ∈ 𝐸)})
Theorem	clwwlknon1loop 28363	If there is a loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ {𝑋} ∈ 𝐸) → (𝑋𝐶1) = {⟨“𝑋”⟩})
Theorem	clwwlknon1nloop 28364	If there is no loop at vertex 𝑋, the set of (closed) walks on 𝑋 of length 1 as words over the set of vertices is empty. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ({𝑋} ∉ 𝐸 → (𝑋𝐶1) = ∅)
Theorem	clwwlknon1sn 28365	The set of (closed) walks on vertex 𝑋 of length 1 as words over the set of vertices is a singleton containing the singleton word consisting of 𝑋 iff there is a loop at 𝑋. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Feb-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋 ∈ 𝑉 → ((𝑋𝐶1) = {⟨“𝑋”⟩} ↔ {𝑋} ∈ 𝐸))
Theorem	clwwlknon1le1 28366	There is at most one (closed) walk on vertex 𝑋 of length 1 as word over the set of vertices. (Contributed by AV, 11-Feb-2022.) (Revised by AV, 25-Mar-2022.)
⊢ (♯‘(𝑋(ClWWalksNOn‘𝐺)1)) ≤ 1
Theorem	clwwlknon2 28367*	The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices. (Contributed by AV, 5-Mar-2022.) (Revised by AV, 25-Mar-2022.)
⊢ 𝐶 = (ClWWalksNOn‘𝐺)    ⇒   ⊢ (𝑋𝐶2) = {𝑤 ∈ (2 ClWWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}
Theorem	clwwlknon2x 28368*	The set of closed walks on vertex 𝑋 of length 2 in a graph 𝐺 as words over the set of vertices, definition of ClWWalksN expanded. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 25-Mar-2022.)
⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑋𝐶2) = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = 2 ∧ {(𝑤‘0), (𝑤‘1)} ∈ 𝐸 ∧ (𝑤‘0) = 𝑋)}
Theorem	s2elclwwlknon2 28369	Sufficient conditions of a doubleton word to represent a closed walk on vertex 𝑋 of length 2. (Contributed by AV, 11-May-2022.)
⊢ 𝐶 = (ClWWalksNOn‘𝐺)    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ {𝑋, 𝑌} ∈ 𝐸) → ⟨“𝑋𝑌”⟩ ∈ (𝑋𝐶2))
Theorem	clwwlknon2num 28370	In a 𝐾-regular graph 𝐺, there are 𝐾 closed walks on vertex 𝑋 of length 2. (Contributed by Alexander van der Vekens, 19-Sep-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 25-Mar-2022.)
⊢ ((𝐺 RegUSGraph 𝐾 ∧ 𝑋 ∈ (Vtx‘𝐺)) → (♯‘(𝑋(ClWWalksNOn‘𝐺)2)) = 𝐾)
Theorem	clwwlknonwwlknonb 28371	A word over vertices represents a closed walk of a fixed length 𝑁 on vertex 𝑋 iff the word concatenated with 𝑋 represents a walk of length 𝑁 on 𝑋 and 𝑋. This theorem would not hold for 𝑁 = 0 and 𝑊 = ∅, see clwwlknwwlksnb 28320. (Contributed by AV, 4-Mar-2022.) (Revised by AV, 27-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁) ↔ (𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑋(𝑁 WWalksNOn 𝐺)𝑋)))
Theorem	clwwlknonex2lem1 28372	Lemma 1 for clwwlknonex2 28374: Transformation of a special half-open integer range into a union of a smaller half-open integer range and an unordered pair. This Lemma would not hold for 𝑁 = 2, i.e., (♯‘𝑊) = 0, because (0..^(((♯‘𝑊) + 2) − 1)) = (0..^((0 + 2) − 1)) = (0..^1) = {0} ≠ {-1, 0} = (∅ ∪ {-1, 0}) = ((0..^(0 − 1)) ∪ {(0 − 1), 0}) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}). (Contributed by AV, 22-Sep-2018.) (Revised by AV, 26-Jan-2022.)
⊢ ((𝑁 ∈ (ℤ≥‘3) ∧ (♯‘𝑊) = (𝑁 − 2)) → (0..^(((♯‘𝑊) + 2) − 1)) = ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}))
Theorem	clwwlknonex2lem2 28373*	Lemma 2 for clwwlknonex2 28374: Transformation of a walk and two edges into a walk extended by two vertices/edges. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 27-Jan-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {(lastS‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (♯‘𝑊) = (𝑁 − 2) ∧ (𝑊‘0) = 𝑋)) ∧ {𝑋, 𝑌} ∈ 𝐸) → ∀𝑖 ∈ ((0..^((♯‘𝑊) − 1)) ∪ {((♯‘𝑊) − 1), (♯‘𝑊)}){(((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘𝑖), (((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩)‘(𝑖 + 1))} ∈ 𝐸)
Theorem	clwwlknonex2 28374	Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk. (Contributed by AV, 22-Sep-2018.) (Revised by AV, 25-Feb-2022.) (Proof shortened by AV, 28-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑁 ClWWalksN 𝐺))
Theorem	clwwlknonex2e 28375	Extending a closed walk 𝑊 on vertex 𝑋 by an additional edge (forth and back) results in a closed walk on vertex 𝑋. (Contributed by AV, 17-Apr-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝑁 ∈ (ℤ≥‘3)) ∧ {𝑋, 𝑌} ∈ 𝐸 ∧ 𝑊 ∈ (𝑋(ClWWalksNOn‘𝐺)(𝑁 − 2))) → ((𝑊 ++ ⟨“𝑋”⟩) ++ ⟨“𝑌”⟩) ∈ (𝑋(ClWWalksNOn‘𝐺)𝑁))
Theorem	clwwlknondisj 28376*	The sets of closed walks on different vertices are disjunct. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
⊢ Disj 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁)
Theorem	clwwlknun 28377*	The set of closed walks of fixed length 𝑁 in a simple graph 𝐺 is the union of the closed walks of the fixed length 𝑁 on each of the vertices of graph 𝐺. (Contributed by Alexander van der Vekens, 7-Oct-2018.) (Revised by AV, 28-May-2021.) (Revised by AV, 3-Mar-2022.) (Proof shortened by AV, 28-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ USGraph → (𝑁 ClWWalksN 𝐺) = ∪ 𝑥 ∈ 𝑉 (𝑥(ClWWalksNOn‘𝐺)𝑁))
Theorem	clwwlkvbij 28378*	There is a bijection between the set of closed walks of a fixed length 𝑁 on a fixed vertex 𝑋 represented by walks (as word) and the set of closed walks (as words) of the fixed length 𝑁 on the fixed vertex 𝑋. The difference between these two representations is that in the first case the fixed 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.) (Revised by AV, 26-Apr-2021.) (Revised by AV, 7-Jul-2022.) (Proof shortened by AV, 2-Nov-2022.)
⊢ ((𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → ∃𝑓 𝑓:{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((lastS‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}–1-1-onto→(𝑋(ClWWalksNOn‘𝐺)𝑁))
16.3.11  Examples for walks, trails and paths
Theorem	0ewlk 28379	The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.)
⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) → ∅ ∈ (𝐺 EdgWalks 𝑆))
Theorem	1ewlk 28380	A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)
⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0* ∧ 𝐼 ∈ dom (iEdg‘𝐺)) → ⟨“𝐼”⟩ ∈ (𝐺 EdgWalks 𝑆))
Theorem	0wlk 28381	A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (∅(Walks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	is0wlk 28382	A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Walks‘𝐺)𝑃)
Theorem	0wlkonlem1 28383	Lemma 1 for 0wlkon 28385 and 0trlon 28389. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → (𝑁 ∈ 𝑉 ∧ 𝑁 ∈ 𝑉))
Theorem	0wlkonlem2 28384	Lemma 2 for 0wlkon 28385 and 0trlon 28389. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → 𝑃 ∈ (𝑉 ↑pm (0...0)))
Theorem	0wlkon 28385	A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(WalksOn‘𝐺)𝑁)𝑃)
Theorem	0wlkons1 28386	A walk of length 0 from a vertex to itself. (Contributed by AV, 17-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(WalksOn‘𝐺)𝑁)⟨“𝑁”⟩)
Theorem	0trl 28387	A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑈 → (∅(Trails‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	is0trl 28388	A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃 = {⟨0, 𝑁⟩} ∧ 𝑁 ∈ 𝑉) → ∅(Trails‘𝐺)𝑃)
Theorem	0trlon 28389	A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(TrailsOn‘𝐺)𝑁)𝑃)
Theorem	0pth 28390	A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (∅(Paths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	0spth 28391	A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑊 → (∅(SPaths‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	0pthon 28392	A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(PathsOn‘𝐺)𝑁)𝑃)
Theorem	0pthon1 28393	A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∅(𝑁(PathsOn‘𝐺)𝑁){⟨0, 𝑁⟩})
Theorem	0pthonv 28394*	For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → ∃𝑓∃𝑝 𝑓(𝑁(PathsOn‘𝐺)𝑁)𝑝)
Theorem	0clwlk 28395	A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.) (Revised by AV, 30-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑋 → (∅(ClWalks‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶𝑉))
Theorem	0clwlkv 28396	Any vertex (more precisely, a pair of an empty set (of edges) and a singleton function to this vertex) determines a closed walk of length 0. (Contributed by AV, 11-Feb-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑋 ∈ 𝑉 ∧ 𝐹 = ∅ ∧ 𝑃:{0}⟶{𝑋}) → 𝐹(ClWalks‘𝐺)𝑃)
Theorem	0clwlk0 28397	There is no closed walk in the empty set (i.e. the null graph). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
⊢ (ClWalks‘∅) = ∅
Theorem	0crct 28398	A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ (𝐺 ∈ 𝑊 → (∅(Circuits‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺)))
Theorem	0cycl 28399	A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ (𝐺 ∈ 𝑊 → (∅(Cycles‘𝐺)𝑃 ↔ 𝑃:(0...0)⟶(Vtx‘𝐺)))
Theorem	1pthdlem1 28400	Lemma 1 for 1pthd 28408. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
⊢ 𝑃 = ⟨“𝑋𝑌”⟩    &   ⊢ 𝐹 = ⟨“𝐽”⟩    ⇒   ⊢ Fun ◡(𝑃 ↾ (1..^(♯‘𝐹)))