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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremwwlkm1edg 26001 Removing the trailing edge from a walk (as word) with at least one edge results in a walk. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
((𝑊 ∈ (𝑉 WWalks 𝐸) ∧ 2 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (𝑉 WWalks 𝐸))

Theoremdisjxwwlks 26002* Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 29-Jul-2018.)
Disj 𝑦 ∈ ((𝑉 WWalksN 𝐸)‘𝑁){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

Theoremwwlknndef 26003 Conditions for WWalksN not being defined. (Contributed by Alexander van der Vekens, 30-Jul-2018.)
((𝑉 ∉ V ∨ 𝐸 ∉ V ∨ 𝑁 ∉ ℕ0) → ((𝑉 WWalksN 𝐸)‘𝑁) = ∅)

Theoremwwlknfi 26004 The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.)
(𝑉 ∈ Fin → ((𝑉 WWalksN 𝐸)‘𝑁) ∈ Fin)

Theoremwlknfi 26005* The number of walks of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑉 ∈ Fin) → {𝑝 ∈ (𝑉 Walks 𝐸) ∣ (#‘(1st𝑝)) = 𝑁} ∈ Fin)

Theoremwlknwwlknvbij 26006* There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
((𝑉 USGrph 𝐸𝑁 ∈ ℕ0𝑋𝑉) → ∃𝑓 𝑓:{𝑝 ∈ (𝑉 Walks 𝐸) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑋})

Theoremwwlkextproplem1 26007 Lemma 1 for wwlkextprop 26010. (Contributed by Alexander van der Vekens, 31-Jul-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))       ((𝑊𝑋𝑁 ∈ ℕ0) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))

Theoremwwlkextproplem2 26008 Lemma 2 for wwlkextprop 26010. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))       ((𝑊𝑋𝑁 ∈ ℕ0) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ ran 𝐸)

Theoremwwlkextproplem3 26009* Lemma 3 for wwlkextprop 26010. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       ((𝑊𝑋 ∧ (𝑊‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)

Theoremwwlkextprop 26010* Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       (𝑁 ∈ ℕ0 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)})

Theoremdisjxwwlkn 26011* Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}

Theoremhashwwlkext 26012* Number of walks (as words) extended by an edge as sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.)
𝑋 = ((𝑉 WWalksN 𝐸)‘(𝑁 + 1))    &   𝑌 = {𝑤 ∈ ((𝑉 WWalksN 𝐸)‘𝑁) ∣ (𝑤‘0) = 𝑃}       (𝑉 ∈ Fin → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}) = Σ𝑦𝑌 (#‘{𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ ran 𝐸)}))

16.1.5.6  Closed walks

Syntaxcclwlk 26013 Extend class notation with Closed Walks (of a graph).
class ClWalks

Syntaxcclwwlk 26014 Extend class notation with Closed Walks (of a graph) as Word over the set of vertices.
class ClWWalks

Syntaxcclwwlkn 26015 Extend class notation with Closed Walks (of a graph) of a fixed length as Word over the set of vertices.
class ClWWalksN

Definitiondf-clwlk 26016* Define the set of all Closed Walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0).

Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26031! (Contributed by Alexander van der Vekens, 12-Mar-2018.)

ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Definitiondf-clwwlk 26017* Define the set of all Closed Walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence 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) as defined in df-clwlk 26016. Notice that the word does not contain the terminating vertex p(n) of the walk, because it is always equal to the first vertex of the closed walk.

Notice that by this definition, a single vertex cannot be represented as closed walk, since the word <" v "> with vertex v represents the walk "vv", which is a (closed) walk of length 1 (if there is an edge/loop from v to v). Therefore, a closed walk corresponds to a closed walk as word in an undirected graph only for walks of length at least 1, see clwlkisclwwlk2 26056. (Contributed by Alexander van der Vekens, 20-Mar-2018.)

ClWWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {𝑤 ∈ Word 𝑣 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝑒 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝑒)})

Definitiondf-clwwlkn 26018* Define the set of all Closed Walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence 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) as defined in df-clwlk 26016. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))

Theoremclwlk 26019* The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWalks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Theoremisclwlk0 26020 Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremisclwlkg 26021 Generalisation of isclwlk0 26020: Properties of a pair of functions to be a closed walk (in an undirected graph) in terms of walks. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theoremisclwlk 26022* Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 ClWalks 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))

Theoremclwlkiswlk 26023 A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(𝐹(𝑉 ClWalks 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremclwlkswlks 26024 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝑊 ∈ (𝑉 ClWalks 𝐸) → 𝑊 ∈ (𝑉 Walks 𝐸))

Theoremclwlksarewlks 26025 Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.)
(𝑉 ClWalks 𝐸) ⊆ (𝑉 Walks 𝐸)

Theoremwlkv0 26026 If there is a walk in an empty graph, it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(𝑊 ∈ (∅ Walks 𝐸) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Theoremwlk0 26027 There is no walk in an empty graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(∅ Walks 𝐸) = ∅

Theoremclwlk0 26028 There is no closed walk in an empty graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.)
(∅ ClWalks 𝐸) = ∅

Theoremclwlkcomp 26029* A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝑉𝑋𝐸𝑌𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (𝑉 ClWalks 𝐸) ↔ ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹))))))

Theoremclwlkcompim 26030* Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (𝑉 ClWalks 𝐸) → ((𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))

Theorem0clwlk 26031 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.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 ClWalks 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theoremclwwlk 26032* The set of closed walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWWalks 𝐸) = {𝑤 ∈ Word 𝑉 ∣ (∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ ran 𝐸)})

Theoremclwwlkn 26033* The set of closed walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → ((𝑉 ClWWalksN 𝐸)‘𝑁) = {𝑤 ∈ (𝑉 ClWWalks 𝐸) ∣ (#‘𝑤) = 𝑁})

Theoremisclwwlk 26034* Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑊 ∈ (𝑉 ClWWalks 𝐸) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ ran 𝐸)))

Theoremisclwwlkn 26035 Properties of a word to represent a closed walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.)
((𝑉𝑋𝐸𝑌𝑁 ∈ ℕ0) → (𝑊 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) ↔ (𝑊 ∈ (𝑉 ClWWalks 𝐸) ∧ (#‘𝑊) = 𝑁)))

Theoremclwwlkprop 26036 Properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.)
(𝑃 ∈ (𝑉 ClWWalks 𝐸) → (𝑉 ∈ V ∧ 𝐸 ∈ V ∧ 𝑃 ∈ Word 𝑉))

Theoremclwwlkgt0 26037 A closed walk in an undirected graph has a length of at least 2. (Contributed by Alexander van der Vekens, 16-Sep-2018.)
(𝑉 USGrph 𝐸 → (𝑃 ∈ (𝑉 ClWWalks 𝐸) → 2 ≤ (#‘𝑃)))

Theoremclwwlknprop 26038 Properties of a closed walk of a fixed length as word. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
(𝑃 ∈ ((𝑉 ClWWalksN 𝐸)‘𝑁) → ((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ 𝑃 ∈ Word 𝑉 ∧ (𝑁 ∈ ℕ0 ∧ (#‘𝑃) = 𝑁)))

Theoremclwwlknndef 26039 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 26040 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 26041 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 26042* 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 26043 Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.)
((𝑉𝑋𝐸𝑌) → (𝑉 ClWWalks 𝐸) ⊆ Word 𝑉)

Theoremclwwlknfi 26044 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 26045* Lemma 1 for clwlkisclwwlklem2a 26051. (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 26046* Lemma 3 for clwlkisclwwlklem2a 26051. (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 26047* Lemma 3 for clwlkisclwwlklem2a 26051. (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 26048* Lemma 4a for clwlkisclwwlklem2a 26051. (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 26049* Lemma 4b for clwlkisclwwlklem2a 26051. (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 26050* Lemma 4 for clwlkisclwwlklem2a 26051. (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 26051* Lemma 2 for clwlkisclwwlklem2 26052. (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 26052* Lemma for clwlkisclwwlk 26055. (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 26053* Lemma for clwlkisclwwlk 26055. (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 26054* Lemma for clwlkisclwwlk 26055. (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 26055* 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 26056* 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 26057 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 26058 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 26059* 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 26060* Lemma 1 for clwwlkbij 26065: 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 26061* Lemma 2 for clwwlkbij 26065: 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 26062* Lemma 3 for clwwlkbij 26065: 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 26063* Lemma 4 for clwwlkbij 26065: 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 26064* Lemma 5 for clwwlkbij 26065: 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 26065* 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 26066* 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 26067* 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 26068 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 26069 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 26070 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 26071* Lemma 1 for clwwisshclwwlem 26072. (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 26072* Lemma for clwwisshclww 26073. (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 26073 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 26074 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 26075 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 26076 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 26077* 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 26078* 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 26079* 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 26080* 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 26081* 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 26082* 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 26083* Lemma 1 for eleclclwwlkn 26098. (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 26084* Lemma 2 for eleclclwwlkn 26098. (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 26085* 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 26086 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 26087* 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 26088 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 26089* 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 26090* 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 26091* 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 26092* 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 26093* 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 26094* 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 26095* 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 26096* 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 26097* 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 26098* 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 26099* 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 26100* 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))

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