P. 298 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 29701-29800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pthiswlk 29701	A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	spthiswlk 29702	A simple path is a walk (in an undirected graph). (Contributed by AV, 16-May-2021.)
⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	pthdivtx 29703	The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(♯‘𝐹)) ∧ 𝐽 ∈ (0...(♯‘𝐹)) ∧ 𝐼 ≠ 𝐽)) → (𝑃‘𝐼) ≠ (𝑃‘𝐽))
Theorem	pthdadjvtx 29704	The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹) ∧ 𝐼 ∈ (0..^(♯‘𝐹))) → (𝑃‘𝐼) ≠ (𝑃‘(𝐼 + 1)))
Theorem	dfpth2 29705	Alternate definition for a pair of classes/functions to be a path (in an undirected graph). (Contributed by AV, 4-Oct-2025.)
⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun ◡(𝑃 ↾ (1...(♯‘𝐹))) ∧ (𝑃‘0) ∉ (𝑃 “ (1..^(♯‘𝐹)))))
Theorem	pthdifv 29706	The vertices of a path are distinct (except the first and last vertex), so the restricted vertex function is one-to-one. (Contributed by AV, 2-Oct-2025.)
⊢ (𝐹(Paths‘𝐺)𝑃 → (𝑃 ↾ (1...(♯‘𝐹))):(1...(♯‘𝐹))–1-1→(Vtx‘𝐺))
Theorem	2pthnloop 29707*	A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 30126. (Contributed by AV, 6-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))2 ≤ (♯‘(𝐼‘(𝐹‘𝑖))))
Theorem	upgr2pthnlp 29708*	A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (♯‘𝐹)) → ∀𝑖 ∈ (0..^(♯‘𝐹))(♯‘(𝐼‘(𝐹‘𝑖))) = 2)
Theorem	spthdifv 29709	The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(SPaths‘𝐺)𝑃 → 𝑃:(0...(♯‘𝐹))–1-1→(Vtx‘𝐺))
Theorem	spthdep 29710	A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ ((𝐹(SPaths‘𝐺)𝑃 ∧ (♯‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))
Theorem	pthdepisspth 29711	A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ ((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃)
Theorem	upgrwlkdvdelem 29712*	Lemma for upgrwlkdvde 29713. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
⊢ ((𝑃:(0...(♯‘𝐹))–1-1→𝑉 ∧ 𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} → Fun ◡𝐹))
Theorem	upgrwlkdvde 29713	In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 29714. (Contributed by AV, 17-Jan-2021.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → Fun ◡𝐹)
Theorem	upgrspthswlk 29714*	The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun ◡𝑝)})
Theorem	upgrwlkdvspth 29715	A walk consisting of different vertices is a simple path. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 29714. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
⊢ ((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun ◡𝑃) → 𝐹(SPaths‘𝐺)𝑃)
Theorem	pthsonfval 29716*	The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(Paths‘𝐺)𝑝)})
Theorem	spthson 29717*	The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(SPathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝 ∧ 𝑓(SPaths‘𝐺)𝑝)})
Theorem	ispthson 29718	Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))
Theorem	isspthson 29719	Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑈 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)))
Theorem	pthsonprop 29720	Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(Paths‘𝐺)𝑃)))
Theorem	spthonprop 29721	Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ∧ 𝐹(SPaths‘𝐺)𝑃)))
Theorem	pthonispth 29722	A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.)
⊢ (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → 𝐹(Paths‘𝐺)𝑃)
Theorem	pthontrlon 29723	A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
⊢ (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃)
Theorem	pthonpth 29724	A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
⊢ (𝐹(Paths‘𝐺)𝑃 → 𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)
Theorem	isspthonpth 29725	A pair of functions is a simple path between two given vertices iff it is a simple path starting and ending at the two vertices. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ (𝐹 ∈ 𝑊 ∧ 𝑃 ∈ 𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(SPaths‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))
Theorem	spthonisspth 29726	A simple path between to vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 18-Jan-2021.)
⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(SPaths‘𝐺)𝑃)
Theorem	spthonpthon 29727	A simple path between two vertices is a path between these vertices. (Contributed by AV, 24-Jan-2021.)
⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → 𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃)
Theorem	spthonepeq 29728	The endpoints of a simple path between two vertices are equal iff the path is of length 0. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 18-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → (𝐴 = 𝐵 ↔ (♯‘𝐹) = 0))
Theorem	uhgrwkspthlem1 29729	Lemma 1 for uhgrwkspth 29731. (Contributed by AV, 25-Jan-2021.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 1) → Fun ◡𝐹)
Theorem	uhgrwkspthlem2 29730	Lemma 2 for uhgrwkspth 29731. (Contributed by AV, 25-Jan-2021.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ ((♯‘𝐹) = 1 ∧ 𝐴 ≠ 𝐵) ∧ ((𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)) → Fun ◡𝑃)
Theorem	uhgrwkspth 29731	Any walk of length 1 between two different vertices is a simple path. (Contributed by AV, 25-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.) (Revised by AV, 7-Jul-2022.)
⊢ ((𝐺 ∈ 𝑊 ∧ (♯‘𝐹) = 1 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))
Theorem	usgr2wlkneq 29732	The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
⊢ (((𝐺 ∈ USGraph ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → (((𝑃‘0) ≠ (𝑃‘1) ∧ (𝑃‘0) ≠ (𝑃‘2) ∧ (𝑃‘1) ≠ (𝑃‘2)) ∧ (𝐹‘0) ≠ (𝐹‘1)))
Theorem	usgr2wlkspthlem1 29733	Lemma 1 for usgr2wlkspth 29735. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 26-Jan-2021.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun ◡𝐹)
Theorem	usgr2wlkspthlem2 29734	Lemma 2 for usgr2wlkspth 29735. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.)
⊢ ((𝐹(Walks‘𝐺)𝑃 ∧ (𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ (𝑃‘0) ≠ (𝑃‘(♯‘𝐹)))) → Fun ◡𝑃)
Theorem	usgr2wlkspth 29735	In a simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 27-Jan-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2 ∧ 𝐴 ≠ 𝐵) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ 𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃))
Theorem	usgr2trlncl 29736	In a simple graph, any trail of length 2 does not start and end at the same vertex. (Contributed by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 → (𝑃‘0) ≠ (𝑃‘2)))
Theorem	usgr2trlspth 29737	In a simple graph, any trail of length 2 is a simple path. (Contributed by AV, 5-Jun-2021.)
⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Trails‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
Theorem	usgr2pthspth 29738	In a simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.) (Revised by AV, 5-Jun-2021.)
⊢ ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → (𝐹(Paths‘𝐺)𝑃 ↔ 𝐹(SPaths‘𝐺)𝑃))
Theorem	usgr2pthlem 29739*	Lemma for usgr2pth 29740. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ 𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) → ((𝐺 ∈ USGraph ∧ (♯‘𝐹) = 2) → ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧}))))
Theorem	usgr2pth 29740*	In a simple graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 31-Oct-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑦 ∧ (𝑃‘2) = 𝑧) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑦} ∧ (𝐼‘(𝐹‘1)) = {𝑦, 𝑧})))))
Theorem	usgr2pth0 29741*	In a simply graph, there is a path of length 2 iff there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.) (Revised by AV, 5-Jun-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝐺 ∈ USGraph → ((𝐹(Paths‘𝐺)𝑃 ∧ (♯‘𝐹) = 2) ↔ (𝐹:(0..^2)–1-1→dom 𝐼 ∧ 𝑃:(0...2)–1-1→𝑉 ∧ ∃𝑥 ∈ 𝑉 ∃𝑦 ∈ (𝑉 ∖ {𝑥})∃𝑧 ∈ (𝑉 ∖ {𝑥, 𝑦})(((𝑃‘0) = 𝑥 ∧ (𝑃‘1) = 𝑧 ∧ (𝑃‘2) = 𝑦) ∧ ((𝐼‘(𝐹‘0)) = {𝑥, 𝑧} ∧ (𝐼‘(𝐹‘1)) = {𝑧, 𝑦})))))
Theorem	pthdlem1 29742*	Lemma 1 for pthd 29745. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 9-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ 𝑅 = ((♯‘𝑃) − 1)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))    ⇒   ⊢ (𝜑 → Fun ◡(𝑃 ↾ (1..^𝑅)))
Theorem	pthdlem2lem 29743*	Lemma for pthdlem2 29744. (Contributed by AV, 10-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ 𝑅 = ((♯‘𝑃) − 1)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))    ⇒   ⊢ ((𝜑 ∧ (♯‘𝑃) ∈ ℕ ∧ (𝐼 = 0 ∨ 𝐼 = 𝑅)) → (𝑃‘𝐼) ∉ (𝑃 “ (1..^𝑅)))
Theorem	pthdlem2 29744*	Lemma 2 for pthd 29745. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 10-Feb-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ 𝑅 = ((♯‘𝑃) − 1)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))    ⇒   ⊢ (𝜑 → ((𝑃 “ {0, 𝑅}) ∩ (𝑃 “ (1..^𝑅))) = ∅)
Theorem	pthd 29745*	Two words representing a trail which also represent a path in a graph. (Contributed by AV, 10-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝜑 → 𝑃 ∈ Word V)    &   ⊢ 𝑅 = ((♯‘𝑃) − 1)    &   ⊢ (𝜑 → ∀𝑖 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^𝑅)(𝑖 ≠ 𝑗 → (𝑃‘𝑖) ≠ (𝑃‘𝑗)))    &   ⊢ (♯‘𝐹) = 𝑅    &   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)    ⇒   ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
17.3.5  Closed walks
Syntax	cclwlks 29746	Extend class notation with closed walks (of a graph).
class ClWalks
Definition	df-clwlks 29747*	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 { 0 , ... , ( n - 1 ) } 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(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0). Notice that by this definition, a single vertex can be considered as a closed walk of length 0, see also 0clwlk 30105. (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)
⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
Theorem	clwlks 29748*	The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Revised by AV, 29-Oct-2021.)
⊢ (ClWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
Theorem	isclwlk 29749	A pair of functions represents a closed walk iff it represents a walk in which the first vertex is equal to the last vertex. (Contributed by Alexander van der Vekens, 24-Jun-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(ClWalks‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = (𝑃‘(♯‘𝐹))))
Theorem	clwlkiswlk 29750	A closed walk is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 16-Feb-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ (𝐹(ClWalks‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	clwlkwlk 29751	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 29752	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 29753*	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 29754*	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 29755*	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 29756*	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 29757*	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 29758	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 29759	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 29760	Extend class notation with circuits (in a graph).
class Circuits
Syntax	ccycls 29761	Extend class notation with cycles (in a graph).
class Cycles
Definition	df-crcts 29762*	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 29763*	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 29764*	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 29765*	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 29766	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 29767	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 29768	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 29769	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 29770	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 29771	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 29772	A circuit is a walk. (Contributed by AV, 6-Apr-2021.)
⊢ (𝐹(Circuits‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃)
Theorem	cyclispth 29773	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 29774	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 29775	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	cyclnumvtx 29776	The number of vertices of a (non-trivial) cycle is the number of edges in the cycle. (Contributed by AV, 5-Oct-2025.)
⊢ ((1 ≤ (♯‘𝐹) ∧ 𝐹(Cycles‘𝐺)𝑃) → (♯‘ran 𝑃) = (♯‘𝐹))
Theorem	cyclnspth 29777	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	pthisspthorcycl 29778	A path is either a simple path or a cycle (or both). (Contributed by BTernaryTau, 20-Oct-2023.)
⊢ (𝐹(Paths‘𝐺)𝑃 → (𝐹(SPaths‘𝐺)𝑃 ∨ 𝐹(Cycles‘𝐺)𝑃))
Theorem	pthspthcyc 29779	A pair ⟨𝐹, 𝑃⟩ represents a path if it represents either a simple path or a cycle. The exclusivity only holds for non-trivial paths (𝐹 ≠ ∅), see cyclnspth 29777. (Contributed by AV, 2-Oct-2025.)
⊢ (𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(SPaths‘𝐺)𝑃 ∨ 𝐹(Cycles‘𝐺)𝑃))
Theorem	cyclispthon 29780	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 29781*	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 29782	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 29783	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 29784	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 29785	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 29786	Lemma for crctcshwlkn0 29797. (Contributed by AV, 13-Mar-2021.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℕ) → ((𝐴 − 𝐵) + 1) ≤ 𝐴)
Theorem	crctcshwlkn0lem2 29787*	Lemma for crctcshwlkn0 29797. (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 29788*	Lemma for crctcshwlkn0 29797. (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 29789*	Lemma for crctcshwlkn0 29797. (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 29790*	Lemma for crctcshwlkn0 29797. (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 29791*	Lemma for crctcshwlkn0 29797. (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 29792*	Lemma for crctcshwlkn0 29797. (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 29793	Lemma for crctcsh 29800. (Contributed by AV, 10-Mar-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐹(Circuits‘𝐺)𝑃)    &   ⊢ 𝑁 = (♯‘𝐹)    ⇒   ⊢ (𝜑 → 𝑁 ∈ ℕ0)
Theorem	crctcshlem2 29794	Lemma for crctcsh 29800. (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 29795*	Lemma for crctcsh 29800. (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 29796*	Lemma for crctcsh 29800. (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 29797*	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 29798*	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 29799*	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 29800*	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)