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Theorem List for Metamath Proof Explorer - 40901-41000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxc1wlks 40901 Extend class notation with 1-walks (of a hypergraph). TODO-AV: should be renamed to Walks after the current definition of Walks becomes obsolete.
class 1Walks

Syntaxcwlks 40902 Extend class notation with walks (of a pseudograph).
class UPWalks

Syntaxcwlkson 40903 Extend class notation with walks between two vertices (within a graph).
class WalksOn

Definitiondf-ewlks 40904* Define the set of all s-walks of edges (in a hypergraph) corresponding to s-walks "on the edge level" discussed in Aksoy et al. For an extended nonnegative integer s, an s-walk is a sequence of hyperedges, e(0), e(1), ... , e(k), where for j=1, ... , k, e(j-1) and e(j) have at least s vertices in common. In contrast to the definition in Aksoy et al., 𝑠 = 0 (a 0-walk is an arbitrary sequence of hyperedges) and 𝑠 = +∞ (then the number of common vertices of two adjacent hyperedges must be infinite) are allowed. Furthermore, it is not forbidden that adjacent hyperedges are equal. (Contributed by AV, 4-Jan-2021.)
EdgWalks = (𝑔 ∈ V, 𝑠 ∈ ℕ0* ↦ {𝑓[(iEdg‘𝑔) / 𝑖](𝑓 ∈ Word dom 𝑖 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑠 ≤ (#‘((𝑖‘(𝑓‘(𝑘 − 1))) ∩ (𝑖‘(𝑓𝑘)))))})

Definitiondf-1wlks 40905* Define the set of all 1-walks (in a hypergraph). Such 1-walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see 1wlk1walk 40948) discussed in Aksoy et al. The predicate 𝐹(1Walks‘𝐺)𝑃 can be read as "The pair 𝐹, 𝑃 represents a walk in a graph 𝐺", see also is1wlk 40918.

The condition {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝𝑘).

According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by AV, 30-Dec-2020.)

1Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})

Definitiondf-wlks 40906* Define the set of all walks (in a pseudograph). TODO-AV: This corresponds to the definition of Walks, but can be removed and the defining theorem upgriswlk 40954 could be used instead.

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n).

Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudograhs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

UPWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Definitiondf-wlkson 40907* Define the collection of walks with particular endpoints (in a hypergraph). The predicate 𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 can be read as "The pair 𝐹, 𝑃 represents a walk from vertex 𝐴 to vertex 𝐵 in a graph 𝐺", see also iswlkOn 40970. This corresponds to the "x0-x(l)-walks", see Definition in [Bollobas] p. 5. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
WalksOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))

Theoremewlksfval 40908* The set of s-walks of edges (in a hypergraph). (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝑆 ∈ ℕ0*) → (𝐺 EdgWalks 𝑆) = {𝑓 ∣ (𝑓 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝑓))𝑆 ≤ (#‘((𝐼‘(𝑓‘(𝑘 − 1))) ∩ (𝐼‘(𝑓𝑘)))))})

Theoremisewlk 40909* Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝑆 ∈ ℕ0*𝐹𝑈) → (𝐹 ∈ (𝐺 EdgWalks 𝑆) ↔ (𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹𝑘)))))))

Theoremewlkprop 40910* Properties of an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹 ∈ (𝐺 EdgWalks 𝑆) → ((𝐺 ∈ V ∧ 𝑆 ∈ ℕ0*) ∧ 𝐹 ∈ Word dom 𝐼 ∧ ∀𝑘 ∈ (1..^(#‘𝐹))𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹𝑘))))))

Theoremewlkinedg 40911 The intersection (common vertices) of two adjacent edges in an s-walk of edges. (Contributed by AV, 4-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝐾 ∈ (1..^(#‘𝐹))) → 𝑆 ≤ (#‘((𝐼‘(𝐹‘(𝐾 − 1))) ∩ (𝐼‘(𝐹𝐾)))))

Theoremewlkle 40912 An s-walk of edges is also a t-walk of edges if t <_ s. (Contributed by AV, 4-Jan-2021.)
((𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 𝑇 ∈ ℕ0*𝑇𝑆) → 𝐹 ∈ (𝐺 EdgWalks 𝑇))

Theoremupgrewlkle2 40913 In a pseudograph, there is no s-walk of edges of length greater than 1 with s>2. (Contributed by AV, 4-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹 ∈ (𝐺 EdgWalks 𝑆) ∧ 1 < (#‘𝐹)) → 𝑆 ≤ 2)

Theorem1wlkslem1 40914 Lemma 1 for 1-walks to substitute the index of the condition for vertices and edges in a 1-walk. (Contributed by AV, 23-Apr-2021.)
(𝐴 = 𝐵 → (if-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃‘(𝐵 + 1)), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃‘(𝐵 + 1))} ⊆ (𝐼‘(𝐹𝐵)))))

Theorem1wlkslem2 40915 Lemma 2 for 1-walks to substitute the index of the condition for vertices and edges in a 1-walk. (Contributed by AV, 23-Apr-2021.)
((𝐴 = 𝐵 ∧ (𝐴 + 1) = 𝐶) → (if-((𝑃𝐴) = (𝑃‘(𝐴 + 1)), (𝐼‘(𝐹𝐴)) = {(𝑃𝐴)}, {(𝑃𝐴), (𝑃‘(𝐴 + 1))} ⊆ (𝐼‘(𝐹𝐴))) ↔ if-((𝑃𝐵) = (𝑃𝐶), (𝐼‘(𝐹𝐵)) = {(𝑃𝐵)}, {(𝑃𝐵), (𝑃𝐶)} ⊆ (𝐼‘(𝐹𝐵)))))

Theorem1wlksfval 40916* The set of 1-walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (1Walks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), (𝐼‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ (𝐼‘(𝑓𝑘))))})

Theoremwlksfval 40917* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.) (Revised by AV, 28-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (UPWalks‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐼𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremis1wlk 40918* Properties of a pair of functions to be a 1-walk. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))

TheoremisWlk 40919* Properties of a pair of functions to be a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(UPWalks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremwlkv 40920 The classes involved in a 1-walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.)
(𝐹(1Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))

Theoremis1wlkg 40921* Generalisation of is1wlk 40918: Conditions for two classes to represent a 1-walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))

TheoremwlkbProp 40922 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 29-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐹(UPWalks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))

Theorem2m1wlk 40923 The two mappings determining a 1-walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉))

Theorem1wlkf 40924 The mapping enumerating the (indices of the) edges of a 1-walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)

Theorem1wlkcl 40925 A 1-walk has length #(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 41483. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
(𝐹(1Walks‘𝐺)𝑃 → (#‘𝐹) ∈ ℕ0)

Theorem1wlkp 40926 The mapping enumerating the vertices of a 1-walk is a function. (Contributed by AV, 5-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(1Walks‘𝐺)𝑃𝑃:(0...(#‘𝐹))⟶𝑉)

Theorem1wlkpwrd 40927 The sequence of vertices of a 1-walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(1Walks‘𝐺)𝑃𝑃 ∈ Word 𝑉)

Theorem1wlklenvp1 40928 The number of vertices of a walk (in an undirected graph) is the number of its edges plus 1. (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 1-May-2021.)
(𝐹(1Walks‘𝐺)𝑃 → (#‘𝑃) = ((#‘𝐹) + 1))

Theorem1wlksv 40929* The class of 1-walks is a set. (Contributed by AV, 15-Jan-2021.)
{⟨𝑓, 𝑝⟩ ∣ 𝑓(1Walks‘𝐺)𝑝} ∈ V

Theorem1wlkn0 40930 The sequence of vertices of a walk cannot be empty, i.e. a walk always consists of at least one vertex. (Contributed by Alexander van der Vekens, 19-Jul-2018.) (Revised by AV, 2-Jan-2021.)
(𝐹(1Walks‘𝐺)𝑃𝑃 ≠ ∅)

Theorem1wlklenvm1 40931 The number of edges of a walk is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.) (Revised by AV, 2-Jan-2021.)
(𝐹(1Walks‘𝐺)𝑃 → (#‘𝐹) = ((#‘𝑃) − 1))

Theorem1wlkvtxeledglem 40932 Lemma for 1wlkvtxeledg 40933: Two adjacent vertices in a 1-walk are incident with an edge. (Contributed by AV, 4-Apr-2021.)
(if-((𝑃𝐾) = (𝑃‘(𝐾 + 1)), (𝐼‘(𝐹𝐾)) = {(𝑃𝐾)}, {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾))) → {(𝑃𝐾), (𝑃‘(𝐾 + 1))} ⊆ (𝐼‘(𝐹𝐾)))

Theorem1wlkvtxeledg 40933* Each pair of adjacent vertices in a 1-walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))

Theorem1wlkvtxiedg 40934* The vertices of a walk are connected by indexed edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(#‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)

Theoremrel1wlk 40935 The set (1Walks‘𝐺) of all 1-walks on 𝐺 is a set of pairs by our definition of a 1-walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Inspired by releupa 26229 contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 19-Feb-2021.)
Rel (1Walks‘𝐺)

Theorem1wlkvv 40936 If there is at least one walk in the graph, all walks are in the universal class of ordered pairs. (Contributed by AV, 2-Jan-2021.)
((1st𝑊)(1Walks‘𝐺)(2nd𝑊) → 𝑊 ∈ (V × V))

Theorem1wlkop 40937 A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)
(𝑊 ∈ (1Walks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)

Theorem1wlkcpr 40938 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
(𝑊 ∈ (1Walks‘𝐺) ↔ (1st𝑊)(1Walks‘𝐺)(2nd𝑊))

Theorem1wlk2f 40939* If there is a 1-walk 𝑊 there is a pair of functions representing this 1-walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑊 ∈ (1Walks‘𝐺) → ∃𝑓𝑝 𝑓(1Walks‘𝐺)𝑝)

Theorem1wlkcomp 40940* A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝐺𝑈𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (1Walks‘𝐺) ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))

Theorem1wlkcompim 40941* Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (1Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))))

Theorem1wlkelwrd 40942 The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 2-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (1Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉))

Theorem1wlkeq 40943* Conditions for two walks (within the same graph) being the same. (Contributed by AV, 1-Jul-2018.) (Revised by AV, 16-May-2019.) (Revised by AV, 14-Apr-2021.)
((𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))

Theoremedginwlk 40944 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺𝑊 ∧ Fun 𝐼𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(#‘𝐹)) → (𝐼‘(𝐹𝐾)) ∈ 𝐸))

Theoremupgredginwlk 40945 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹 ∈ Word dom 𝐼) → (𝐾 ∈ (0..^(#‘𝐹)) → (𝐼‘(𝐹𝐾)) ∈ 𝐸))

Theoremiedginwlk 40946 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 23-Apr-2021.)
𝐼 = (iEdg‘𝐺)       ((Fun 𝐼𝐹(1Walks‘𝐺)𝑃𝑋 ∈ (0..^(#‘𝐹))) → (𝐼‘(𝐹𝑋)) ∈ ran 𝐼)

Theorem1wlkl1loop 40947 A 1-walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
(((Fun (iEdg‘𝐺) ∧ 𝐹(1Walks‘𝐺)𝑃) ∧ ((#‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

Theorem1wlk1walk 40948* A 1-walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.)
𝐼 = (iEdg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(#‘𝐹))1 ≤ (#‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹𝑘)))))

Theorem1wlk1ewlk 40949 A 1-walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.)
(𝐹(1Walks‘𝐺)𝑃𝐹 ∈ (𝐺 EdgWalks 1))

Theoremifpprsnss 40950 An unordered pair is a singleton or a subset of itself. This theorem is helpful to convert theorems about walks in arbitrary graphs into theorems about walks in pseudographs. (Contributed by AV, 27-Feb-2021.)
(𝑃 = {𝐴, 𝐵} → if-(𝐴 = 𝐵, 𝑃 = {𝐴}, {𝐴, 𝐵} ⊆ 𝑃))

Theoremwlk1wlk 40951 A walk is a 1-walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 27-Feb-2021.)
(𝐹(UPWalks‘𝐺)𝑃𝐹(1Walks‘𝐺)𝑃)

Theoremupgr1wlkwlk 40952 In a pseudograph, a 1-walk is a walk. (Contributed by AV, 30-Dec-2020.) (Proof shortened by AV, 2-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → 𝐹(UPWalks‘𝐺)𝑃)

Theoremupgr1wlkwlkb 40953 In a pseudograph, the definitions for a 1-walk and a walk are equivalent. (Contributed by AV, 30-Dec-2020.)
(𝐺 ∈ UPGraph → (𝐹(1Walks‘𝐺)𝑃𝐹(UPWalks‘𝐺)𝑃))

Theoremupgriswlk 40954* Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹𝑈𝑃𝑍) → (𝐹(1Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremupgrwlkedg 40955* The edges of a walk in a pseudograph join exactly the two corresponding adjacent vertices in the walk. (Contributed by AV, 27-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})

Theoremupgr1wlkcompim 40956* Implications for the properties of the components of a walk in a pseudograph. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 14-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝐺 ∈ UPGraph ∧ 𝑊 ∈ (1Walks‘𝐺)) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))

Theorem1wlkvtxedg 40957* The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
𝐸 = (Edg‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(#‘𝐹))∃𝑒𝐸 {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)

Theoremupgr1wlkvtxedg 40958* The pairs of connected vertices of a walk are edges in a pseudograph. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
𝐸 = (Edg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸)

Theoremuspgr2wlkeq 40959* Conditions for two walks within the same simple pseudograph being the same. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 3-Jul-2018.) (Revised by AV, 14-Apr-2021.)
((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ 𝑁 = (#‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (#‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))

Theoremuspgr2wlkeq2 40960 Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by Alexander van der Vekens, 25-Aug-2018.) (Revised by AV, 14-Apr-2021.)
(((𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ0) ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (1Walks‘𝐺) ∧ (#‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))

Theoremuspgr2wlkeqi 40961 Conditions for two walks within the same simple pseudograph to be identical. It is sufficient that the vertices (in the same order) are identical. (Contributed by AV, 6-May-2021.)
((𝐺 ∈ USPGraph ∧ (𝐴 ∈ (1Walks‘𝐺) ∧ 𝐵 ∈ (1Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)

Theoremumgr1wlknloop 40962* In a multigraph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.) (Revised by AV, 3-Jan-2021.)
((𝐺 ∈ UMGraph ∧ 𝐹(1Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

TheoremwlkRes 40963* Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval 41032) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 30-Dec-2020.) (Proof shortened by AV, 15-Jan-2021.)
(𝑓(𝑊𝐺)𝑝𝑓(1Walks‘𝐺)𝑝)       {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊𝐺)𝑝𝜑)} ∈ V

Theorem1wlkv0 40964 If there is a walk in the null graph (a class without vertices), it would be the pair consisting of empty sets. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
(((Vtx‘𝐺) = ∅ ∧ 𝑊 ∈ (1Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Theoremg01wlk0 40965 There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
((Vtx‘𝐺) = ∅ → (1Walks‘𝐺) = ∅)

Theorem01wlk0 40966 There is no walk for the empty set, i.e. in a null graph. (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
(1Walks‘∅) = ∅

Theorem1wlk0prc 40967 There is no walk in a null graph (a class without vertices). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
((𝑆 ∉ V ∧ (Vtx‘𝑆) = (Vtx‘𝐺)) → (1Walks‘𝐺) = ∅)

Theorem1wlklenvclwlk 40968 The number of vertices in a walk equals the length of the walk after it is "closed" (i.e. enhanced by an edge from its last vertex to its first vertex). (Contributed by Alexander van der Vekens, 29-Jun-2018.) (Revised by AV, 2-May-2021.)
((𝑊 ∈ Word (Vtx‘𝐺) ∧ 1 ≤ (#‘𝑊)) → (⟨𝐹, (𝑊 ++ ⟨“(𝑊‘0)”⟩)⟩ ∈ (1Walks‘𝐺) → (#‘𝐹) = (#‘𝑊)))

Theoremwlkson 40969* The set of walks between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 30-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(WalksOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(1Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})

TheoremiswlkOn 40970 Properties of a pair of functions to be a walk between two given vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 31-Dec-2020.) (Revised by AV, 22-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ↔ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

TheoremwlkOnprop 40971 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 31-Dec-2020.) (Proof shortened by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(1Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

Theorem1wlkpvtx 40972 A 1-walk connects vertices. (Contributed by AV, 22-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(#‘𝐹)) → (𝑃𝑁) ∈ 𝑉))

Theorem1wlkepvtx 40973 The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(1Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(#‘𝐹)) ∈ 𝑉))

TheoremwlkOniswlk 40974 A walk between two vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 2-Jan-2021.)
(𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(1Walks‘𝐺)𝑃)

TheoremwlkOnwlk 40975 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.) (Revised by AV, 2-Jan-2021.) (Proof shortened by AV, 31-Jan-2021.)
(𝐹(1Walks‘𝐺)𝑃𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃)

TheoremwlkOnwlk1l 40976 A walk is a walk from its first vertex to its last vertex. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 22-Mar-2021.)
(𝜑𝐹(1Walks‘𝐺)𝑃)       (𝜑𝐹((𝑃‘0)(WalksOn‘𝐺)( lastS ‘𝑃))𝑃)

Theoremwlksoneq1eq2 40977 Two walks with identical sequences of vertices start and end at the same vertices. (Contributed by AV, 14-May-2021.)
((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐻(𝐶(WalksOn‘𝐺)𝐷)𝑃) → (𝐴 = 𝐶𝐵 = 𝐷))

TheoremwlkOnl1iedg 40978* If there is a walk between two vertices 𝐴 and 𝐵 at least of length 1, then the start vertex 𝐴 is incident with an edge. (Contributed by AV, 4-Apr-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃 ∧ (#‘𝐹) ≠ 0) → ∃𝑒 ∈ ran 𝐼 𝐴𝑒)

TheoremwlkOn2n0 40979 The length of a walk between two different vertices is not 0 (i.e. is at least 1). (Contributed by AV, 3-Apr-2021.)
((𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐴𝐵) → (#‘𝐹) ≠ 0)

Theorem2Wlklem 40980* Lemma for upgr2wlk 40981 and 2wlklemA 25822. Identical with is2wlk 25833. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
(∀𝑘 ∈ {0, 1} (𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ↔ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))

Theoremupgr2wlk 40981 Properties of a pair of functions to be a walk of length 2 in a pseudograph. Note that the vertices need not to be distinct and the edges can be loops or multiedges. (Contributed by Alexander van der Vekens, 16-Feb-2018.) (Revised by AV, 3-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(1Walks‘𝐺)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐼𝑃:(0...2)⟶𝑉 ∧ ((𝐼‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐼‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

Theorem1wlkreslem0 40982 Lemma for 1wlkres 40984. TODO-AV: Will become obsolete if 𝐻 = (𝐹 ↾ (0..^𝑁)) is replaced by 𝐻 = (𝐹 substr ⟨0, 𝑁⟩) or 𝐻 = (𝐹 prefix 𝑁) in 1wlkres 40984 and trlres 41013. (Contributed by AV, 5-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)

Theorem1wlkreslem 40983 Lemma for 1wlkres 40984. (Contributed by AV, 5-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))

Theorem1wlkres 40984 The restriction 𝐻, 𝑄 of a 1-walk 𝐹, 𝑃 to an initial segment of the 1-walk (of length 𝑁) forms a 1-walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 41488. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(1Walks‘𝑆)𝑄)

Theoremred1wlklem 40985 Lemma for red1wlk 40986. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
((𝐹 ∈ Word 𝑆 ∧ 1 ≤ (#‘𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉) → (𝑃 ↾ (0..^(#‘𝐹))):(0...(#‘(𝐹 ↾ (0..^((#‘𝐹) − 1)))))⟶𝑉)

Theoremred1wlk 40986 A 1-walk ending at the last but one vertex of the walk is a 1-walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
((𝐹(1Walks‘𝐺)𝑃 ∧ 1 ≤ (#‘𝐹)) → (𝐹 ↾ (0..^((#‘𝐹) − 1)))(1Walks‘𝐺)(𝑃 ↾ (0..^(#‘𝐹))))

Theorem1wlkp1lem1 40987 Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)       (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)

Theorem1wlkp1lem2 40988 Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})       (𝜑 → (#‘𝐻) = (𝑁 + 1))

Theorem1wlkp1lem3 40989 Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})       (𝜑 → ((iEdg‘𝑆)‘(𝐻𝑁)) = ((𝐼 ∪ {⟨𝐵, 𝐸⟩})‘𝐵))

Theorem1wlkp1lem4 40990 Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)       (𝜑 → (𝑆 ∈ V ∧ 𝐻 ∈ V ∧ 𝑄 ∈ V))

Theorem1wlkp1lem5 40991* Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)       (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑄𝑘) = (𝑃𝑘))

Theorem1wlkp1lem6 40992* Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)       (𝜑 → ∀𝑘 ∈ (0..^𝑁)((𝑄𝑘) = (𝑃𝑘) ∧ (𝑄‘(𝑘 + 1)) = (𝑃‘(𝑘 + 1)) ∧ ((iEdg‘𝑆)‘(𝐻𝑘)) = (𝐼‘(𝐹𝑘))))

Theorem1wlkp1lem7 40993 Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)       (𝜑 → {(𝑄𝑁), (𝑄‘(𝑁 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑁)))

Theorem1wlkp1lem8 40994* Lemma for 1wlkp1 40995. (Contributed by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐻))if-((𝑄𝑘) = (𝑄‘(𝑘 + 1)), ((iEdg‘𝑆)‘(𝐻𝑘)) = {(𝑄𝑘)}, {(𝑄𝑘), (𝑄‘(𝑘 + 1))} ⊆ ((iEdg‘𝑆)‘(𝐻𝑘))))

Theorem1wlkp1 40995 Append one path segment (edge) 𝐸 from vertex (𝑃𝑁) to a vertex 𝐶 to a 1-walk 𝐹, 𝑃 to become a 1-walk 𝐻, 𝑄 of the supergraph 𝑆 obtained by adding the new edge to the graph 𝐺. Formerly proven directly for Eulerian paths (for pseudographs), see eupthp1 41489. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Prove shortened by AV, 18-Apr-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → Fun 𝐼)    &   (𝜑𝐼 ∈ Fin)    &   (𝜑𝐵 ∈ V)    &   (𝜑𝐶𝑉)    &   (𝜑 → ¬ 𝐵 ∈ dom 𝐼)    &   (𝜑𝐹(1Walks‘𝐺)𝑃)    &   𝑁 = (#‘𝐹)    &   (𝜑𝐸 ∈ (Edg‘𝐺))    &   (𝜑 → {(𝑃𝑁), 𝐶} ⊆ 𝐸)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ∪ {⟨𝐵, 𝐸⟩}))    &   𝐻 = (𝐹 ∪ {⟨𝑁, 𝐵⟩})    &   𝑄 = (𝑃 ∪ {⟨(𝑁 + 1), 𝐶⟩})    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   ((𝜑𝐶 = (𝑃𝑁)) → 𝐸 = {𝐶})       (𝜑𝐻(1Walks‘𝑆)𝑄)

Theorem1wlkdlem1 40996* Lemma 1 for 1wlkd 41000. (Contributed by AV, 7-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   (𝜑𝐹 ∈ Word V)    &   (𝜑 → (#‘𝑃) = ((#‘𝐹) + 1))    &   (𝜑 → ∀𝑘 ∈ (0...(#‘𝐹))(𝑃𝑘) ∈ 𝑉)       (𝜑𝑃:(0...(#‘𝐹))⟶𝑉)

Theorem1wlkdlem2 40997* Lemma 2 for 1wlkd 41000. (Contributed by AV, 7-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   (𝜑𝐹 ∈ Word V)    &   (𝜑 → (#‘𝑃) = ((#‘𝐹) + 1))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))       (𝜑 → (((#‘𝐹) ∈ ℕ → (𝑃‘(#‘𝐹)) ∈ (𝐼‘(𝐹‘((#‘𝐹) − 1)))) ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ∈ (𝐼‘(𝐹𝑘))))

Theorem1wlkdlem3 40998* Lemma 3 for 1wlkd 41000. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   (𝜑𝐹 ∈ Word V)    &   (𝜑 → (#‘𝑃) = ((#‘𝐹) + 1))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))       (𝜑𝐹 ∈ Word dom 𝐼)

Theorem1wlkdlem4 40999* Lemma 4 for 1wlkd 41000. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 23-Jan-2021.)
(𝜑𝑃 ∈ Word V)    &   (𝜑𝐹 ∈ Word V)    &   (𝜑 → (#‘𝑃) = ((#‘𝐹) + 1))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))       (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))if-((𝑃𝑘) = (𝑃‘(𝑘 + 1)), (𝐼‘(𝐹𝑘)) = {(𝑃𝑘)}, {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))

Theorem1wlkd 41000* Two words representing a walk in a graph. (Contributed by AV, 7-Feb-2021.)
(𝜑𝑃 ∈ Word V)    &   (𝜑𝐹 ∈ Word V)    &   (𝜑 → (#‘𝑃) = ((#‘𝐹) + 1))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))    &   (𝜑 → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))    &   (𝜑𝐺𝑊)    &   𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑 → ∀𝑘 ∈ (0...(#‘𝐹))(𝑃𝑘) ∈ 𝑉)       (𝜑𝐹(1Walks‘𝐺)𝑃)

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