P. 296 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29501-29600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	clwlkwlk 29501	Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝑊 ∈ (ClWalks‘𝐺) → 𝑊 ∈ (Walks‘𝐺))
Theorem	clwlkswks 29502	Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Feb-2021.)
⊢ (ClWalks‘𝐺) ⊆ (Walks‘𝐺)
Theorem	isclwlke 29503*	Properties of a pair of functions to be a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ 𝑋 → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))
Theorem	isclwlkupgr 29504*	Properties of a pair of functions to be a closed walk (in a pseudograph). (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 11-Apr-2021.) (Revised by AV, 28-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝐹(ClWalks‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))
Theorem	clwlkcomp 29505*	A closed walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ ((𝐺 ∈ 𝑋 ∧ 𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (ClWalks‘𝐺) ↔ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))))
Theorem	clwlkcompim 29506*	Implications for the properties of the components of a closed walk. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 17-Feb-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ (𝑊 ∈ (ClWalks‘𝐺) → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ (∀𝑘 ∈ (0..^(♯‘𝐹))if-((𝑃‘𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘)}, {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘))) ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹)))))
Theorem	upgrclwlkcompim 29507*	Implications for the properties of the components of a closed walk in a pseudograph. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 2-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (ClWalks‘𝐺)) → ((𝐹 ∈ Word dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉) ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Theorem	clwlkcompbp 29508	Basic properties of the components of a closed walk. (Contributed by AV, 23-May-2022.)
⊢ 𝐹 = (1st ‘𝑊)    &   ⊢ 𝑃 = (2nd ‘𝑊)    ⇒   ⊢ (𝑊 ∈ (ClWalks‘𝐺) → (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Theorem	clwlkl1loop 29509	A closed walk of length 1 is a loop. (Contributed by AV, 22-Apr-2021.)
⊢ ((Fun (iEdg‘𝐺) ∧ 𝐹(ClWalks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → ((𝑃‘0) = (𝑃‘1) ∧ {(𝑃‘0)} ∈ (Edg‘𝐺)))
17.3.6  Circuits and cycles
Syntax	ccrcts 29510	Extend class notation with circuits (in a graph).
class Circuits
Syntax	ccycls 29511	Extend class notation with cycles (in a graph).
class Cycles
Definition	df-crcts 29512*	Define the set of all circuits (in an undirected graph). According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...". Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit is also 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). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ Circuits = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
Definition	df-cycls 29513*	Define the set of all (simple) cycles (in an undirected graph). According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex." According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle." See Definition of [Bollobas] p. 5. However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ Cycles = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
Theorem	crcts 29514*	The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ (Circuits‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
Theorem	cycls 29515*	The set of cycles (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ (Cycles‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Paths‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
Theorem	iscrct 29516	Sufficient and necessary conditions for a pair of functions to be a circuit (in an undirected graph): A pair of function "is" (represents) a circuit iff it is a closed trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ (𝐹(Circuits‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Theorem	iscycl 29517	Sufficient and necessary conditions for a pair of functions to be a cycle (in an undirected graph): A pair of function "is" (represents) a cycle iff it is a closed path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.) (Revised by AV, 30-Oct-2021.)
⊢ (𝐹(Cycles‘𝐺)𝑃 ↔ (𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Theorem	crctprop 29518	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))))
Theorem	cyclprop 29519	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))))
Theorem	crctisclwlk 29520	A circuit is a closed walk. (Contributed by AV, 17-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(ClWalks‘𝐺)𝑃)
Theorem	crctistrl 29521	A circuit is a trail. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Trails‘𝐺)𝑃)
Theorem	crctiswlk 29522	A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	cyclispth 29523	A cycle is a path. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Paths‘𝐺)𝑃)
Theorem	cycliswlk 29524	A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 31-Jan-2021.)
⊢ (𝐹(Cycles‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	cycliscrct 29525	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)𝑃 → 𝐹(Circuits‘𝐺)𝑃)
Theorem	cyclnspth 29526	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))
Theorem	cyclispthon 29527	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)𝑃 → 𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘0))𝑃)
Theorem	lfgrn1cycl 29528*	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))
Theorem	usgr2trlncrct 29529	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))
Theorem	umgrn1cycl 29530	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)
Theorem	uspgrn2crct 29531	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)
Theorem	usgrn2cycl 29532	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)
Theorem	crctcshwlkn0lem1 29533	Lemma for crctcshwlkn0 29544. (Contributed by AV, 13-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴 − 𝐵) + 1) ≤ 𝐴)
Theorem	crctcshwlkn0lem2 29534*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ (0...(𝑁 − 𝑆))) → (𝑄‘𝐽) = (𝑃‘(𝐽 + 𝑆)))
Theorem	crctcshwlkn0lem3 29535*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ ((𝜑 ∧ 𝐽 ∈ (((𝑁 − 𝑆) + 1)...𝑁)) → (𝑄‘𝐽) = (𝑃‘((𝐽 + 𝑆) − 𝑁)))
Theorem	crctcshwlkn0lem4 29536*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝐴)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))    ⇒   ⊢ (𝜑 → ∀𝑗 ∈ (0..^(𝑁 − 𝑆))if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
Theorem	crctcshwlkn0lem5 29537*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝐴)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))    ⇒   ⊢ (𝜑 → ∀𝑗 ∈ (((𝑁 − 𝑆) + 1)..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
Theorem	crctcshwlkn0lem6 29538*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝐴)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘0))    ⇒   ⊢ ((𝜑 ∧ 𝐽 = (𝑁 − 𝑆)) → if-((𝑄‘𝐽) = (𝑄‘(𝐽 + 1)), (𝐼‘(𝐻‘𝐽)) = {(𝑄‘𝐽)}, {(𝑄‘𝐽), (𝑄‘(𝐽 + 1))} ⊆ (𝐼‘(𝐻‘𝐽))))
Theorem	crctcshwlkn0lem7 29539*	Lemma for crctcshwlkn0 29544. (Contributed by AV, 12-Mar-2021.)
⊢ (𝜑 → 𝑆 ∈ (1..^𝑁))    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝐹 ∈ Word 𝐴)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^𝑁)if-((𝑃‘𝑖) = (𝑃‘(𝑖 + 1)), (𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖)}, {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ⊆ (𝐼‘(𝐹‘𝑖))))    &   ⊢ (𝜑 → (𝑃‘𝑁) = (𝑃‘0))    ⇒   ⊢ (𝜑 → ∀𝑗 ∈ (0..^𝑁)if-((𝑄‘𝑗) = (𝑄‘(𝑗 + 1)), (𝐼‘(𝐻‘𝑗)) = {(𝑄‘𝑗)}, {(𝑄‘𝑗), (𝑄‘(𝑗 + 1))} ⊆ (𝐼‘(𝐻‘𝑗))))
Theorem	crctcshlem1 29540	Lemma for crctcsh 29547. (Contributed by AV, 10-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ0)
Theorem	crctcshlem2 29541	Lemma for crctcsh 29547. (Contributed by AV, 10-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    ⇒   ⊢ (𝜑 → (♯‘𝐻) = 𝑁)
Theorem	crctcshlem3 29542*	Lemma for crctcsh 29547. (Contributed by AV, 10-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ (𝜑 → (𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))
Theorem	crctcshlem4 29543*	Lemma for crctcsh 29547. (Contributed by AV, 10-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ ((𝜑 ∧ 𝑆 = 0) → (𝐻 = 𝐹 ∧ 𝑄 = 𝑃))
Theorem	crctcshwlkn0 29544*	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)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ ((𝜑 ∧ 𝑆 ≠ 0) → 𝐻(Walks‘𝐺)𝑄)
Theorem	crctcshwlk 29545*	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)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ (𝜑 → 𝐻(Walks‘𝐺)𝑄)
Theorem	crctcshtrl 29546*	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)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ (𝜑 → 𝐻(Trails‘𝐺)𝑄)
Theorem	crctcsh 29547*	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)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    &   ⊢ (𝜑 → 𝑆 ∈ (0..^𝑁))    &   ⊢ 𝐻 = (𝐹 cyclShift 𝑆)    &   ⊢ 𝑄 = (𝑥 ∈ (0...𝑁) ↦ if(𝑥 ≤ (𝑁 − 𝑆), (𝑃‘(𝑥 + 𝑆)), (𝑃‘((𝑥 + 𝑆) − 𝑁))))    ⇒   ⊢ (𝜑 → 𝐻(Circuits‘𝐺)𝑄)
17.3.7  Walks as words In general, a walk is an alternating sequence of vertices and edges, as defined in df-wlks 29325: 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 14462, 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 29553 and df-wwlksn 29554, 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).
Syntax	cwwlks 29548	Extend class notation with walks (in a graph) as word over the set of vertices.
class WWalks
Syntax	cwwlksn 29549	Extend class notation with walks (in a graph) of a fixed length as word over the set of vertices.
class WWalksN
Syntax	cwwlksnon 29550	Extend class notation with walks between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WWalksNOn
Syntax	cwwspthsn 29551	Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN
Syntax	cwwspthsnon 29552	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
Definition	df-wwlks 29553*	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 29325. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 29347. (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)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})
Definition	df-wwlksn 29554*	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 29325. (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)})
Definition	df-wwlksnon 29555*	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 ∧ (𝑤‘𝑛) = 𝑏)}))
Definition	df-wspthsn 29556*	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})
Definition	df-wspthsnon 29557*	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}))
Theorem	wwlks 29558*	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)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}
Theorem	iswwlks 29559*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
Theorem	wwlksn 29560*	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)})
Theorem	iswwlksn 29561	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))))
Theorem	wwlksnprcl 29562	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 𝑉 ∧ 𝑁 ∈ ℕ0) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑊) = 𝑁))
Theorem	iswwlksnx 29563*	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)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1))))
Theorem	wwlkbp 29564	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))
Theorem	wwlknbp 29565	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 ∈ ℕ0 ∧ 𝑊 ∈ Word 𝑉))
Theorem	wwlknp 29566*	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){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
Theorem	wwlknbp1 29567	Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0 ∧ 𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
Theorem	wwlknvtx 29568*	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)(𝑊‘𝑖) ∈ (Vtx‘𝐺))
Theorem	wwlknllvtx 29569	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 ∧ (𝑊‘𝑁) ∈ 𝑉))
Theorem	wwlknlsw 29570	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) → (𝑊‘𝑁) = (lastS‘𝑊))
Theorem	wspthsn 29571*	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}
Theorem	iswspthn 29572*	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))
Theorem	wspthnp 29573*	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 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
Theorem	wwlksnon 29574*	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 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}))
Theorem	wspthsnon 29575*	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 ∈ ℕ0 ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
Theorem	iswwlksnon 29576*	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 ∧ (𝑤‘𝑁) = 𝐵)}
Theorem	wwlksnon0 29577	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 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
Theorem	wwlksonvtx 29578	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) → (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
Theorem	iswspthsnon 29579*	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}
Theorem	wwlknon 29580	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 ∧ (𝑊‘𝑁) = 𝐵))
Theorem	wspthnon 29581*	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))
Theorem	wspthnonp 29582*	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 ∈ ℕ0 ∧ 𝐺 ∈ V) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
Theorem	wspthneq1eq2 29583	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 = 𝐶 ∧ 𝐵 = 𝐷))
Theorem	wwlksn0s 29584*	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}
Theorem	wwlkssswrd 29585	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
Theorem	wwlksn0 29586*	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) → ∃𝑣 ∈ 𝑉 𝑊 = ⟨“𝑣”⟩)
Theorem	0enwwlksnge1 29587	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) = ∅)
Theorem	wwlkswwlksn 29588	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))
Theorem	wwlkssswwlksn 29589	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)
Theorem	wlkiswwlks1 29590	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)𝑃 → 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlklnwwlkln1 29591	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)))
Theorem	wlkiswwlks2lem1 29592*	Lemma 1 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1))
Theorem	wlkiswwlks2lem2 29593*	Lemma 2 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ (((♯‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))
Theorem	wlkiswwlks2lem3 29594*	Lemma 3 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
Theorem	wlkiswwlks2lem4 29595*	Lemma 4 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}))
Theorem	wlkiswwlks2lem5 29596*	Lemma 5 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → 𝐹 ∈ Word dom 𝐸))
Theorem	wlkiswwlks2lem6 29597*	Lemma 6 for wlkiswwlks2 29598. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}))    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))})))
Theorem	wlkiswwlks2 29598*	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))
Theorem	wlkiswwlks 29599*	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)𝑃 ↔ 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlkiswwlksupgr2 29600*	A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 29598 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10471) 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 29598). (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))