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Theorem List for Metamath Proof Explorer - 26701-26800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcwwspthsn 26701 Extend class notation with simple paths (in a graph) of a fixed length as word over the set of vertices.
class WSPathsN

Syntaxcwwspthsnon 26702 Extend class notation with simple paths between two vertices (in a graph) of a fixed length as word over the set of vertices.
class WSPathsNOn

Definitiondf-wwlks 26703* Define the set of all walks (in an undirected graph) as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n-1) p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 26476. 𝑤 = ∅ has to be excluded because a walk always consists of at least one vertex, see wlkn0 26497. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalks = (𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))})

Definitiondf-wwlksn 26704* Define the set of all walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 26476. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})

Definitiondf-wwlksnon 26705* Define the collection of walks of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 15-Feb-2018.) (Revised by AV, 11-May-2021.)
WWalksNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤𝑛) = 𝑏)}))

Definitiondf-wspthsn 26706* Define the collection of simple paths of a fixed length as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ∃𝑓 𝑓(SPaths‘𝑔)𝑤})

Definitiondf-wspthsnon 26707* Define the collection of simple paths of a fixed length with particular endpoints as word over the set of vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 11-May-2021.)
WSPathsNOn = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))

Theoremwwlks 26708* The set of walks (in an undirected graph) as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (WWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈ (0..^((#‘𝑤) − 1)){(𝑤𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}

Theoremiswwlks 26709* A word over the set of vertices representing a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (WWalks‘𝐺) ↔ (𝑊 ≠ ∅ ∧ 𝑊 ∈ Word 𝑉 ∧ ∀𝑖 ∈ (0..^((#‘𝑊) − 1)){(𝑊𝑖), (𝑊‘(𝑖 + 1))} ∈ 𝐸))

Theoremwwlksn 26710* The set of walks (in an undirected graph) of a fixed length as words over the set of vertices. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑁 WWalksN 𝐺) = {𝑤 ∈ (WWalks‘𝐺) ∣ (#‘𝑤) = (𝑁 + 1)})

Theoremiswwlksn 26711 A word over the set of vertices representing a walk of a fixed length (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ (𝑊 ∈ (WWalks‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1))))

Theoremiswwlksnx 26712* 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 26713 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 26714 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 26715* 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))

Theoremwspthsn 26716* 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 26717* 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 26718* 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 26719* 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 26720* 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 26721* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤𝑁) = 𝐵)})

Theoremiswspthsnon 26722* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) = {𝑤 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∣ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑤})

Theoremwwlknon 26723 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.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝑁 WWalksN 𝐺) ∧ (𝑊‘0) = 𝐴 ∧ (𝑊𝑁) = 𝐵)))

Theoremwspthnon 26724* 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.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝑊 ∈ (𝐴(𝑁 WSPathsNOn 𝐺)𝐵) ↔ (𝑊 ∈ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∧ ∃𝑓 𝑓(𝐴(SPathsOn‘𝐺)𝐵)𝑊)))

Theoremwspthnonp 26725* Properties of a set being a simple path of a fixed length between two vertices as word. (Contributed by AV, 14-May-2021.)
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 26726 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 26727* 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 26728 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 26729* 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 26730 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 26731 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 26732 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)

Theoremwwlknbp2 26733 Other basic properties of a set being a walk of length n (represented by a word). (Contributed by Alexander van der Vekens, 3-Oct-2018.) (Revised by AV, 12-Apr-2021.)
(𝑊 ∈ (𝑁 WWalksN 𝐺) → (𝑊 ∈ Word (Vtx‘𝐺) ∧ (#‘𝑊) = (𝑁 + 1)))

Theoremwlkiswwlks1 26734 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 26735 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 26736* Lemma 1 for wlkiswwlks2 26742. (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 26737* Lemma 2 for wlkiswwlks2 26742. (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 26738* Lemma 3 for wlkiswwlks2 26742. (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 26739* Lemma 4 for wlkiswwlks2 26742. (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 26740* Lemma 5 for wlkiswwlks2 26742. (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 26741* Lemma 6 for wlkiswwlks2 26742. (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 26742* 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 26743* 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 26744* A walk as word corresponds to the sequence of vertices in a walk in a pseudograph. This variant of wlkiswwlks2 26742 does not require 𝐺 to be a simple pseudograph, but it requires the Axiom of Choice (ac6 9287) 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 26742). (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 26745* A walk as word corresponds to a walk in a pseudograph. This variant of wlkiswwlks 26743 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)))

Theoremwlkpwwlkf1ouspgr 26746* 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))

Theoremwlkisowwlkupgr 26747* The set of walks as words and the set of (ordinary) walks are isomorphic in a simple pseudograph. (Contributed by AV, 6-May-2021.)
(𝐺 ∈ USPGraph → ∃𝑓 𝑓:(Walks‘𝐺)–1-1-onto→(WWalks‘𝐺))

Theoremwwlksm1edg 26748 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.)
((𝑊 ∈ (WWalks‘𝐺) ∧ 2 ≤ (#‘𝑊)) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ∈ (WWalks‘𝐺))

Theoremwlklnwwlkln2lem 26749* Lemma for wlklnwwlkln2 26750 and wlklnwwlklnupgr2 26752. Formerly part of proof for wlklnwwlkln2 26750. (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 26750* 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 26751* 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 26752* 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 26750 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 26753* A walk of length 𝑁 as word corresponds to a walk with length 𝑁 in a pseudograph. This variant of wlkiswwlks 40197 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 26754 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))

Theoremwlknwwlksnfun 26755* Lemma 1 for wlknwwlksnbij2 26759. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ UPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlknwwlksninj 26756* Lemma 2 for wlknwwlksnbij2 26759. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlknwwlksnsur 26757* Lemma 3 for wlknwwlksnbij2 26759. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlknwwlksnbij 26758* Lemma 4 for wlknwwlksnbij2 26759. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}    &   𝑊 = (𝑁 WWalksN 𝐺)    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlknwwlksnbij2 26759* 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 in a simple pseudograph. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁}–1-1-onto→(𝑁 WWalksN 𝐺))

Theoremwlknwwlksnen 26760* 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, 15-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) → {𝑝 ∈ (Walks‘𝐺) ∣ (#‘(1st𝑝)) = 𝑁} ≈ (𝑁 WWalksN 𝐺))

Theoremwlknwwlksneqs 26761* 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)))

Theoremwlkwwlkfun 26762* Lemma 1 for wlkwwlkbij2 26766. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 15-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ UPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇𝑊)

Theoremwlkwwlkinj 26763* Lemma 2 for wlkwwlkbij2 26766. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Proof shortened by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1𝑊)

Theoremwlkwwlksur 26764* Lemma 3 for wlkwwlkbij2 26766. (Contributed by Alexander van der Vekens, 23-Jul-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇onto𝑊)

Theoremwlkwwlkbij 26765* Lemma 4 for wlkwwlkbij2 26766. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.)
𝑇 = {𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}    &   𝑊 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}    &   𝐹 = (𝑡𝑇 ↦ (2nd𝑡))       ((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → 𝐹:𝑇1-1-onto𝑊)

Theoremwlkwwlkbij2 26766* There is a bijection between the set of walks of a fixed length, starting at a fixed vertex, and the set of walks represented as words of the same length, starting at the same vertex. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 16-Apr-2021.)
((𝐺 ∈ USPGraph ∧ 𝑃𝑉𝑁 ∈ ℕ0) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑃)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃})

Theoremwwlkseq 26767* 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 26768 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.)
(𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ (𝑁 WWalksN 𝐺)))

Theoremwwlksnext 26769 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 26770 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.)
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 26771* 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.)
𝐸 = (Edg‘𝐺)       (𝑁 ∈ ℕ0 → (𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺) → ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))

Theoremwwlksnredwwlkn0 26772* 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.)
𝐸 = (Edg‘𝐺)       ((𝑁 ∈ ℕ0𝑊 ∈ ((𝑁 + 1) WWalksN 𝐺)) → ((𝑊‘0) = 𝑃 ↔ ∃𝑦 ∈ (𝑁 WWalksN 𝐺)((𝑊 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑊)} ∈ 𝐸)))

Theoremwwlksnextwrd 26773* Lemma for wwlksnextbij 26778. (Contributed by Alexander van der Vekens, 5-Aug-2018.) (Revised by AV, 18-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝐷 = {𝑤 ∈ Word 𝑉 ∣ ((#‘𝑤) = (𝑁 + 2) ∧ (𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}       (𝑊 ∈ (𝑁 WWalksN 𝐺) → 𝐷 = {𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)})

Theoremwwlksnextfun 26774* Lemma for wwlksnextbij 26778. (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷𝑅)

Theoremwwlksnextinj 26775* Lemma for wwlksnextbij 26778. (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 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}    &   𝑅 = {𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}    &   𝐹 = (𝑡𝐷 ↦ ( lastS ‘𝑡))       (𝑁 ∈ ℕ0𝐹:𝐷1-1𝑅)

Theoremwwlksnextsur 26776* Lemma for wwlksnextbij 26778. (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 substr ⟨0, (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 26777* Lemma for wwlksnextbij 26778. (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 substr ⟨0, (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 26778* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → ∃𝑓 𝑓:{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}–1-1-onto→{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸})

Theoremwwlksnexthasheq 26779* 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.) (Proof shortened by AV, 5-May-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       (𝑊 ∈ (𝑁 WWalksN 𝐺) → (#‘{𝑤 ∈ ((𝑁 + 1) WWalksN 𝐺) ∣ ((𝑤 substr ⟨0, (𝑁 + 1)⟩) = 𝑊 ∧ {( lastS ‘𝑊), ( lastS ‘𝑤)} ∈ 𝐸)}) = (#‘{𝑛𝑉 ∣ {( lastS ‘𝑊), 𝑛} ∈ 𝐸}))

Theoremdisjxwwlksn 26780* 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.)
𝑉 = (Vtx‘𝐺)    &   𝐸 = (Edg‘𝐺)       Disj 𝑦 ∈ (𝑁 WWalksN 𝐺){𝑥 ∈ Word 𝑉 ∣ ((𝑥 substr ⟨0, 𝑁⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}

Theoremwwlksnndef 26781 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 26782 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.)
((Vtx‘𝐺) ∈ Fin → (𝑁 WWalksN 𝐺) ∈ Fin)

Theoremwlksnfi 26783* 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 26784* 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, 30-Sep-2018.) (Revised by AV, 20-Apr-2021.)
((𝐺 ∈ USGraph ∧ 𝑁 ∈ ℕ0𝑋 ∈ (Vtx‘𝐺)) → ∃𝑓 𝑓:{𝑝 ∈ (Walks‘𝐺) ∣ ((#‘(1st𝑝)) = 𝑁 ∧ ((2nd𝑝)‘0) = 𝑋)}–1-1-onto→{𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑋})

Theoremwwlksnextproplem1 26785 Lemma 1 for wwlksnextprop 26788. (Contributed by Alexander van der Vekens, 31-Jul-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)       ((𝑊𝑋𝑁 ∈ ℕ0) → ((𝑊 substr ⟨0, (𝑁 + 1)⟩)‘0) = (𝑊‘0))

Theoremwwlksnextproplem2 26786 Lemma 2 for wwlksnextprop 26788. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝑊𝑋𝑁 ∈ ℕ0) → {( lastS ‘(𝑊 substr ⟨0, (𝑁 + 1)⟩)), ( lastS ‘𝑊)} ∈ 𝐸)

Theoremwwlksnextproplem3 26787* Lemma 3 for wwlksnextprop 26788. (Contributed by Alexander van der Vekens, 1-Aug-2018.) (Revised by AV, 20-Apr-2021.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       ((𝑊𝑋 ∧ (𝑊‘0) = 𝑃𝑁 ∈ ℕ0) → (𝑊 substr ⟨0, (𝑁 + 1)⟩) ∈ 𝑌)

Theoremwwlksnextprop 26788* 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.)
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 substr ⟨0, (𝑁 + 1)⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)})

Theoremdisjxwwlkn 26789* 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.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       Disj 𝑦𝑌 {𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}

Theoremhashwwlksnext 26790* 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.)
𝑋 = ((𝑁 + 1) WWalksN 𝐺)    &   𝐸 = (Edg‘𝐺)    &   𝑌 = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ (𝑤‘0) = 𝑃}       ((Vtx‘𝐺) ∈ Fin → (#‘{𝑥𝑋 ∣ ∃𝑦𝑌 ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}) = Σ𝑦𝑌 (#‘{𝑥𝑋 ∣ ((𝑥 substr ⟨0, 𝑀⟩) = 𝑦 ∧ (𝑦‘0) = 𝑃 ∧ {( lastS ‘𝑦), ( lastS ‘𝑥)} ∈ 𝐸)}))

Theoremwwlksnwwlksnon 26791* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑊 ∈ (𝑁 WWalksN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏)))

Theoremwspthsnwspthsnon 26792* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ0𝐺𝑈) → (𝑊 ∈ (𝑁 WSPathsN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉 𝑊 ∈ (𝑎(𝑁 WSPathsNOn 𝐺)𝑏)))

Theoremwwlksnon0 26793 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) = ∅)

Theoremwspthsnonn0vne 26794 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 26795 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 26796 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 26797 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.)
((Vtx‘𝐺) ∈ Fin → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) ∈ Fin)

Theoremwspthsswwlknon 26798 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 26799 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 26800* 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.)
𝑉 = (Vtx‘𝐺)       ((𝑁 ∈ ℕ ∧ 𝐺𝑈) → (𝑁 WSPathsN 𝐺) = 𝑥𝑉 𝑦 ∈ (𝑉 ∖ {𝑥})(𝑥(𝑁 WSPathsNOn 𝐺)𝑦))

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