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Theorem List for Metamath Proof Explorer - 28201-28300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremwwlksnprcl 28201 Derivation of the length of a word 𝑊 whose concatenation with a singleton word represents a walk of a fixed length 𝑁 (a partial reverse closure theorem). (Contributed by AV, 4-Mar-2022.)
((𝑊 ∈ Word 𝑉𝑁 ∈ ℕ0) → ((𝑊 ++ ⟨“𝑋”⟩) ∈ (𝑁 WWalksN 𝐺) → (♯‘𝑊) = 𝑁))
 
Theoremiswwlksnx 28202* Properties of a word to represent a walk of a fixed length, definition of WWalks expanded. (Contributed by AV, 28-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((♯‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸 ∧ (♯‘𝑊) = (𝑁 + 1))))
 
Theoremwwlkbp 28203 Basic properties of a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) → (𝐺 ∈ V ∧ 𝑊 ∈ Word 𝑉))
 
Theoremwwlknbp 28204 Basic properties of a walk of a fixed length (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 16-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 20-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝐺 ∈ V ∧ 𝑁 ∈ ℕ0𝑊 ∈ Word 𝑉))
 
Theoremwwlknp 28205* Properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 17-Jun-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word 𝑉 ∧ (♯‘𝑊) = (𝑁 + 1) ∧ ∀𝑖 ∈ (0..^𝑁){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))
 
Theoremwwlknbp1 28206 Other basic properties of a walk of a fixed length as word. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑁 ∈ ℕ0𝑊 ∈ Word (Vtx‘𝐺) ∧ (♯‘𝑊) = (𝑁 + 1)))
 
Theoremwwlknvtx 28207* The symbols of a word 𝑊 representing a walk of a fixed length 𝑁 are vertices. (Contributed by AV, 16-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → ∀𝑖 ∈ (0...𝑁)(𝑊𝑖) ∈ (Vtx‘𝐺))
 
Theoremwwlknllvtx 28208 If a word 𝑊 represents a walk of a fixed length 𝑁, then the first and the last symbol of the word is a vertex. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → ((𝑊‘0) ∈ 𝑉 ∧ (𝑊𝑁) ∈ 𝑉))
 
Theoremwwlknlsw 28209 If a word represents a walk of a fixed length, then the last symbol of the word is the endvertex of the walk. (Contributed by AV, 8-Mar-2022.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊𝑁) = (lastS‘𝑊))
 
Theoremwspthsn 28210* The set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑁 WSPathsN 𝐺) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ∃𝑓 𝑓(SPaths‘𝐺)𝑤}
 
Theoremiswspthn 28211* An element of the set of simple paths of a fixed length as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwspthnp 28212* Properties of a set being a simple path of a fixed length as word. (Contributed by AV, 18-May-2021.)
(𝑊 ∈ (𝑁 WSPathsN 𝐺) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ 𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ ∃𝑓 𝑓(SPaths‘𝐺)𝑊))
 
Theoremwwlksnon 28213* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑁) = 𝑏)}))
 
Theoremwspthsnon 28214* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎𝑉, 𝑏𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}))
 
Theoremiswwlksnon 28215* The set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)}
 
Theoremwwlksnon0 28216 Sufficient conditions for a set of walks of a fixed length between two vertices to be empty. (Contributed by AV, 15-May-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (¬ ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)
 
Theoremwwlksonvtx 28217 If a word 𝑊 represents a walk of length 2 on two classes 𝐴 and 𝐶, these classes are vertices. (Contributed by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐶) → (𝐴𝑉𝐶𝑉))
 
Theoremiswspthsnon 28218* The set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤}
 
Theoremwwlknon 28219 An element of the set of walks of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 14-Mar-2022.)
(𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵))
 
Theoremwspthnon 28220* An element of the set of simple paths of a fixed length between two vertices as word. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 12-May-2021.) (Revised by AV, 15-Mar-2022.)
(𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊))
 
Theoremwspthnonp 28221* Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.) (Proof shortened by AV, 15-Mar-2022.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) → ((𝑁 ∈ ℕ0𝐺 ∈ V) ∧ (𝐴𝑉𝐵𝑉) ∧ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))
 
Theoremwspthneq1eq2 28222 Two simple paths with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
((𝑃 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ∧ 𝑃 ∈ (𝐶(𝑁 WSPathsNOn 𝐺)𝐷)) → (𝐴 = 𝐶𝐵 = 𝐷))
 
Theoremwwlksn0s 28223* The set of all walks as words of length 0 is the set of all words of length 1 over the vertices. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(0 WWalksN 𝐺) = {𝑤 ∈ Word (Vtx‘𝐺) ∣ (♯‘𝑤) = 1}
 
Theoremwwlkssswrd 28224 Walks (represented by words) are words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 9-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (WWalks‘𝐺) ⊆ Word 𝑉
 
Theoremwwlksn0 28225* A walk of length 0 is represented by a singleton word. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.) (Proof shortened by AV, 21-May-2021.)
𝑉 = (Vtx‘𝐺)       (𝑊 ∈ (0 WWalksN 𝐺) → ∃𝑣𝑉 𝑊 = ⟨“𝑣”⟩)
 
Theorem0enwwlksnge1 28226 In graphs without edges, there are no walks of length greater than 0. (Contributed by Alexander van der Vekens, 26-Jul-2018.) (Revised by AV, 7-May-2021.)
(((Edg‘𝐺) = ∅ ∧ 𝑁 ∈ ℕ) → (𝑁 WWalksN 𝐺) = ∅)
 
Theoremwwlkswwlksn 28227 A walk of a fixed length as word is a walk (in an undirected graph) as word. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝑊 ∈ (WWalks‘𝐺))
 
Theoremwwlkssswwlksn 28228 The walks of a fixed length as words are walks (in an undirected graph) as words. (Contributed by Alexander van der Vekens, 17-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝑁 WWalksN 𝐺) ⊆ (WWalks‘𝐺)
 
Theoremwlkiswwlks1 28229 The sequence of vertices in a walk is a walk as word in a pseudograph. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 9-Apr-2021.)
(𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlklnwwlkln1 28230 The sequence of vertices in a walk of length 𝑁 is a walk as word of length 𝑁 in a pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → ((𝐹(Walks‘𝐺)𝑃 ∧ (♯‘𝐹) = 𝑁) → 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlkiswwlks2lem1 28231* Lemma 1 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (♯‘𝐹) = ((♯‘𝑃) − 1))
 
Theoremwlkiswwlks2lem2 28232* Lemma 2 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       (((♯‘𝑃) ∈ ℕ0𝐼 ∈ (0..^((♯‘𝑃) − 1))) → (𝐹𝐼) = (𝐸‘{(𝑃𝐼), (𝑃‘(𝐼 + 1))}))
 
Theoremwlkiswwlks2lem3 28233* Lemma 3 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 20-Jul-2018.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))       ((𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → 𝑃:(0...(♯‘𝐹))⟶𝑉)
 
Theoremwlkiswwlks2lem4 28234* Lemma 4 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 20-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))}))
 
Theoremwlkiswwlks2lem5 28235* Lemma 5 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸𝐹 ∈ Word dom 𝐸))
 
Theoremwlkiswwlks2lem6 28236* Lemma 6 for wlkiswwlks2 28237. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
𝐹 = (𝑥 ∈ (0..^((♯‘𝑃) − 1)) ↦ (𝐸‘{(𝑃𝑥), (𝑃‘(𝑥 + 1))}))    &   𝐸 = (iEdg‘𝐺)       ((𝐺 ∈ USPGraph ∧ 𝑃 ∈ Word 𝑉 ∧ 1 ≤ (♯‘𝑃)) → (∀𝑖 ∈ (0..^((♯‘𝑃) − 1)){(𝑃𝑖), (𝑃‘(𝑖 + 1))} ∈ ran 𝐸 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑖 ∈ (0..^(♯‘𝐹))(𝐸‘(𝐹𝑖)) = {(𝑃𝑖), (𝑃‘(𝑖 + 1))})))
 
Theoremwlkiswwlks2 28237* A walk as word corresponds to the sequence of vertices in a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
 
Theoremwlkiswwlks 28238* A walk as word corresponds to a walk in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlkiswwlksupgr2 28239* A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 28237 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 10234) 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 28237). (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))
 
Theoremwlkiswwlkupgr 28240* A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks 28238 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 10-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓 𝑓(Walks‘𝐺)𝑃𝑃 ∈ (WWalks‘𝐺)))
 
Theoremwlkswwlksf1o 28241* The mapping of (ordinary) walks to their sequences of vertices is a bijection in a simple pseudograph. (Contributed by AV, 6-May-2021.)
𝐹 = (𝑤 ∈ (Walks‘𝐺) ↦ (2nd𝑤))       (𝐺 ∈ USPGraph → 𝐹:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))
 
Theoremwlkswwlksen 28242 The set of walks as words and the set of (ordinary) walks are equinumerous in a simple pseudograph. (Contributed by AV, 6-May-2021.) (Revised by AV, 5-Aug-2022.)
(𝐺 ∈ USPGraph → (Walks‘𝐺) ≈ (WWalks‘𝐺))
 
Theoremwwlksm1edg 28243 Removing the trailing edge from a walk (as word) with at least one edge results in a walk. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 19-Apr-2021.) (Revised by AV, 26-Oct-2022.)
((𝑊 ∈ (WWalks‘𝐺) ∧ 2 ≤ (♯‘𝑊)) → (𝑊 prefix ((♯‘𝑊) − 1)) ∈ (WWalks‘𝐺))
 
Theoremwlklnwwlkln2lem 28244* Lemma for wlklnwwlkln2 28245 and wlklnwwlklnupgr2 28247. Formerly part of proof for wlklnwwlkln2 28245. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝜑 → (𝑃 ∈ (WWalks‘𝐺) → ∃𝑓 𝑓(Walks‘𝐺)𝑃))       (𝜑 → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlkln2 28245* A walk of length 𝑁 as word corresponds to the sequence of vertices in a walk of length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlkn 28246* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a simple pseudograph. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ USPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlklnwwlklnupgr2 28247* 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 28245 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (𝑃 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁)))
 
Theoremwlklnwwlknupgr 28248* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of wlkiswwlks 28238 does not require 𝐺 to be a simple pseudograph, but it requires (indirectly) the Axiom of Choice for its proof. (Contributed by Alexander van der Vekens, 21-Jul-2018.) (Revised by AV, 12-Apr-2021.)
(𝐺 ∈ UPGraph → (∃𝑓(𝑓(Walks‘𝐺)𝑃 ∧ (♯‘𝑓) = 𝑁) ↔ 𝑃 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwlknewwlksn 28249 If a walk in a pseudograph has length 𝑁, then the sequence of the vertices of the walk is a word representing the walk as word of length 𝑁. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 11-Apr-2021.)
(((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (Walks‘𝐺)) ∧ (𝑁 ∈ ℕ0 ∧ (♯‘(1st𝑊)) = 𝑁)) → (2nd𝑊) ∈ (𝑁 WWalksN 𝐺))
 
Theoremwlknwwlksnbij 28250* The mapping (𝑡𝑇 ↦ (2nd𝑡)) is a bijection between the set of walks of a fixed length and the set of walks represented by words of the same length in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)
 
Theoremwlknwwlksnen 28251* In a simple pseudograph, the set of walks of a fixed length and the set of walks represented by words are equinumerous. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 5-Aug-2022.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))
 
Theoremwlknwwlksneqs 28252* The set of walks of a fixed length and the set of walks represented by words have the same size. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → (♯‘{𝑝 ∈ (Walks‘𝐺) ∣ (♯‘(1st𝑝)) = 𝑁}) = (♯‘(𝑁 WWalksN 𝐺)))
 
Theoremwwlkseq 28253* Equality of two walks (as words). (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
((𝑊 ∈ (WWalks‘𝐺) ∧ 𝑇 ∈ (WWalks‘𝐺)) → (𝑊 = 𝑇 ↔ ((♯‘𝑊) = (♯‘𝑇) ∧ ∀𝑖 ∈ (0..^(♯‘𝑊))(𝑊𝑖) = (𝑇𝑖))))
 
Theoremwwlksnred 28254 Reduction of a walk (as word) by removing the trailing edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Revised by AV, 26-Oct-2022.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 prefix (𝑁 + 1)) ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwwlksnext 28255 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 4-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑇 ∈ (𝑁 WWalksN 𝐺) ∧ 𝑆𝑉 ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸) → (𝑇 ++ ⟨“𝑆”⟩) ∈ ((𝑁 + 1) WWalksN 𝐺))
 
Theoremwwlksnextbi 28256 Extension of a walk (as word) by adding an edge/vertex. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 16-Apr-2021.) (Proof shortened by AV, 27-Oct-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (((𝑁 ∈ ℕ0𝑆𝑉) ∧ (𝑇 ∈ Word 𝑉𝑊 = (𝑇 ++ ⟨“𝑆”⟩) ∧ {(lastS‘𝑇), 𝑆} ∈ 𝐸)) → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) ↔ 𝑇 ∈ (𝑁 WWalksN 𝐺)))
 
Theoremwwlksnredwwlkn 28257* 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)))
 
Theoremwwlksnredwwlkn0 28258* 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 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑊)} ∈ 𝐸)))
 
Theoremwwlksnextwrd 28259* Lemma for wwlksnextbij 28264. (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)})
 
Theoremwwlksnextfun 28260* Lemma for wwlksnextbij 28264. (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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷𝑅)
 
Theoremwwlksnextinj 28261* Lemma for wwlksnextbij 28264. (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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)
 
Theoremwwlksnextsurj 28262* Lemma for wwlksnextbij 28264. (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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))       (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷onto𝑅)
 
Theoremwwlksnextbij0 28263* Lemma for wwlksnextbij 28264. (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 = {𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ (lastS‘𝑡))       (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐹:𝐷1-1-onto𝑅)
 
Theoremwwlksnextbij 28264* 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→{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸})
 
Theoremwwlksnexthasheq 28265* 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)}) = (♯‘{𝑛𝑉 ∣ {(lastS‘𝑊), 𝑛} ∈ 𝐸}))
 
Theoremdisjxwwlksn 28266* 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)}
 
Theoremwwlksnndef 28267 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) = ∅)
 
Theoremwwlksnfi 28268 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)
 
Theoremwlksnfi 28269* 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) ∣ (♯‘(1st𝑝)) = 𝑁} ∈ Fin)
 
Theoremwlksnwwlknvbij 28270* 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) ∣ ((♯‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})
 
Theoremwwlksnextproplem1 28271 Lemma 1 for wwlksnextprop 28274. (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)       ((𝑊𝑋𝑁 ∈ ℕ0) → ((𝑊 prefix (𝑁 + 1))‘0) = (𝑊‘0))
 
Theoremwwlksnextproplem2 28272 Lemma 2 for wwlksnextprop 28274. (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)       ((𝑊𝑋𝑁 ∈ ℕ0) → {(lastS‘(𝑊 prefix (𝑁 + 1))), (lastS‘𝑊)} ∈ 𝐸)
 
Theoremwwlksnextproplem3 28273* Lemma 3 for wwlksnextprop 28274. (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}       ((𝑊𝑋 ∧ (𝑊‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑊 prefix (𝑁 + 1)) ∈ 𝑌)
 
Theoremwwlksnextprop 28274* 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 → {𝑥𝑋 ∣ (𝑥‘0) = 𝑃} = {𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix (𝑁 + 1)) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)})
 
Theoremdisjxwwlkn 28275* 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 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}
 
Theoremhashwwlksnext 28276* 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 → (♯‘{𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}) = Σ𝑦𝑌 (♯‘{𝑥𝑋 ∣ ((𝑥 prefix 𝑀) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {(lastS‘𝑦), (lastS‘𝑥)} ∈ 𝐸)}))
 
Theoremwwlksnwwlksnon 28277* 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) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏))
 
Theoremwspthsnwspthsnon 28278* 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) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏))
 
Theoremwspthsnonn0vne 28279 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) ≠ ∅) → 𝑋𝑌)
 
Theoremwspthsswwlkn 28280 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)
 
Theoremwspthnfi 28281 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)
 
Theoremwwlksnonfi 28282 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)
 
Theoremwspthsswwlknon 28283 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)
 
Theoremwspthnonfi 28284 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)
 
Theoremwspniunwspnon 28285* 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 ∧ 𝐺𝑈) → (𝑁 WSPathsN 𝐺) = 𝑥𝑉 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))
 
Theoremwspn0 28286 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) = ∅)
 
16.3.8  Walks/paths of length 2 (as length 3 strings)
 
Theorem2wlkdlem1 28287 Lemma 1 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (♯‘𝑃) = ((♯‘𝐹) + 1)
 
Theorem2wlkdlem2 28288 Lemma 2 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩       (0..^(♯‘𝐹)) = {0, 1}
 
Theorem2wlkdlem3 28289 Lemma 3 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ((𝑃‘0) = 𝐴 ∧ (𝑃‘1) = 𝐵 ∧ (𝑃‘2) = 𝐶))
 
Theorem2wlkdlem4 28290* Lemma 4 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))       (𝜑 → ∀𝑘 ∈ (0...(♯‘𝐹))(𝑃𝑘) ∈ 𝑉)
 
Theorem2wlkdlem5 28291* Lemma 5 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))
 
Theorem2pthdlem1 28292* Lemma 1 for 2pthd 28302. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝑃))∀𝑗 ∈ (1..^(♯‘𝐹))(𝑘𝑗 → (𝑃𝑘) ≠ (𝑃𝑗)))
 
Theorem2wlkdlem6 28293 Lemma 6 for 2wlkd 28298. (Contributed by AV, 23-Jan-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → (𝐵 ∈ (𝐼𝐽) ∧ 𝐵 ∈ (𝐼𝐾)))
 
Theorem2wlkdlem7 28294 Lemma 7 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → (𝐽 ∈ V ∧ 𝐾 ∈ V))
 
Theorem2wlkdlem8 28295 Lemma 8 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ((𝐹‘0) = 𝐽 ∧ (𝐹‘1) = 𝐾))
 
Theorem2wlkdlem9 28296 Lemma 9 for 2wlkd 28298. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼‘(𝐹‘0)) ∧ {𝐵, 𝐶} ⊆ (𝐼‘(𝐹‘1))))
 
Theorem2wlkdlem10 28297* Lemma 10 for 3wlkd 28531. (Contributed by AV, 14-Feb-2021.)
𝑃 = ⟨“𝐴𝐵𝐶”⟩    &   𝐹 = ⟨“𝐽𝐾”⟩    &   (𝜑 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))       (𝜑 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))
 
Theorem2wlkd 28298 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(Walks‘𝐺)𝑃)
 
Theorem2wlkond 28299 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝜑𝐹(𝐴(WalksOn‘𝐺)𝐶)𝑃)
 
Theorem2trld 28300 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 → (𝐴𝑉𝐵𝑉𝐶𝑉))    &   (𝜑 → (𝐴𝐵𝐵𝐶))    &   (𝜑 → ({𝐴, 𝐵} ⊆ (𝐼𝐽) ∧ {𝐵, 𝐶} ⊆ (𝐼𝐾)))    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐽𝐾)       (𝜑𝐹(Trails‘𝐺)𝑃)
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