P. 301 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 30001-30100   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	wwlkssswrd 30001	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 30002*	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 30003	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 30004	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 30005	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 30006	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 30007	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 30008*	Lemma 1 for wlkiswwlks2 30014. (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 30009*	Lemma 2 for wlkiswwlks2 30014. (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 30010*	Lemma 3 for wlkiswwlks2 30014. (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 30011*	Lemma 4 for wlkiswwlks2 30014. (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 30012*	Lemma 5 for wlkiswwlks2 30014. (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 30013*	Lemma 6 for wlkiswwlks2 30014. (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 30014*	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 30015*	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 30016*	A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 30014 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10427) 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 30014). (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))
Theorem	wlkiswwlkupgr 30017*	A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks 30015 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
⊢ (𝐺 ∈ UPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑃 ↔ 𝑃 ∈ (WWalks‘𝐺)))
Theorem	wlkswwlksf1o 30018*	The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)
⊢ 𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd ‘𝑤))    ⇒   ⊢ (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
Theorem	wlkswwlksen 30019	The set of walks as words and the set of (ordinary) walks are equinumerous in a simple pseudograph. (Contributed by AV, 6-May-2021.) (Revised by AV, 5-Aug-2022.)
⊢ (𝐺 ∈ USPGraph → (Walks‘𝐺) ≈ (WWalks‘𝐺))
Theorem	wwlksm1edg 30020	Removing the trailing edge from a walk (as word) with at least one edge results in a walk. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ ((𝑊 ∈ (WWalks‘𝐺) ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix ((♯‘𝑊) − 1)) ∈ (WWalks‘𝐺))
Theorem	wlklnwwlkln2lem 30021*	Lemma for wlklnwwlkln2 30022 and wlklnwwlklnupgr2 30024. Formerly part of proof for wlklnwwlkln2 30022. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝜑 → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))    ⇒   ⊢ (𝜑 → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlkln2 30022*	A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝐺 ∈ USPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlkn 30023*	A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wlklnwwlklnupgr2 30024*	A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a pseudograph. This variant of wlklnwwlkln2 30022 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝐺 ∈ UPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
Theorem	wlklnwwlknupgr 30025*	A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of wlklnwwlkn 30023 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
⊢ (𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wlknewwlksn 30026	If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)
⊢ (((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st ‘𝑊)) = 𝑁)) → (2nd ‘𝑊) ∈ (𝑁 WWalksN 𝐺))
Theorem	wlknwwlksnbij 30027*	The mapping (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
⊢ 𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}    &   ⊢ 𝑊 = (𝑁 WWalksN 𝐺)    &   ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (2nd ‘𝑡))    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇–1-1-onto→𝑊)
Theorem	wlknwwlksnen 30028*	In a simple pseudograph, the set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))
Theorem	wlknwwlksneqs 30029*	The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁}) = (♯‘(𝑁 WWalksN 𝐺)))
Theorem	wwlkseq 30030*	Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
⊢ ((𝑊 ∈ (WWalks‘𝐺) ∧ 𝑇 ∈ (WWalks‘𝐺)) → (𝑊 = 𝑇 ↔ ((♯‘𝑊) = (♯‘𝑇) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊‘𝑖) = (𝑇‘𝑖))))
Theorem	wwlksnred 30031	Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
Theorem	wwlksnext 30032	Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆 ∈ 𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
Theorem	wwlksnextbi 30033	Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Proof shortened by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (((𝑁 ∈ ℕ0 ∧ 𝑆 ∈ 𝑉) ∧ (𝑇 ∈ Word 𝑉 ∧ 𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
Theorem	wwlksnredwwlkn 30034*	For each walk (as word) of length at least 1 there is a shorter walk (as word). (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
Theorem	wwlksnredwwlkn0 30035*	For each walk (as word) of length at least 1 there is a shorter walk (as word) starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ0 ∧ 𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
Theorem	wwlksnextwrd 30036*	Lemma for wwlksnextbij 30041. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)})
Theorem	wwlksnextfun 30037*	Lemma for wwlksnextbij 30041. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷⟶𝑅)
Theorem	wwlksnextinj 30038*	Lemma for wwlksnextbij 30041. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑁 ∈ ℕ0 → 𝐹:𝐷–1-1→𝑅)
Theorem	wwlksnextsurj 30039*	Lemma for wwlksnextbij 30041. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–onto→𝑅)
Theorem	wwlksnextbij0 30040*	Lemma for wwlksnextbij 30041. (Contributed by Alexander van der Vekens, 7-Aug-2018.) (Revised by AV, 18-Apr-2021.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((♯‘𝑤) = (𝑁 + 2) ∧ (𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}    &   ⊢ 𝑅 = {𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   ⊢ 𝐹 = (𝑡 ∈ 𝐷 ↦ (lastS‘𝑡))    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷–1-1-onto→𝑅)
Theorem	wwlksnextbij 30041*	There is a bijection between the extensions of a walk (as word) by an edge and the set of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 18-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
Theorem	wwlksnexthasheq 30042*	The number of the extensions of a walk (as word) by an edge equals the number of vertices being connected to the trailing vertex of the walk. (Contributed by Alexander van der Vekens, 23-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) → (♯‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 prefix (𝑁 + 1)) = 𝑊 ∧ {(lastS‘𝑊), (lastS‘𝑤)} ∈ 𝐸)}) = (♯‘{𝑛 ∈ 𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
Theorem	disjxwwlksn 30043*	Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 29-Jul-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 27-Oct-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 prefix 𝑁) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
Theorem	wwlksnndef 30044	Conditions for WWalksN not being defined. (Contributed by Alexander van der Vekens, 30-Jul-2018.) (Revised by AV, 19-Apr-2021.)
⊢ ((𝐺 ∉ V ∨ 𝑁 ∉ ℕ0) → (𝑁 WWalksN 𝐺) = ∅)
Theorem	wwlksnfi 30045	The number of walks represented by words of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 30-Jul-2018.) (Revised by AV, 19-Apr-2021.) (Proof shortened by JJ, 18-Nov-2022.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)
Theorem	wlksnfi 30046*	The number of walks of fixed length is finite if the number of vertices is finite (in the graph). (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 20-Apr-2021.)
⊢ ((𝐺 ∈ FinUSGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st ‘𝑝)) = 𝑁} ∈ Fin)
Theorem	wlksnwwlknvbij 30047*	There is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length and starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 5-Aug-2022.)
⊢ ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((♯‘(1st ‘𝑝)) = 𝑁 ∧ ((2nd ‘𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
Theorem	wwlksnextproplem1 30048	Lemma 1 for wwlksnextprop 30051. (Contributed by Alexander van der Vekens, 31-Jul-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
Theorem	wwlksnextproplem2 30049	Lemma 2 for wwlksnextprop 30051. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ 𝑁 ∈ ℕ0) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
Theorem	wwlksnextproplem3 30050*	Lemma 3 for wwlksnextprop 30051. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ ((𝑊 ∈ 𝑋 ∧ (𝑊‘0) = 𝑃 ∧ 𝑁 ∈ ℕ0) → (𝑊 prefix (𝑁 + 1)) ∈ 𝑌)
Theorem	wwlksnextprop 30051*	Adding additional properties to the set of walks (as words) of a fixed length starting at a fixed vertex. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 29-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ (𝑁 ∈ ℕ0 → {𝑥 ∈ 𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
Theorem	disjxwwlkn 30052*	Sets of walks (as words) extended by an edge are disjunct if each set contains extensions of distinct walks. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ Disj 𝑦 ∈ 𝑌 {𝑥 ∈ 𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
Theorem	hashwwlksnext 30053*	Number of walks (as words) extended by an edge as a sum over the prefixes. (Contributed by Alexander van der Vekens, 21-Aug-2018.) (Revised by AV, 20-Apr-2021.) (Revised by AV, 26-Oct-2022.)
⊢ 𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    &   ⊢ 𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    ⇒   ⊢ ((Vtx‘𝐺) ∈ Fin → (♯‘{𝑥 ∈ 𝑋 ∣ ∃𝑦 ∈ 𝑌 ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}) = Σ𝑦 ∈ 𝑌 (♯‘{𝑥 ∈ 𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
Theorem	wwlksnwwlksnon 30054*	A walk of fixed length is a walk of fixed length between two vertices. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
Theorem	wspthsnwspthsnon 30055*	A simple path of fixed length is a simple path of fixed length between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-May-2021.) (Revised by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑎 ∈ 𝑉 ∃𝑏 ∈ 𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏))
Theorem	wspthsnonn0vne 30056	If the set of simple paths of length at least 1 between two vertices is not empty, the two vertices must be different. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.)
⊢ ((𝑁 ∈ ℕ ∧ (𝑋(𝑁 WSPathsNOn 𝐺)𝑌) ≠ ∅) → 𝑋 ≠ 𝑌)
Theorem	wspthsswwlkn 30057	The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length. (Contributed by AV, 18-May-2021.)
⊢ (𝑁 WSPathsN 𝐺) ⊆ (𝑁 WWalksN 𝐺)
Theorem	wspthnfi 30058	In a finite graph, the set of simple paths of a fixed length is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 18-May-2021.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝑁 WSPathsN 𝐺) ∈ Fin)
Theorem	wwlksnonfi 30059	In a finite graph, the set of walks of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ Fin)
Theorem	wspthsswwlknon 30060	The set of simple paths of a fixed length between two vertices is a subset of the set of walks of the fixed length between the two vertices. (Contributed by AV, 15-May-2021.)
⊢ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ⊆ (𝐴(𝑁 WWalksNOn 𝐺)𝐵)
Theorem	wspthnonfi 30061	In a finite graph, the set of simple paths of a fixed length between two vertices is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.) (Revised by AV, 15-May-2021.)
⊢ ((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∈ Fin)
Theorem	wspniunwspnon 30062*	The set of nonempty simple paths of fixed length is the double union of the simple paths of the fixed length between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝑁 ∈ ℕ ∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsN 𝐺) = ∪ 𝑥 ∈ 𝑉 ∪ 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))
Theorem	wspn0 30063	If there are no vertices, then there are no simple paths (of any length), too. (Contributed by Alexander van der Vekens, 11-Mar-2018.) (Revised by AV, 16-May-2021.) (Proof shortened by AV, 13-Mar-2022.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ (𝑉 = ∅ → (𝑁 WSPathsN 𝐺) = ∅)
17.3.8  Walks/paths of length 2 (as length 3 strings)
Theorem	2wlkdlem1 30064	Lemma 1 for 2wlkd 30075. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    ⇒   ⊢ (♯‘𝑃) = ((♯‘𝐹) + 1)
Theorem	2wlkdlem2 30065	Lemma 2 for 2wlkd 30075. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    ⇒   ⊢ (0..^(♯‘𝐹)) = {0, 1}
Theorem	2wlkdlem3 30066	Lemma 3 for 2wlkd 30075. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    ⇒   ⊢ (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))
Theorem	2wlkdlem4 30067*	Lemma 4 for 2wlkd 30075. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃‘𝑘) ∈ 𝑉)
Theorem	2wlkdlem5 30068*	Lemma 5 for 2wlkd 30075. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃‘𝑘) ≠ (𝑃‘(𝑘 + 1)))
Theorem	2pthdlem1 30069*	Lemma 1 for 2pthd 30079. (Contributed by AV, 14-Feb-2021.)
⊢ 𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   ⊢ 𝐹 = ⟨“𝐽𝐾”⟩    &   ⊢ (𝜑 → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉))    &   ⊢ (𝜑 → (𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘 ≠ 𝑗 → (𝑃‘𝑘) ≠ (𝑃‘𝑗)))
Theorem	2wlkdlem6 30070	Lemma 6 for 2wlkd 30075. (Contributed by AV, 23-Jan-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)))    ⇒   ⊢ (𝜑 → (𝐵 ∈ (𝐼‘𝐽) ∧ 𝐵 ∈ (𝐼‘𝐾)))
Theorem	2wlkdlem7 30071	Lemma 7 for 2wlkd 30075. (Contributed by AV, 14-Feb-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)))    ⇒   ⊢ (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))
Theorem	2wlkdlem8 30072	Lemma 8 for 2wlkd 30075. (Contributed by AV, 14-Feb-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)))    ⇒   ⊢ (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾))
Theorem	2wlkdlem9 30073	Lemma 9 for 2wlkd 30075. (Contributed by AV, 14-Feb-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)))    ⇒   ⊢ (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1))))
Theorem	2wlkdlem10 30074*	Lemma 10 for 3wlkd 30311. (Contributed by AV, 14-Feb-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)))    ⇒   ⊢ (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹‘𝑘)))
Theorem	2wlkd 30075	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 30076	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 30077	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 30078	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 30079	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 30080	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 30081	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 30082*	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 30083	Lemma for umgr2adedgwlk 30084, umgr2adedgspth 30087, 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 30084	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 30085	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 30086	Alternate proof for umgr2adedgwlkon 30085, using umgr2adedgwlk 30084, 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 30087	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 30088*	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 30089*	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 30090*	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 30091*	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 30092	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 30093	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 30094*	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	s3wwlks2on 30095*	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	sps3wwlks2on 30096*	A length 3 string which represents a walk of length 2 between two vertices. Concerns simple pseudographs, in contrast to s3wwlks2on 30095 and does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 28-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ∃𝑓(𝑓(Walks‘𝐺)⟨“𝐴𝐵𝐶”⟩ ∧ (♯‘𝑓) = 2)))
Theorem	usgrwwlks2on 30097	A walk of length 2 between two vertices as word in a simple graph. This theorem is analogous to umgrwwlks2on 30098 except it talks about simple graphs and therefore does not require the Axiom of Choice for its proof. (Contributed by Ender Ting, 29-Jan-2026.)
⊢ 𝑉 = (Vtx‘𝐺)    &   ⊢ 𝐸 = (Edg‘𝐺)    ⇒   ⊢ ((𝐺 ∈ USGraph ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (⟨“𝐴𝐵𝐶”⟩ ∈ (𝐴(2 WWalksNOn 𝐺)𝐶) ↔ ({𝐴, 𝐵} ∈ 𝐸 ∧ {𝐵, 𝐶} ∈ 𝐸)))
Theorem	umgrwwlks2on 30098	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 30099	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 30100*	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))))