P. 270 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	2wlkd 26901	Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.) (Revised by AV, 23-Jan-2021.) (Proof shortened by AV, 14-Feb-2021.) (Revised by AV, 24-Mar-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝜑 → 𝐹(Walks‘𝐺)𝑃)
Theorem	2wlkond 26902	A walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    ⇒   ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)
Theorem	2trld 26903	Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    ⇒   ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃)
Theorem	2trlond 26904	A trail of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    ⇒   ⊢ (𝜑 → 𝐹(𝐴(TrailsOn‘𝐺)𝐶)𝑃)
Theorem	2pthd 26905	A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    ⇒   ⊢ (𝜑 → 𝐹(Paths‘𝐺)𝑃)
Theorem	2spthd 26906	A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.) (Proof shortened by AV, 30-Oct-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃)
Theorem	2pthond 26907	A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Proof shortened by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    &   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼‘𝐾)))    &   ⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹(𝐴(SPathsOn‘𝐺)𝐶)𝑃)
Theorem	2pthon3v 26908*	For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝐺 ∈ UHGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) ∧ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)) → ∃𝑓∃𝑝(𝑓(𝐴(SPathsOn‘𝐺)𝐶)𝑝 ∧ (#‘𝑓) = 2))
Theorem	umgr2adedgwlklem 26909	Lemma for umgr2adedgwlk 26910, umgr2adedgspth 26913, etc. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) ∧ (𝐴 ∈ (Vtx‘𝐺) ∧ 𝐵 ∈ (Vtx‘𝐺) ∧ 𝐶 ∈ (Vtx‘𝐺))))
Theorem	umgr2adedgwlk 26910	In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})    ⇒   ⊢ (𝜑 → (𝐹(Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2 ∧ (𝐴 = (𝑃‘0) ∧ 𝐵 = (𝑃‘1) ∧ 𝐶 = (𝑃‘2))))
Theorem	umgr2adedgwlkon 26911	In a multigraph, two adjacent edges form a walk between two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})    ⇒   ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)
Theorem	umgr2adedgwlkonALT 26912	Alternate proof for umgr2adedgwlkon 26911, using umgr2adedgwlk 26910, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})    ⇒   ⊢ (𝜑 → 𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)
Theorem	umgr2adedgspth 26913	In a multigraph, two adjacent edges with different endvertices form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐼 = (iEdg‘𝐺)    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ (𝜑 → 𝐺 ∈ UMGraph)    &   ⊢ (𝜑 → ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸))    &   ⊢ (𝜑 → (𝐼‘𝐽) = {𝐴, 𝐵})    &   ⊢ (𝜑 → (𝐼‘𝐾) = {𝐵, 𝐶})    &   ⊢ (𝜑 → 𝐴 ≠ 𝐶)    ⇒   ⊢ (𝜑 → 𝐹(SPaths‘𝐺)𝑃)
Theorem	umgr2wlk 26914*	In a multigraph, there is a walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝐴 = (𝑝‘0) ∧ 𝐵 = (𝑝‘1) ∧ 𝐶 = (𝑝‘2))))
Theorem	umgr2wlkon 26915*	For each pair of adjacent edges in a multigraph, there is a walk of length 2 between the not common vertices of the edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸) → ∃𝑓∃𝑝 𝑓(𝐴(WalksOn‘𝐺)𝐶)𝑝)
Theorem	elwwlks2s3 26916*	A walk of length 2 as word is a length 3 string. (Contributed by AV, 18-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
Theorem	midwwlks2s3 26917*	There is a vertex between the endpoints of a walk of length 2 between two vertices as length 3 string. (Contributed by AV, 10-Jan-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (2 WWalksN 𝐺) → ∃𝑏 ∈ 𝑉 (𝑊‘1) = 𝑏)
Theorem	wwlks2onv 26918	If a length 3 string represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.) (Proof shortened by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐵 ∈ 𝑈 ∧ ⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))
Theorem	elwwlks2ons3im 26919	A walk as word of length 2 between two vertices is a length 3 string and its second symbol is a vertex. (Contributed by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) → (𝑊 = ⟨“𝐴(𝑊‘1)𝐶”⟩ ∧ (𝑊‘1) ∈ 𝑉))
Theorem	elwwlks2ons3 26920*	For each walk of length 2 between two vertices, there is a third vertex in the middle of the walk. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶)))
Theorem	elwwlks2ons3OLD 26921*	Obsolete version of elwwlks2ons3 26920 as of 13-Mar-2022. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ 𝑈 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶))))
Theorem	s3wwlks2on 26922*	A length 3 string which represents a walk of length 2 between two vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (#‘𝑓) = 2)))
Theorem	umgrwwlks2on 26923	A walk of length 2 between two vertices as word in a multigraph. This theorem would also hold for pseudographs, but to prove this the cases 𝐴 = 𝐵 and/or 𝐵 = 𝐶 must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 12-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
Theorem	wwlks2onsym 26924	There is a walk of length 2 from one vertex to another vertex iff there is a walk of length 2 from the other vertex to the first vertex. (Contributed by AV, 7-Jan-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UMGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ⟨“𝐶𝐵𝐴”⟩ ∈ (𝐶(2 WWalksNOn 𝐺)𝐴)))
Theorem	elwwlks2on 26925*	A walk of length 2 between two vertices as length 3 string. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ∃𝑓(𝑓(Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))
Theorem	elwspths2on 26926*	A simple path of length 2 between two vertices (in a graph) as length 3 string. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 12-May-2021.) (Proof shortened by AV, 16-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (𝑊 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 (𝑊 = ⟨“𝐴𝑏𝐶”⟩ ∧ ⟨“𝐴𝑏𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶))))
Theorem	wpthswwlks2on 26927	For two different vertices, a walk of length 2 between these vertices is a simple path of length 2 between these vertices in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (Revised by AV, 16-Mar-2022.)
⊢ ((𝐺 ∈ USGraph ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))
Theorem	wpthswwlks2onOLD 26928	Obsolete version of wpthswwlks2on 26927 as of 16-Mar-2022. (Contributed by Alexander van der Vekens, 2-Mar-2018.) (Revised by AV, 13-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ∧ 𝐴 ≠ 𝐵) → (𝐴(2 WSPathsNOn 𝐺)𝐵) = (𝐴(2 WWalksNOn 𝐺)𝐵))
Theorem	2wspdisj 26929*	All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 9-Jan-2022.)
⊢ Disj 𝑏 ∈ (𝑉 ∖ {𝐴})(𝐴(2 WSPathsNOn 𝐺)𝑏)
Theorem	2wspiundisj 26930*	All simple paths of length 2 from a fixed vertex to another vertex are disjunct. (Contributed by Alexander van der Vekens, 5-Mar-2018.) (Revised by AV, 14-May-2021.) (Proof shortened by AV, 9-Jan-2022.)
⊢ Disj 𝑎 ∈ 𝑉 ∪ 𝑏 ∈ (𝑉 ∖ {𝑎})(𝑎(2 WSPathsNOn 𝐺)𝑏)
Theorem	usgr2wspthons3 26931	A simple path of length 2 between two vertices represented as length 3 string corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 16-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ (𝐴 ≠ 𝐶 ∧ {𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
Theorem	usgr2wspthon 26932*	A simple path of length 2 between two vertices corresponds to two adjacent edges in a simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.) (Revised by AV, 17-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝑇 ∈ (𝐴(2 WSPathsNOn 𝐺)𝐶) ↔ ∃𝑏 ∈ 𝑉 ((𝑇 = ⟨“𝐴𝑏𝐶”⟩ ∧ 𝐴 ≠ 𝐶) ∧ ({𝐴, 𝑏} ∈ 𝐸 ∧ {𝑏, 𝐶} ∈ 𝐸))))
Theorem	elwwlks2 26933*	A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.) (Proof shortened by AV, 14-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Theorem	elwspths2spth 26934*	A simple path of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 28-Feb-2018.) (Revised by AV, 18-May-2021.) (Proof shortened by AV, 16-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 ∃𝑐 ∈ 𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓∃𝑝(𝑓(SPaths‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
16.3.9  Walks in regular graphs
Theorem	rusgrnumwwlkl1 26935*	In a k-regular graph, there are k walks (as word) of length 1 starting at each vertex. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑃 ∈ 𝑉) → (#‘{𝑤 ∈ (1 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = 𝐾)
Theorem	rusgrnumwwlkslem 26936*	Lemma for rusgrnumwwlks 26941. (Contributed by Alexander van der Vekens, 23-Aug-2018.)
⊢ (𝑌 ∈ {𝑤 ∈ 𝑍 ∣ (𝑤‘0) = 𝑃} → {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ 𝜓)} = {𝑤 ∈ 𝑋 ∣ (𝜑 ∧ (𝑌‘0) = 𝑃 ∧ 𝜓)})
Theorem	rusgrnumwwlklem 26937*	Lemma for rusgrnumwwlk 26942 etc. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0) → (𝑃𝐿𝑁) = (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}))
Theorem	rusgrnumwwlkb0 26938*	Induction base 0 for rusgrnumwwlk 26942. Here, we do not need the regularity of the graph yet. (Contributed by Alexander van der Vekens, 24-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿0) = 1)
Theorem	rusgrnumwwlkb1 26939*	Induction base 1 for rusgrnumwwlk 26942. (Contributed by Alexander van der Vekens, 28-Jul-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ 𝑃 ∈ 𝑉) → (𝑃𝐿1) = 𝐾)
Theorem	rusgr0edg 26940*	Special case for 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.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺RegUSGraph0 ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ) → (𝑃𝐿𝑁) = 0)
Theorem	rusgrnumwwlks 26941*	Induction step for rusgrnumwwlk 26942. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → ((𝑃𝐿𝑁) = (𝐾↑𝑁) → (𝑃𝐿(𝑁 + 1)) = (𝐾↑(𝑁 + 1))))
Theorem	rusgrnumwwlk 26942*	In a 𝐾-regular graph, the number of walks of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. By definition, (𝑁 WWalksN 𝐺) is the set of walks (as words) with length 𝑁, and (𝑃𝐿𝑁) is the number of walks with length 𝑁 starting at the vertex 𝑃. Because of the 𝐾-regularity, a walk can be continued in 𝐾 different ways at the end vertex of the walk, and this repeated 𝑁 times. This theorem even holds for 𝑁 = 0: in this case, the walk consists of only one vertex 𝑃, so the number of walks of length 𝑁 = 0 starting with 𝑃 is (𝐾↑0) = 1. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐿 = (𝑣 ∈ 𝑉, 𝑛 ∈ ℕ0 ↦ (#‘{𝑤 ∈ (𝑛 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑣}))    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (𝑃𝐿𝑁) = (𝐾↑𝑁))
Theorem	rusgrnumwwlkg 26943*	In a 𝐾-regular graph, the number of walks (as words) of a fixed length 𝑁 from a fixed vertex is 𝐾 to the power of 𝑁. Closed form of rusgrnumwwlk 26942. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}) = (𝐾↑𝑁))
Theorem	rusgrnumwlkg 26944*	In a k-regular graph, the number of walks of a fixed length n from a fixed vertex is k to the power of n. This theorem corresponds to statement 11 in [Huneke] p. 2: "The total number of walks v(0) v(1) ... v(n-2) from a fixed vertex v = v(0) is k^(n-2) as G is k-regular.". This theorem even holds for n=0: then the walk consists of only one vertex v(0), so the number of walks of length n=0 starting with v=v(0) is 1=k^0. (Contributed by Alexander van der Vekens, 24-Aug-2018.) (Revised by AV, 7-May-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺RegUSGraph𝐾 ∧ (𝑉 ∈ Fin ∧ 𝑃 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘{𝑤 ∈ (Walks‘𝐺) ∣ ((#‘(1st ‘𝑤)) = 𝑁 ∧ ((2nd ‘𝑤)‘0) = 𝑃)}) = (𝐾↑𝑁))
Theorem	clwwlknclwwlkdif 26945*	The set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑉 and ending not at this vertex is the difference between the set 𝐶 of walks of length 𝑁 starting with this vertex 𝑋 and the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 16-Mar-2022.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)    &   ⊢ 𝐶 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋}    ⇒   ⊢ 𝐴 = (𝐶 ∖ 𝐵)
Theorem	clwwlknclwwlkdifnum 26946*	In a 𝐾-regular graph, the size of the set 𝐴 of walks of length 𝑁 starting with a fixed vertex 𝑋 and ending not at this vertex is the difference between 𝐾 to the power of 𝑁 and the size of the set 𝐵 of closed walks of length 𝑁 anchored at this vertex 𝑋. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Revised by AV, 8-Mar-2022.) (Proof shortened by AV, 16-Mar-2022.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = (𝑋(𝑁 WWalksNOn 𝐺)𝑋)    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ0)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))
Theorem	clwwlknclwwlkdifsOLD 26947	Obsolete version of clwwlknclwwlkdif 26945 as of 8-Mar-2022. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}    ⇒   ⊢ 𝐴 = ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋} ∖ 𝐵)
Theorem	clwwlknclwwlkdifnumOLD 26948*	Obsolete version of clwwlknclwwlkdifnum 26946 as of 8-Mar-2022. (Contributed by Alexander van der Vekens, 30-Sep-2018.) (Revised by AV, 7-May-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝐴 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑋 ∧ ( lastS ‘𝑤) ≠ 𝑋)}    &   ⊢ 𝐵 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (( lastS ‘𝑤) = (𝑤‘0) ∧ (𝑤‘0) = 𝑋)}    &   ⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (((𝐺RegUSGraph𝐾 ∧ 𝑉 ∈ Fin) ∧ (𝑋 ∈ 𝑉 ∧ 𝑁 ∈ ℕ)) → (#‘𝐴) = ((𝐾↑𝑁) − (#‘𝐵)))
16.3.10  Closed walks as words In general, a closed walk is an alternating sequence of vertices and edges, as defined in df-clwlks 26723: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n), with p(n) = p(0). Often, it is sufficient to refer to a walk by the (cyclic) sequence of its vertices, i.e omitting its edges in its representation: p(0) p(1) ... p(n-1) p(0), see the corresponding remark on cycles (which are special closed walks) in [Diestel] p. 7. As for "walks as words" in general, the concept of a Word, see df-word 13331, is also used in definitions df-clwwlk 26950 and df-clwwlkn 26983, and the representation of a closed walk as the sequence of its vertices is called "closed walk as word". In contrast to "walks as words", the terminating vertex p(n) of a closed walk is omitted in the representation of a closed walk as word, see definitions df-clwwlk 26950 and df-clwwlkn 26983, because it is always equal to the first vertex of the closed walk. This represenation has the advantage that the vertices can be cyclically shifted without changing the represented closed walk. Furthermore, the length of a closed walk (i.e. the number of its edges) equals the number of symbols/vertices of the word representing the closed walk. To avoid to handle the degenerate case of representing a (closed) walk of length 0 by the empty word, this case is excluded within the definition (𝑤 ≠ ∅). This is because a walk of length 0 is anchored at an arbitrary vertex by the general definition for closed walks, see 0clwlkv 27109, which neither can be reflected by the empty word nor by a singleton word ⟨“𝑣”⟩ with vertex v : ⟨“𝑣”⟩ represents the walk "𝑣 𝑣", which is a (closed) walk of length 1 (if there is an edge/loop from 𝑣 to 𝑣), see loopclwwlkn1b 27005. Therefore, a closed walk corresponds to a closed walk as word only for walks of length at least 1, see clwlkclwwlk2 26969. Although the set ClWWalksN of all closed walks of a fixed length as words over the set of vertices is defined as function over ℕ0, the fixed length is usually not 0, because (0 ClWWalksN 𝐺) = ∅ (see clwwlkn0 26989). Analogous to (𝐴(𝑁 WWalksNOn 𝐺)𝐵), the set of walks of a fixed length 𝑁 between two vertices 𝐴 and 𝐵, the set (𝑋(ClWWalksNOn‘𝐺)𝑁) of closed walks of a fixed length 𝑁 anchored at a fixed vertex 𝑋 is defined by df-clwwlknon 27061. This definition is also based on ℕ0 instead of ℕ, with (𝑋(ClWWalksNOn‘𝐺)0) = ∅ (see clwwlk0on0 27067). clwwlknon1le1 27076 states that there is at most one (closed) walk of length 1 on a vertex, which would consist of a loop (see clwwlknon1loop 27073). And in a 𝐾-regular graph, there are 𝐾 closed walks of length 2 on each vertex, see clwwlknon2num 27079.
16.3.10.1  Closed walks as words
Syntax	cclwwlk 26949	Extend class notation with closed walks (in an undirected graph) as word over the set of vertices.
class ClWWalks
Definition	df-clwwlk 26950*	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-clwlks 26723. 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. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ ClWWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ (Edg‘𝑔))})
Theorem	clwwlk 26951*	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.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (ClWWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑤), (𝑤‘0)} ∈ 𝐸)}
Theorem	isclwwlk 26952*	Properties of a word to represent a closed walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (ClWWalks‘𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅) ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸))
Theorem	clwwlkbp 26953	Basic properties of a closed walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (ClWWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉 ∧ 𝑊 ≠ ∅))
Theorem	clwwlkgt0 26954	There is no empty closed walk (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)
⊢ (𝑊 ∈ (ClWWalks‘𝐺) → 0 < (#‘𝑊))
Theorem	clwwlksswrd 26955	Closed walks (represented by words) are words. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 25-Apr-2021.)
⊢ (ClWWalks‘𝐺) ⊆ Word (Vtx‘𝐺)
Theorem	clwwlk1loop 26956	A closed walk of length 1 is a loop. See also clwlkl1loop 26734. (Contributed by AV, 24-Apr-2021.)
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 1) → {(𝑊‘0), (𝑊‘0)} ∈ (Edg‘𝐺))
Theorem	umgrclwwlkge2 26957	A closed walk in a multigraph has a length of at least 2 (because it cannot have a loop). (Contributed by Alexander van der Vekens, 16-Sep-2018.) (Revised by AV, 24-Apr-2021.)
⊢ (𝐺 ∈ UMGraph → (𝑃 ∈ (ClWWalks‘𝐺) → 2 ≤ (#‘𝑃)))
Theorem	clwlkclwwlklem2a1 26958*	Lemma 1 for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∀𝑖 ∈ (0..^((#‘𝑃) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸))
Theorem	clwlkclwwlklem2a2 26959*	Lemma 2 for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (#‘𝐹) = ((#‘𝑃) − 1))
Theorem	clwlkclwwlklem2a3 26960*	Lemma 3 for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 21-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (𝑃‘(#‘𝐹)) = ( lastS ‘𝑃))
Theorem	clwlkclwwlklem2fv1 26961*	Lemma 4a for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ (((#‘𝑃) ∈ ℕ0 ∧ 𝐼 ∈ (0..^((#‘𝑃) − 2))) → (𝐹‘𝐼) = (◡𝐸‘{(𝑃‘𝐼), (𝑃‘(𝐼 + 1))}))
Theorem	clwlkclwwlklem2fv2 26962*	Lemma 4b for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 22-Jun-2018.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ (((#‘𝑃) ∈ ℕ0 ∧ 2 ≤ (#‘𝑃)) → (𝐹‘((#‘𝑃) − 2)) = (◡𝐸‘{(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)}))
Theorem	clwlkclwwlklem2a4 26963*	Lemma 4 for clwlkclwwlklem2a 26964. (Contributed by Alexander van der Vekens, 21-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ 𝐼 ∈ (0..^((#‘𝑃) − 1))) → ({(𝑃‘𝐼), (𝑃‘(𝐼 + 1))} ∈ ran 𝐸 → (𝐸‘(𝐹‘𝐼)) = {(𝑃‘𝐼), (𝑃‘(𝐼 + 1))})))
Theorem	clwlkclwwlklem2a 26964*	Lemma for clwlkclwwlklem2 26966. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ 𝐹 = (𝑥 ∈ (0..^((#‘𝑃) − 1)) ↦ if(𝑥 < ((#‘𝑃) − 2), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘(𝑥 + 1))}), (◡𝐸‘{(𝑃‘𝑥), (𝑃‘0)})))    ⇒   ⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ((𝐹 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))))
Theorem	clwlkclwwlklem1 26965*	Lemma 1 for clwlkclwwlk 26968. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → ((( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸)) → ∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓)))))
Theorem	clwlkclwwlklem2 26966*	Lemma 2 for clwlkclwwlk 26968. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ (((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝐹 ∈ Word dom 𝐸) ∧ (𝑃:(0...(#‘𝐹))⟶𝑉 ∧ 2 ≤ (#‘𝑃)) ∧ (∀𝑖 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∧ (𝑃‘0) = (𝑃‘(#‘𝐹)))) → (( lastS ‘𝑃) = (𝑃‘0) ∧ ∀𝑖 ∈ (0..^((#‘𝐹) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝐹) − 1)), (𝑃‘0)} ∈ ran 𝐸))
Theorem	clwlkclwwlklem3 26967*	Lemma 3 for clwlkclwwlk 26968. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 11-Apr-2021.)
⊢ ((𝐸:dom 𝐸–1-1→𝑅 ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓((𝑓 ∈ Word dom 𝐸 ∧ 𝑃:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓‘𝑖)) = {(𝑃‘𝑖), (𝑃‘(𝑖 + 1))}) ∧ (𝑃‘0) = (𝑃‘(#‘𝑓))) ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (∀𝑖 ∈ (0..^((((#‘𝑃) − 1) − 0) − 1)){(𝑃‘𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 ∧ {(𝑃‘((#‘𝑃) − 2)), (𝑃‘0)} ∈ ran 𝐸))))
Theorem	clwlkclwwlk 26968*	A closed walk as word of length at least 2 corresponds to a closed walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 2 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)𝑃 ↔ (( lastS ‘𝑃) = (𝑃‘0) ∧ (𝑃 substr ⟨0, ((#‘𝑃) − 1)⟩) ∈ (ClWWalks‘𝐺))))
Theorem	clwlkclwwlk2 26969*	A closed walk corresponds to a closed walk as word in a simple pseudograph. (Contributed by Alexander van der Vekens, 22-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 7-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (iEdg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (#‘𝑃)) → (∃𝑓 𝑓(ClWalks‘𝐺)(𝑃 ++ ⟨“(𝑃‘0)”⟩) ↔ 𝑃 ∈ (ClWWalks‘𝐺)))
Theorem	clwwisshclwwslemlem 26970*	Lemma for clwwisshclwwslem 26971. (Contributed by Alexander van der Vekens, 23-Mar-2018.)
⊢ (((𝐿 ∈ (ℤ≥‘2) ∧ 𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) ∧ ∀𝑖 ∈ (0..^(𝐿 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝑅 ∧ {(𝑊‘(𝐿 − 1)), (𝑊‘0)} ∈ 𝑅) → {(𝑊‘((𝐴 + 𝐵) mod 𝐿)), (𝑊‘(((𝐴 + 1) + 𝐵) mod 𝐿))} ∈ 𝑅)
Theorem	clwwisshclwwslem 26971*	Lemma for clwwisshclwws 26972. (Contributed by AV, 24-Mar-2018.) (Revised by AV, 28-Apr-2021.)
⊢ ((𝑊 ∈ Word 𝑉 ∧ 𝑁 ∈ (1..^(#‘𝑊))) → ((∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) → ∀𝑗 ∈ (0..^((#‘(𝑊 cyclShift 𝑁)) − 1)){((𝑊 cyclShift 𝑁)‘𝑗), ((𝑊 cyclShift 𝑁)‘(𝑗 + 1))} ∈ 𝐸))
Theorem	clwwisshclwws 26972	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 AV, 28-Apr-2021.)
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0..^(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺))
Theorem	clwwisshclwwsn 26973	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.) (Revised by AV, 29-Apr-2021.)
⊢ ((𝑊 ∈ (ClWWalks‘𝐺) ∧ 𝑁 ∈ (0...(#‘𝑊))) → (𝑊 cyclShift 𝑁) ∈ (ClWWalks‘𝐺))
Theorem	erclwwlkrel 26974	∼ is a relation. (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ Rel ∼
Theorem	erclwwlkeq 26975*	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 AV, 29-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 ↔ (𝑈 ∈ (ClWWalks‘𝐺) ∧ 𝑊 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑊))𝑈 = (𝑊 cyclShift 𝑛))))
Theorem	erclwwlkeqlen 26976*	If two classes are equivalent regarding ∼, then they are words of the same length. (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ ((𝑈 ∈ 𝑋 ∧ 𝑊 ∈ 𝑌) → (𝑈 ∼ 𝑊 → (#‘𝑈) = (#‘𝑊)))
Theorem	erclwwlkref 26977*	∼ is a reflexive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 25-Mar-2018.) (Revised by AV, 29-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∈ (ClWWalks‘𝐺) ↔ 𝑥 ∼ 𝑥)
Theorem	erclwwlksym 26978*	∼ is a symmetric relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 8-Apr-2018.) (Revised by AV, 29-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ (𝑥 ∼ 𝑦 → 𝑦 ∼ 𝑥)
Theorem	erclwwlktr 26979*	∼ is a transitive relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ ((𝑥 ∼ 𝑦 ∧ 𝑦 ∼ 𝑧) → 𝑥 ∼ 𝑧)
Theorem	erclwwlk 26980*	∼ is an equivalence relation over the set of closed walks (defined as words). (Contributed by Alexander van der Vekens, 10-Apr-2018.) (Revised by AV, 30-Apr-2021.)
⊢ ∼ = {⟨𝑢, 𝑤⟩ ∣ (𝑢 ∈ (ClWWalks‘𝐺) ∧ 𝑤 ∈ (ClWWalks‘𝐺) ∧ ∃𝑛 ∈ (0...(#‘𝑤))𝑢 = (𝑤 cyclShift 𝑛))}    ⇒   ⊢ ∼ Er (ClWWalks‘𝐺)
16.3.10.2  Closed walks of a fixed length as words
Syntax	cclwwlkn 26981	Extend class notation with closed walks (in an undirected graph) of a fixed length as word over the set of vertices.
class ClWWalksN
Syntax	cclwwlknold 26982	Obsolete version of ClWWalksN as of 22-Mar-2022.
class ClWWalksNOLD
Definition	df-clwwlkn 26983*	Define the set of all closed walks of a fixed length 𝑛 as words over the set of vertices in a graph 𝑔. If 0 < 𝑛, 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-clwlks 26723. For 𝑛 = 0, the set is empty, see clwwlkn0 26989. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)
⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛})
Definition	df-clwwlknOLD 26984*	Obsolete version of df-clwwlkn 26983 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (New usage is discouraged.)
⊢ ClWWalksNOLD = (𝑛 ∈ ℕ, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (#‘𝑤) = 𝑛})
Theorem	clwwlkn 26985*	The set of closed walks of a fixed length 𝑁 as words over the set of vertices in a graph 𝐺. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2021.)
⊢ (𝑁 ClWWalksN 𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁}
Theorem	clwwlknOLD 26986*	Obsolete version of clwwlkn 26985 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → (𝑁ClWWalksNOLD𝐺) = {𝑤 ∈ (ClWWalks‘𝐺) ∣ (#‘𝑤) = 𝑁})
Theorem	isclwwlkn 26987	A word over the set of vertices representing a closed walk of a fixed length. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2021.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁))
Theorem	isclwwlknOLD 26988	Obsolete version of isclwwlkn 26987 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁ClWWalksNOLD𝐺) ↔ (𝑊 ∈ (ClWWalks‘𝐺) ∧ (#‘𝑊) = 𝑁)))
Theorem	clwwlkn0 26989	There is no closed walk of length 0 (i.e. a closed walk without any edge) represented by a word of vertices. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.)
⊢ (0 ClWWalksN 𝐺) = ∅
Theorem	clwwlkneq0 26990	Sufficient conditions for ClWWalksN to be empty. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 24-Feb-2022.)
⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ) → (𝑁 ClWWalksN 𝐺) = ∅)
Theorem	clwwlkn0OLD 26991	Obsolete version of clwwlkn0 26989 as of 22-Mar-2022. (Contributed by Alexander van der Vekens, 15-Sep-2018.) (Revised by AV, 24-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (0 ClWWalksN 𝐺) = ∅
Theorem	clwwlkclwwlkn 26992	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.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ (ClWWalks‘𝐺))
Theorem	clwwlksclwwlkn 26993	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.) (Revised by AV, 12-Apr-2021.)
⊢ (𝑁 ClWWalksN 𝐺) ⊆ (ClWWalks‘𝐺)
Theorem	clwwlknlen 26994	The length of a word representing a closed walk of a fixed length is this fixed length. (Contributed by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (#‘𝑊) = 𝑁)
Theorem	clwwlknnn 26995	The length of a closed walk of a fixed length as word is a positive integer. (Contributed by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑁 ∈ ℕ)
Theorem	clwwlknwrd 26996	A closed walk of a fixed length as word is a word over the vertices. (Contributed by AV, 30-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ Word 𝑉)
Theorem	clwwlknbp 26997	Basic properties of a closed walk of a fixed length as word. (Contributed by AV, 30-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁))
Theorem	isclwwlknx 26998*	Characterization of a word representing a closed walk of a fixed length, definition of ClWWalks expanded. (Contributed by AV, 25-Apr-2021.) (Proof shortened by AV, 22-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ → (𝑊 ∈ (𝑁 ClWWalksN 𝐺) ↔ ((𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸) ∧ (#‘𝑊) = 𝑁)))
Theorem	clwwlknp 26999*	Properties of a set being a closed walk (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 24-Apr-2021.) (Proof shortened by AV, 23-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → ((𝑊 ∈ Word 𝑉 ∧ (#‘𝑊) = 𝑁) ∧ ∀𝑖 ∈ (0..^(𝑁 − 1)){(𝑊‘𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ {( lastS ‘𝑊), (𝑊‘0)} ∈ 𝐸))
Theorem	clwwlknwwlksn 27000	A word representing a closed walk of length 𝑁 also represents a walk of length 𝑁 − 1. The walk is one edge shorter than the closed walk, because the last edge connecting the last with the first vertex is missing. For example, if ⟨“𝑎𝑏𝑐”⟩ ∈ (3 ClWWalksN 𝐺) represents a closed walk "abca" of length 3, then ⟨“𝑎𝑏𝑐”⟩ ∈ (2 WWalksN 𝐺) represents a walk "abc" (not closed if 𝑎 ≠ 𝑐) of length 2, and ⟨“𝑎𝑏𝑐𝑎”⟩ ∈ (3 WWalksN 𝐺) represents also a closed walk "abca" of length 3. (Contributed by AV, 24-Jan-2022.) (Revised by AV, 22-Mar-2022.)
⊢ (𝑊 ∈ (𝑁 ClWWalksN 𝐺) → 𝑊 ∈ ((𝑁 − 1) WWalksN 𝐺))