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Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcrctprop 27501 The properties of a circuit: A circuit is a closed trail. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuits‘𝐺)𝑃 → (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
 
Theoremcyclprop 27502 The properties of a cycle: A cycle is a closed path. (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃 → (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
 
Theoremcrctisclwlk 27503 A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(ClWalks‘𝐺)𝑃)
 
Theoremcrctistrl 27504 A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)
 
Theoremcrctiswlk 27505 A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
(𝐹(Circuits‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
 
Theoremcyclispth 27506 A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Paths‘𝐺)𝑃)
 
Theoremcycliswlk 27507 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)
 
Theoremcycliscrct 27508 A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹(Circuits‘𝐺)𝑃)
 
Theoremcyclnspth 27509 A (non-trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹 ≠ ∅ → (𝐹(Cycles‘𝐺)𝑃 → ¬ 𝐹(SPaths‘𝐺)𝑃))
 
Theoremcyclispthon 27510 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Cycles‘𝐺)𝑃𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘0))𝑃)
 
Theoremlfgrn1cycl 27511* In a loop-free graph there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} → (𝐹(Cycles‘𝐺)𝑃 → (♯‘𝐹) ≠ 1))
 
Theoremusgr2trlncrct 27512 In a simple graph, any trail of length 2 is not a circuit. (Contributed by AV, 5-Jun-2021.)
((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → ¬ 𝐹(Circuits‘𝐺)𝑃))
 
Theoremumgrn1cycl 27513 In a multigraph graph (with no loops!) there are no cycles with length 1 (consisting of one edge). (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 2-Feb-2021.)
((𝐺 ∈ UMGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘𝐹) ≠ 1)
 
Theoremuspgrn2crct 27514 In a simple pseudograph there are no circuits with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 3-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐺 ∈ USPGraph ∧ 𝐹(Circuits‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)
 
Theoremusgrn2cycl 27515 In a simple graph there are no cycles with length 2 (consisting of two edges). (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 4-Feb-2021.)
((𝐺 ∈ USGraph ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘𝐹) ≠ 2)
 
Theoremcrctcshwlkn0lem1 27516 Lemma for crctcshwlkn0 27527. (Contributed by AV, 13-Mar-2021.)
((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴𝐵) + 1) ≤ 𝐴)
 
Theoremcrctcshwlkn0lem2 27517* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (0...(𝑁𝑆))) → (𝑄𝐽) = (𝑃‘(𝐽 + 𝑆)))
 
Theoremcrctcshwlkn0lem3 27518* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝐽 ∈ (((𝑁𝑆) + 1)...𝑁)) → (𝑄𝐽) = (𝑃‘((𝐽 + 𝑆) − 𝑁)))
 
Theoremcrctcshwlkn0lem4 27519* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (0..^(𝑁𝑆))if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
 
Theoremcrctcshwlkn0lem5 27520* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))       (𝜑 → ∀𝑗 ∈ (((𝑁𝑆) + 1)..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
 
Theoremcrctcshwlkn0lem6 27521* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       ((𝜑𝐽 = (𝑁𝑆)) → if-((𝑄𝐽) = (𝑄‘(𝐽 + 1)), (𝐼‘(𝐻𝐽)) = {(𝑄𝐽)}, {(𝑄𝐽), (𝑄‘(𝐽 + 1))} ⊆ (𝐼‘(𝐻𝐽))))
 
Theoremcrctcshwlkn0lem7 27522* Lemma for crctcshwlkn0 27527. (Contributed by AV, 12-Mar-2021.)
(𝜑𝑆 ∈ (1..^𝑁))    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝐹 ∈ Word 𝐴)    &   (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹𝑖)) = {(𝑃𝑖)}, {(𝑃𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹𝑖))))    &   (𝜑 → (𝑃𝑁) = (𝑃‘0))       (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻𝑗)) = {(𝑄𝑗)}, {(𝑄𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻𝑗))))
 
Theoremcrctcshlem1 27523 Lemma for crctcsh 27530. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)       (𝜑𝑁 ∈ ℕ0)
 
Theoremcrctcshlem2 27524 Lemma for crctcsh 27530. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)       (𝜑 → (♯‘𝐻) = 𝑁)
 
Theoremcrctcshlem3 27525* Lemma for crctcsh 27530. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
 
Theoremcrctcshlem4 27526* Lemma for crctcsh 27530. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 = 0) → (𝐻 = 𝐹𝑄 = 𝑃))
 
Theoremcrctcshwlkn0 27527* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       ((𝜑𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄)
 
Theoremcrctcshwlk 27528* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a walk 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Walks‘𝐺)𝑄)
 
Theoremcrctcshtrl 27529* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a trail 𝐻, 𝑄. (Contributed by AV, 14-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Trails‘𝐺)𝑄)
 
Theoremcrctcsh 27530* Cyclically shifting the indices of a circuit 𝐹, 𝑃 results in a circuit 𝐻, 𝑄. (Contributed by AV, 10-Mar-2021.) (Proof shortened by AV, 31-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Circuits‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)    &   (𝜑𝑆 ∈ (0..^𝑁))    &   𝐻 = (𝐹 cyclShift 𝑆)    &   𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))       (𝜑𝐻(Circuits‘𝐺)𝑄)
 
16.3.7  Walks as words

In general, a walk is an alternating sequence of vertices and edges, as defined in df-wlks 27309: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). Often, it is sufficient to refer to a walk by the natural sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(n), see the corresponding remark in [Diestel] p. 6. The concept of a Word, see df-word 13852, is the appropriate way to define such a sequence (being finite and starting at index 0) of vertices. Therefore, it is used in definitions df-wwlks 27536 and df-wwlksn 27537, and the representation of a walk as sequence of its vertices is called "walk as word".

Only for simple pseudographs, however, the edges can be uniquely reconstructed from such a representation. In other cases, there could be more than one edge between two adjacent vertices in the walk (in a multigraph), or two adjacent vertices could be connected by two different hyperedges involving additional vertices (in a hypergraph).

 
Syntaxcwwlks 27531 Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks
 
Syntaxcwwlksn 27532 Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN
 
Syntaxcwwlksnon 27533 Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn
 
Syntaxcwwspthsn 27534 Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN
 
Syntaxcwwspthsnon 27535 Extend class notation with simple paths between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WSPathsNOn
 
Definitiondf-wwlks 27536* Define the set of all 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) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 27309. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 27330. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
 
Definitiondf-wwlksn 27537* Define the set of all 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) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 27309. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
 
Definitiondf-wwlksnon 27538* Define the collection of walks of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))
 
Definitiondf-wspthsn 27539* Define the collection of simple paths of a fixed length as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})
 
Definitiondf-wspthsnon 27540* Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))
 
Theoremwwlks 27541* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (WWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}
 
Theoremiswwlks 27542* A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlksn 27543* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (♯‘𝑤) = (𝑁 + 1)})
 
Theoremiswwlksn 27544 A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1))))
 
Theoremwwlksnprcl 27545 Derivation of the length of a word 𝑊 whose concatenation with a singleton word represents a walk of a fixed length 𝑁 (a partial reverse closure theorem). (Contributed by AV, 4-Mar-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑊) = 𝑁))
 
Theoremiswwlksnx 27546* Properties of a word to represent a walk of a fixed length, definition of WWalks expanded. (Contributed by AV, 28-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1))))
 
Theoremwwlkbp 27547 Basic properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉))
 
Theoremwwlknbp 27548 Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 20-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
 
Theoremwwlknp 27549* Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlknbp1 27550 Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
 
Theoremwwlknvtx 27551* The symbols of a word 𝑊 representing a walk of a fixed length 𝑁 are vertices. (Contributed by AV, 16-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑖 ∈ (0...𝑁)(𝑊𝑖) ∈ (Vtx‘𝐺))
 
Theoremwwlknllvtx 27552 If a word 𝑊 represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊𝑁) ∈ 𝑉))
 
Theoremwwlknlsw 27553 If a word represents a walk of a fixed length, then the last symbol of the word is the endvertex of the walk. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊𝑁) = (lastS‘𝑊))
 
Theoremwspthsn 27554* The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
 
Theoremiswspthn 27555* An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwspthnp 27556* Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwwlksnon 27557* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
 
Theoremwspthsnon 27558* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
 
Theoremiswwlksnon 27559* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}
 
Theoremwwlksnon0 27560 Sufficient conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
 
Theoremwwlksonvtx 27561 If a word 𝑊 represents a walk of length 2 on two classes 𝐴 and 𝐶, these classes are vertices. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐶𝑉))
 
Theoremiswspthsnon 27562* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
 
Theoremwwlknon 27563 An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
(𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵))
 
Theoremwspthnon 27564* An element of the set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 15-Mar-2022.)
(𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
 
Theoremwspthnonp 27565* Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
 
Theoremwspthneq1eq2 27566 Two simple paths with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
((𝑃 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ 𝑃 ∈ (𝐶(𝑁 WSPathsNOn 𝐺)𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremwwlksn0s 27567* The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}
 
Theoremwwlkssswrd 27568 Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (WWalks‘𝐺) ⊆ Word 𝑉
 
Theoremwwlksn0 27569* A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (0 WWalksN 𝐺) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)
 
Theorem0enwwlksnge1 27570 In graphs without edges, there are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.)
(((Edg‘𝐺) = ∅ ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅)
 
Theoremwwlkswwlksn 27571 A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ (WWalks‘𝐺))
 
Theoremwwlkssswwlksn 27572 The walks of a fixed length as words are walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑁 WWalksN 𝐺) ⊆ (WWalks‘𝐺)
 
Theoremwlkiswwlks1 27573 The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
(𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlklnwwlkln1 27574 The sequence of vertices in a walk of length 𝑁 is a walk as word of length 𝑁 in a pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 𝑁) → 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlkiswwlks2lem1 27575* Lemma 1 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1))
 
Theoremwlkiswwlks2lem2 27576* Lemma 2 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       (((♯‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))
 
Theoremwlkiswwlks2lem3 27577* Lemma 3 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
 
Theoremwlkiswwlks2lem4 27578* Lemma 4 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
 
Theoremwlkiswwlks2lem5 27579* Lemma 5 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸𝐹 ∈ Word dom 𝐸))
 
Theoremwlkiswwlks2lem6 27580* Lemma 6 for wlkiswwlks2 27581. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
 
Theoremwlkiswwlks2 27581* A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
 
Theoremwlkiswwlks 27582* A walk as word corresponds to a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlkiswwlksupgr2 27583* A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 27581 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9891) for its proof. Notice that only the existence of a function 𝑓 can be proven, but, in general, it cannot be "constructed" (as in wlkiswwlks2 27581). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
 
Theoremwlkiswwlkupgr 27584* A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks 27582 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlkswwlksf1o 27585* The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)
𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))       (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
 
Theoremwlkswwlksen 27586 The set of walks as words and the set of (ordinary) walks are equinumerous in a simple pseudograph. (Contributed by AV, 6-May-2021.) (Revised by AV, 5-Aug-2022.)
(𝐺 ∈ USPGraph → (Walks‘𝐺) ≈ (WWalks‘𝐺))
 
Theoremwwlksm1edg 27587 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.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 26-Oct-2022.)
((𝑊 ∈ (WWalks‘𝐺) ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix ((♯‘𝑊) − 1)) ∈ (WWalks‘𝐺))
 
Theoremwlklnwwlkln2lem 27588* Lemma for wlklnwwlkln2 27589 and wlklnwwlklnupgr2 27591. Formerly part of proof for wlklnwwlkln2 27589. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝜑 → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))       (𝜑 → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlkln2 27589* A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlkn 27590* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlklnwwlklnupgr2 27591* A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a pseudograph. This variant of wlklnwwlkln2 27589 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlknupgr 27592* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of wlkiswwlks 27582 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlknewwlksn 27593 If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)
(((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalksN 𝐺))
 
Theoremwlknwwlksnbij 27594* The mapping (𝑡𝑇 ↦ (2nd𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
 
Theoremwlknwwlksnen 27595* In a simple pseudograph, the set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))
 
Theoremwlknwwlksneqs 27596* The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (♯‘(𝑁 WWalksN 𝐺)))
 
Theoremwwlkseq 27597* Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
((𝑊 ∈ (WWalks‘𝐺) ∧ 𝑇 ∈ (WWalks‘𝐺)) → (𝑊 = 𝑇 ↔ ((♯‘𝑊) = (♯‘𝑇) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) = (𝑇𝑖))))
 
Theoremwwlksnred 27598 Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Revised by AV, 26-Oct-2022.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwwlksnext 27599 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
 
Theoremwwlksnextbi 27600 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Proof shortened by AV, 27-Oct-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
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