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

Theoremewlkprop 26901* 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 26902 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 26903 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 26904 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)

Theoremwkslem1 26905 Lemma 1 for walks to substitute the index of the condition for vertices and edges in a 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)))))

Theoremwkslem2 26906 Lemma 2 for walks to substitute the index of the condition for vertices and edges in a 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)))))

Theoremwksfval 26907* The set of walks (in an undirected graph). (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (Walks‘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))))})

Theoremiswlk 26908* Properties of a pair of functions to be a walk. (Contributed by AV, 30-Dec-2020.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       ((𝐺𝑊𝐹𝑈𝑃𝑍) → (𝐹(Walks‘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))))))

Theoremwlkprop 26909* Properties of a walk. (Contributed by AV, 5-Nov-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐹(Walks‘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)))))

Theoremwlkv 26910 The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 3-Feb-2021.)
(𝐹(Walks‘𝐺)𝑃 → (𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V))

Theoremiswlkg 26911* Generalization of iswlk 26908: Conditions for two classes to represent a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.) (Revised by AV, 1-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺𝑊 → (𝐹(Walks‘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))))))

Theoremwlkf 26912 The mapping enumerating the (indices of the) edges of a walk is a word over the indices of the edges of the graph. (Contributed by AV, 5-Apr-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(Walks‘𝐺)𝑃𝐹 ∈ Word dom 𝐼)

Theoremwlkcl 26913 A walk has length ♯(𝐹), which is an integer. Formerly proven for an Eulerian path, see eupthcl 27575. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)
(𝐹(Walks‘𝐺)𝑃 → (♯‘𝐹) ∈ ℕ0)

Theoremwlkp 26914 The mapping enumerating the vertices of a walk is a function. (Contributed by AV, 5-Apr-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃𝑃:(0...(♯‘𝐹))⟶𝑉)

Theoremwlkpwrd 26915 The sequence of vertices of a walk is a word over the set of vertices. (Contributed by AV, 27-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃𝑃 ∈ Word 𝑉)

Theoremwlklenvp1 26916 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(Walks‘𝐺)𝑃 → (♯‘𝑃) = ((♯‘𝐹) + 1))

Theoremwksv 26917* The class of walks is a set. (Contributed by AV, 15-Jan-2021.)
{⟨𝑓, 𝑝⟩ ∣ 𝑓(Walks‘𝐺)𝑝} ∈ V

Theoremwlkn0 26918 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(Walks‘𝐺)𝑃𝑃 ≠ ∅)

Theoremwlklenvm1 26919 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(Walks‘𝐺)𝑃 → (♯‘𝐹) = ((♯‘𝑃) − 1))

Theoremifpsnprss 26920 Lemma for wlkvtxeledg 26921: Two adjacent (not necessarily different) vertices 𝐴 and 𝐵 in a walk are incident with an edge 𝐸. (Contributed by AV, 4-Apr-2021.) (Revised by AV, 5-Nov-2021.)
(if-(𝐴 = 𝐵, 𝐸 = {𝐴}, {𝐴, 𝐵} ⊆ 𝐸) → {𝐴, 𝐵} ⊆ 𝐸)

Theoremwlkvtxeledg 26921* Each pair of adjacent vertices in a walk is a subset of an edge. (Contributed by AV, 28-Jan-2021.) (Proof shortened by AV, 4-Apr-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘)))

Theoremwlkvtxiedg 26922* 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(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒 ∈ ran 𝐼{(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)

Theoremrelwlk 26923 The set (Walks‘𝐺) of all walks on 𝐺 is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 19-Feb-2021.)
Rel (Walks‘𝐺)

Theoremwlkvv 26924 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)(Walks‘𝐺)(2nd𝑊) → 𝑊 ∈ (V × V))

Theoremwlkop 26925 A walk is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.) (Revised by AV, 1-Jan-2021.)
(𝑊 ∈ (Walks‘𝐺) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)

Theoremwlkcpr 26926 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.) (Revised by AV, 2-Jan-2021.)
(𝑊 ∈ (Walks‘𝐺) ↔ (1st𝑊)(Walks‘𝐺)(2nd𝑊))

Theoremwlk2f 26927* If there is a walk 𝑊 there is a pair of functions representing this walk. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑊 ∈ (Walks‘𝐺) → ∃𝑓𝑝 𝑓(Walks‘𝐺)𝑝)

Theoremwlkcomp 26928* 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 ∈ (Walks‘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))))))

Theoremwlkcompim 26929* 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 ∈ (Walks‘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)))))

Theoremwlkelwrd 26930 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 ∈ (Walks‘𝐺) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉))

Theoremwlkeq 26931* 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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑥 ∈ (0..^𝑁)((1st𝐴)‘𝑥) = ((1st𝐵)‘𝑥) ∧ ∀𝑥 ∈ (0...𝑁)((2nd𝐴)‘𝑥) = ((2nd𝐵)‘𝑥))))

Theoremedginwlk 26932 The value of the edge function for an index of an edge within a walk is an edge. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 9-Dec-2021.)
𝐼 = (iEdg‘𝐺)    &   𝐸 = (Edg‘𝐺)       ((Fun 𝐼𝐹 ∈ Word dom 𝐼𝐾 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝐾)) ∈ 𝐸)

Theoremupgredginwlk 26933 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 26934 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(Walks‘𝐺)𝑃𝑋 ∈ (0..^(♯‘𝐹))) → (𝐼‘(𝐹𝑋)) ∈ ran 𝐼)

Theoremwlkl1loop 26935 A walk of length 1 from a vertex to itself is a loop. (Contributed by AV, 23-Apr-2021.)
(((Fun (iEdg‘𝐺) ∧ 𝐹(Walks‘𝐺)𝑃) ∧ ((♯‘𝐹) = 1 ∧ (𝑃‘0) = (𝑃‘1))) → {(𝑃‘0)} ∈ (Edg‘𝐺))

Theoremwlk1walk 26936* A walk is a 1-walk "on the edge level" according to Aksoy et al. (Contributed by AV, 30-Dec-2020.)
𝐼 = (iEdg‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (1..^(♯‘𝐹))1 ≤ (♯‘((𝐼‘(𝐹‘(𝑘 − 1))) ∩ (𝐼‘(𝐹𝑘)))))

Theoremwlk1ewlk 26937 A walk is an s-walk "on the edge level" (with s=1) according to Aksoy et al. (Contributed by AV, 5-Jan-2021.)
(𝐹(Walks‘𝐺)𝑃𝐹 ∈ (𝐺 EdgWalks 1))

Theoremupgriswlk 26938* Properties of a pair of functions to be a walk in a pseudograph. (Contributed by AV, 2-Jan-2021.) (Revised by AV, 28-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremupgrwlkedg 26939* 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(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})

Theoremupgrwlkcompim 26940* 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 ∈ (Walks‘𝐺)) → (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))

Theoremwlkvtxedg 26941* 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(Walks‘𝐺)𝑃 → ∀𝑘 ∈ (0..^(♯‘𝐹))∃𝑒𝐸 {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ 𝑒)

Theoremupgrwlkvtxedg 26942* 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(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∈ 𝐸)

Theoremuspgr2wlkeq 26943* 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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ 𝑁 = (♯‘(1st𝐴))) → (𝐴 = 𝐵 ↔ (𝑁 = (♯‘(1st𝐵)) ∧ ∀𝑦 ∈ (0...𝑁)((2nd𝐴)‘𝑦) = ((2nd𝐵)‘𝑦))))

Theoremuspgr2wlkeq2 26944 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 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐴)) = 𝑁) ∧ (𝐵 ∈ (Walks‘𝐺) ∧ (♯‘(1st𝐵)) = 𝑁)) → ((2nd𝐴) = (2nd𝐵) → 𝐴 = 𝐵))

Theoremuspgr2wlkeqi 26945 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 ∈ (Walks‘𝐺) ∧ 𝐵 ∈ (Walks‘𝐺)) ∧ (2nd𝐴) = (2nd𝐵)) → 𝐴 = 𝐵)

Theoremumgrwlknloop 26946* 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(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

TheoremwlkRes 26947* Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval 27023) 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(Walks‘𝐺)𝑝)       {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑊𝐺)𝑝𝜑)} ∈ V

Theoremwlkv0 26948 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 ∈ (Walks‘𝐺)) → ((1st𝑊) = ∅ ∧ (2nd𝑊) = ∅))

Theoremg0wlk0 26949 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) = ∅ → (Walks‘𝐺) = ∅)

Theorem0wlk0 26950 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.)
(Walks‘∅) = ∅

Theoremwlk0prc 26951 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)) → (Walks‘𝐺) = ∅)

Theoremwlklenvclwlk 26952 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)”⟩)⟩ ∈ (Walks‘𝐺) → (♯‘𝐹) = (♯‘𝑊)))

Theoremwlkson 26953* 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(Walks‘𝐺)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(♯‘𝑓)) = 𝐵)})

Theoremiswlkon 26954 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(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))

Theoremwlkonprop 26955 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(Walks‘𝐺)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(♯‘𝐹)) = 𝐵)))

Theoremwlkpvtx 26956 A walk connects vertices. (Contributed by AV, 22-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(♯‘𝐹)) → (𝑃𝑁) ∈ 𝑉))

Theoremwlkepvtx 26957 The endpoints of a walk are vertices. (Contributed by AV, 31-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → ((𝑃‘0) ∈ 𝑉 ∧ (𝑃‘(♯‘𝐹)) ∈ 𝑉))

Theoremwlkoniswlk 26958 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(Walks‘𝐺)𝑃)

Theoremwlkonwlk 26959 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(Walks‘𝐺)𝑃𝐹((𝑃‘0)(WalksOn‘𝐺)(𝑃‘(♯‘𝐹)))𝑃)

Theoremwlkonwlk1l 26960 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(Walks‘𝐺)𝑃)       (𝜑𝐹((𝑃‘0)(WalksOn‘𝐺)(lastS‘𝑃))𝑃)

Theoremwlksoneq1eq2 26961 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 26962* 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 26963 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 26964* Lemma for theorems for walks of length 2. (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 26965 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.) (Revised by AV, 28-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → ((𝐹(Walks‘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)}))))

Theoremwlkreslem0OLD 26966 Obsolete as of 30-Nov-2022. (Contributed by AV, 5-Mar-2021.) (Proof shortened by AV, 3-Nov-2022.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹 ∈ Word 𝑆𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)

Theoremwlkreslem0OLDOLD 26967 Obsolete proof of wlkreslem0OLD 26966 as of 12-Oct-2022. (Contributed by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐹 ∈ Word 𝑆𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 ↾ (0..^𝑁))) = 𝑁)

Theoremwlkreslem 26968 Lemma for wlkres 26969. (Contributed by AV, 5-Mar-2021.) (Revised by AV, 30-Nov-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Walks‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(♯‘𝐹)))    &   (𝜑 → (Vtx‘𝑆) = 𝑉)       (𝜑𝑆 ∈ V)

Theoremwlkres 26969 The restriction 𝐻, 𝑄 of a walk 𝐹, 𝑃 to an initial segment of the walk (of length 𝑁) forms a walk on the subgraph 𝑆 consisting of the edges in the initial segment. Formerly proven directly for Eulerian paths, see eupthres 27581. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Walks‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(♯‘𝐹)))    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝐻 = (𝐹 prefix 𝑁)    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(Walks‘𝑆)𝑄)

TheoremwlkreslemOLD 26970 Obsolete version of wlkreslem 26968 as of 30-Nov-2022. (Contributed by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Walks‘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))

TheoremwlkresOLD 26971 Obsolete version of wlkres 26969 as of 30-Nov-2022. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 5-Mar-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Walks‘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(Walks‘𝑆)𝑄)

Theoremredwlklem 26972 Lemma for redwlk 26973. (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)

Theoremredwlk 26973 A walk ending at the last but one vertex of the walk is a walk. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 29-Jan-2021.)
((𝐹(Walks‘𝐺)𝑃 ∧ 1 ≤ (♯‘𝐹)) → (𝐹 ↾ (0..^((♯‘𝐹) − 1)))(Walks‘𝐺)(𝑃 ↾ (0..^(♯‘𝐹))))

Theoremwlkp1lem1 26974 Lemma for wlkp1 26982. (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(Walks‘𝐺)𝑃)    &   𝑁 = (♯‘𝐹)       (𝜑 → ¬ (𝑁 + 1) ∈ dom 𝑃)

Theoremwlkp1lem2 26975 Lemma for wlkp1 26982. (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(Walks‘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))

Theoremwlkp1lem3 26976 Lemma for wlkp1 26982. (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(Walks‘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))

Theoremwlkp1lem4 26977 Lemma for wlkp1 26982. (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(Walks‘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))

Theoremwlkp1lem5 26978* Lemma for wlkp1 26982. (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(Walks‘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))

Theoremwlkp1lem6 26979* Lemma for wlkp1 26982. (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(Walks‘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))))

Theoremwlkp1lem7 26980 Lemma for wlkp1 26982. (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(Walks‘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)))

Theoremwlkp1lem8 26981* Lemma for wlkp1 26982. (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(Walks‘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))))

Theoremwlkp1 26982 Append one path segment (edge) 𝐸 from vertex (𝑃𝑁) to a vertex 𝐶 to a walk 𝐹, 𝑃 to become a 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 27582. (Contributed by Mario Carneiro, 7-Apr-2015.) (Revised by AV, 6-Mar-2021.) (Proof 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(Walks‘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(Walks‘𝑆)𝑄)

Theoremwlkdlem1 26983* Lemma 1 for wlkd 26987. (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)

Theoremwlkdlem2 26984* Lemma 2 for wlkd 26987. (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))))

Theoremwlkdlem3 26985* Lemma 3 for wlkd 26987. (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)

Theoremwlkdlem4 26986* Lemma 4 for wlkd 26987. (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))))

Theoremwlkd 26987* 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(Walks‘𝐺)𝑃)

16.3.2  Walks for loop-free graphs

Theoremlfgrwlkprop 26988* Two adjacent vertices in a walk are different in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹(Walks‘𝐺)𝑃𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theoremlfgriswlk 26989* Conditions for a pair of functions to be a walk in a loop-free graph. (Contributed by AV, 28-Jan-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐺𝑊𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)}) → (𝐹(Walks‘𝐺)𝑃 ↔ (𝐹 ∈ Word dom 𝐼𝑃:(0...(♯‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(♯‘𝐹))((𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)) ∧ {(𝑃𝑘), (𝑃‘(𝑘 + 1))} ⊆ (𝐼‘(𝐹𝑘))))))

Theoremlfgrwlknloop 26990* In a loop-free graph, each walk has no loops! (Contributed by AV, 2-Feb-2021.)
𝐼 = (iEdg‘𝐺)    &   𝑉 = (Vtx‘𝐺)       ((𝐼:dom 𝐼⟶{𝑥 ∈ 𝒫 𝑉 ∣ 2 ≤ (♯‘𝑥)} ∧ 𝐹(Walks‘𝐺)𝑃) → ∀𝑘 ∈ (0..^(♯‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

16.3.3  Trails

Syntaxctrls 26991 Extend class notation with trails (within a graph).
class Trails

Syntaxctrlson 26992 Extend class notation with trails between two vertices (within a graph).
class TrailsOn

Definitiondf-trls 26993* Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also 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 Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)

Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun 𝑓)})

Definitiondf-trlson 26994* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 28-Dec-2020.)
TrailsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(WalksOn‘𝑔)𝑏)𝑝𝑓(Trails‘𝑔)𝑝)}))

Theoremreltrls 26995 The set (Trails‘𝐺) of all trails on 𝐺 is a set of pairs by our definition of a trail, and so is a relation. (Contributed by AV, 29-Oct-2021.)
Rel (Trails‘𝐺)

Theoremtrlsfval 26996* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)
(Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑓)}

Theoremistrl 26997 Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 28-Dec-2020.) (Revised by AV, 29-Oct-2021.)
(𝐹(Trails‘𝐺)𝑃 ↔ (𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝐹))

Theoremtrliswlk 26998 A trail is a walk. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 29-Oct-2021.)
(𝐹(Trails‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremtrlf1 26999 The enumeration 𝐹 of a trail 𝐹, 𝑃 is injective. (Contributed by AV, 20-Feb-2021.) (Proof shortened by AV, 29-Oct-2021.)
𝐼 = (iEdg‘𝐺)       (𝐹(Trails‘𝐺)𝑃𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼)

Theoremtrlreslem 27000 Lemma for trlres 27001. Formerly part of proof of eupthres 27581. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(♯‘𝐹)))    &   𝐻 = (𝐹 prefix 𝑁)       (𝜑𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))

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