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Theorem List for Metamath Proof Explorer - 26601-26700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremumgrwlknloop 26601* 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 26602* Restrictions of walks (i.e. special kinds of walks, for example paths - see pthsfval 26673) 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 26603 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 26604 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 26605 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 26606 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 26607 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 26608* 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 26609 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 26610 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 26611 A walk connects vertices. (Contributed by AV, 22-Feb-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(Walks‘𝐺)𝑃 → (𝑁 ∈ (0...(#‘𝐹)) → (𝑃𝑁) ∈ 𝑉))

Theoremwlkepvtx 26612 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 26613 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 26614 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 26615 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 26616 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 26617* 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 26618 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 26619* 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 26620 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)}))))

Theoremwlkreslem0 26621 Lemma for wlkres 26623. TODO-AV: Will become obsolete if 𝐻 = (𝐹 ↾ (0..^𝑁)) is replaced by 𝐻 = (𝐹 substr ⟨0, 𝑁⟩) or 𝐻 = (𝐹 prefix 𝑁) in wlkres 26623 and trlres 26653. (Contributed by AV, 5-Mar-2021.)
((𝐹 ∈ Word 𝑆𝑁 ∈ (0...(#‘𝐹))) → (#‘(𝐹 ↾ (0..^𝑁))) = 𝑁)

Theoremwlkreslem 26622 Lemma for wlkres 26623. (Contributed by AV, 5-Mar-2021.)
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))

Theoremwlkres 26623 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 27193. (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(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 26624 Lemma for redwlk 26625. (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 26625 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 26626 Lemma for wlkp1 26634. (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 26627 Lemma for wlkp1 26634. (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 26628 Lemma for wlkp1 26634. (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 26629 Lemma for wlkp1 26634. (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 26630* Lemma for wlkp1 26634. (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 26631* Lemma for wlkp1 26634. (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 26632 Lemma for wlkp1 26634. (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 26633* Lemma for wlkp1 26634. (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 26634 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 27194. (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 26635* Lemma 1 for wlkd 26639. (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 26636* Lemma 2 for wlkd 26639. (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 26637* Lemma 3 for wlkd 26639. (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 26638* Lemma 4 for wlkd 26639. (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 26639* 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 26640* 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 26641* 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 26642* 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 26643 Extend class notation with trails (within a graph).
class Trails

Syntaxctrlson 26644 Extend class notation with trails between two vertices (within a graph).
class TrailsOn

Definitiondf-trls 26645* 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 26646* 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 26647 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 26648* 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 26649 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 26650 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 26651 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 26652 Lemma for trlres 26653. Formerly part of proof of eupthres 27193. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   𝐻 = (𝐹 ↾ (0..^𝑁))       (𝜑𝐻:(0..^(#‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))))

Theoremtrlres 26653 The restriction 𝐻, 𝑄 of a trail 𝐹, 𝑃 to an initial segment of the trail (of length 𝑁) forms a trail on the subgraph 𝑆 consisting of the edges in the initial segment. (Contributed by AV, 6-Mar-2021.) (Proof shortened by AV, 29-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)    &   (𝜑𝐹(Trails‘𝐺)𝑃)    &   (𝜑𝑁 ∈ (0..^(#‘𝐹)))    &   𝐻 = (𝐹 ↾ (0..^𝑁))    &   (𝜑 → (Vtx‘𝑆) = 𝑉)    &   (𝜑 → (iEdg‘𝑆) = (𝐼 ↾ (𝐹 “ (0..^𝑁))))    &   𝑄 = (𝑃 ↾ (0...𝑁))       (𝜑𝐻(Trails‘𝑆)𝑄)

Theoremupgrtrls 26654* The set of trails in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → (Trails‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ ((𝑓 ∈ Word dom 𝐼 ∧ Fun 𝑓) ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐼‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremupgristrl 26655* Properties of a pair of functions to be a trail in a pseudograph, definition of walks expanded. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ ((𝐹 ∈ Word dom 𝐼 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremupgrf1istrl 26656* Properties of a pair of a one-to-one function into the set of indices of edges and a function into the set of vertices to be a trail in a pseudograph. (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 29-Oct-2021.)
𝑉 = (Vtx‘𝐺)    &   𝐼 = (iEdg‘𝐺)       (𝐺 ∈ UPGraph → (𝐹(Trails‘𝐺)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐼𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremwksonproplem 26657* Lemma for theorems for properties of walks between two vertices, e.g. trlsonprop 26660. (Contributed by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)    &   (((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝐴(𝑊𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(𝑂𝐺)𝐵)𝑃𝐹(𝑄𝐺)𝑃)))    &   𝑊 = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑂𝑔)𝑏)𝑝𝑓(𝑄𝑔)𝑝)}))    &   (((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ 𝑓(𝑄𝐺)𝑝) → 𝑓(Walks‘𝐺)𝑝)       (𝐹(𝐴(𝑊𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(𝑂𝐺)𝐵)𝑃𝐹(𝑄𝐺)𝑃)))

Theoremtrlsonfval 26658* The set of trails between two vertices. (Contributed by Alexander van der Vekens, 4-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 15-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(TrailsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(WalksOn‘𝐺)𝐵)𝑝𝑓(Trails‘𝐺)𝑝)})

Theoremistrlson 26659 Properties of a pair of functions to be a trail between two given vertices. (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))

Theoremtrlsonprop 26660 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.) (Proof shortened by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)))

Theoremtrlsonistrl 26661 A trail between two vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 7-Jan-2021.)
(𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Trails‘𝐺)𝑃)

Theoremtrlsonwlkon 26662 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Revised by AV, 7-Jan-2021.)
(𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(𝐴(WalksOn‘𝐺)𝐵)𝑃)

Theoremtrlontrl 26663 A trail is a trail between its endpoints. (Contributed by AV, 31-Jan-2021.)
(𝐹(Trails‘𝐺)𝑃𝐹((𝑃‘0)(TrailsOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃)

16.3.4  Paths and simple paths

Syntaxcpths 26664 Extend class notation with paths (of a graph).
class Paths

Syntaxcspths 26665 Extend class notation with simple paths (of a graph).
class SPaths

Syntaxcpthson 26666 Extend class notation with paths between two vertices (within a graph).
class PathsOn

Syntaxcspthson 26667 Extend class notation with simple paths between two vertices (within a graph).
class SPathsOn

Definitiondf-pths 26668* Define the set of all paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see upgrwlkdvspth 26691).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the path 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, 9-Jan-2021.)

Paths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})

Definitiondf-spths 26669* Define the set of all simple paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path 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, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.)

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

Definitiondf-pthson 26670* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)
PathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(Paths‘𝑔)𝑝)}))

Definitiondf-spthson 26671* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 9-Jan-2021.)
SPathsOn = (𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(TrailsOn‘𝑔)𝑏)𝑝𝑓(SPaths‘𝑔)𝑝)}))

Theoremrelpths 26672 The set (Paths‘𝐺) of all paths on 𝐺 is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021.)
Rel (Paths‘𝐺)

Theorempthsfval 26673* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
(Paths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)}

Theoremspthsfval 26674* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
(SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝐺)𝑝 ∧ Fun 𝑝)}

Theoremispth 26675 Conditions for a pair of classes/functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
(𝐹(Paths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅))

Theoremisspth 26676 Conditions for a pair of classes/functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Revised by AV, 29-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃 ↔ (𝐹(Trails‘𝐺)𝑃 ∧ Fun 𝑃))

Theorempthistrl 26677 A path is a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹(Trails‘𝐺)𝑃)

Theoremspthispth 26678 A simple path is a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.) (Revised by AV, 9-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃𝐹(Paths‘𝐺)𝑃)

Theorempthiswlk 26679 A path is a walk (in an undirected graph). (Contributed by AV, 6-Feb-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theoremspthiswlk 26680 A simple path is a walk (in an undirected graph). (Contributed by AV, 16-May-2021.)
(𝐹(SPaths‘𝐺)𝑃𝐹(Walks‘𝐺)𝑃)

Theorempthdivtx 26681 The inner vertices of a path are distinct from all other vertices. (Contributed by AV, 5-Feb-2021.) (Proof shortened by AV, 31-Oct-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ (𝐼 ∈ (1..^(#‘𝐹)) ∧ 𝐽 ∈ (0...(#‘𝐹)) ∧ 𝐼𝐽)) → (𝑃𝐼) ≠ (𝑃𝐽))

Theorempthdadjvtx 26682 The adjacent vertices of a path of length at least 2 are distinct. (Contributed by AV, 5-Feb-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹) ∧ 𝐼 ∈ (0..^(#‘𝐹))) → (𝑃𝐼) ≠ (𝑃‘(𝐼 + 1)))

Theorem2pthnloop 26683* A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon 27129. (Contributed by AV, 6-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))2 ≤ (#‘(𝐼‘(𝐹𝑖))))

Theoremupgr2pthnlp 26684* A path of length at least 2 in a pseudograph does not contain a loop. (Contributed by AV, 6-Feb-2021.)
𝐼 = (iEdg‘𝐺)       ((𝐺 ∈ UPGraph ∧ 𝐹(Paths‘𝐺)𝑃 ∧ 1 < (#‘𝐹)) → ∀𝑖 ∈ (0..^(#‘𝐹))(#‘(𝐼‘(𝐹𝑖))) = 2)

Theoremspthdifv 26685 The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.) (Revised by AV, 5-Jun-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐹(SPaths‘𝐺)𝑃𝑃:(0...(#‘𝐹))–1-1→(Vtx‘𝐺))

Theoremspthdep 26686 A simple path (at least of length 1) has different start and end points (in an undirected graph). (Contributed by AV, 31-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
((𝐹(SPaths‘𝐺)𝑃 ∧ (#‘𝐹) ≠ 0) → (𝑃‘0) ≠ (𝑃‘(#‘𝐹)))

Theorempthdepisspth 26687 A path with different start and end points is a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 31-Oct-2017.) (Revised by AV, 12-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
((𝐹(Paths‘𝐺)𝑃 ∧ (𝑃‘0) ≠ (𝑃‘(#‘𝐹))) → 𝐹(SPaths‘𝐺)𝑃)

Theoremupgrwlkdvdelem 26688* Lemma for upgrwlkdvde 26689. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Proof shortened by AV, 17-Jan-2021.)
((𝑃:(0...(#‘𝐹))–1-1𝑉𝐹 ∈ Word dom 𝐼) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐼‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))} → Fun 𝐹))

Theoremupgrwlkdvde 26689 In a pseudograph, all edges of a walk consisting of different vertices are different. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26690. (Contributed by AV, 17-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → Fun 𝐹)

Theoremupgrspthswlk 26690* The set of simple paths in a pseudograph, expressed as walk. Notice that this theorem would not hold for arbitrary hypergraphs, since a walk with distinct vertices does not need to be a trail: let E = { p0, p1, p2 } be a hyperedge, then ( p0, e, p1, e, p2 ) is walk with distinct vertices, but not with distinct edges. Therefore, E is not a trail and, by definition, also no path. (Contributed by AV, 11-Jan-2021.) (Proof shortened by AV, 17-Jan-2021.) (Proof shortened by AV, 30-Oct-2021.)
(𝐺 ∈ UPGraph → (SPaths‘𝐺) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝐺)𝑝 ∧ Fun 𝑝)})

Theoremupgrwlkdvspth 26691 A walk consisting of different vertices is a simple path. Notice that this theorem would not hold for arbitrary hypergraphs, see the counterexample given in the comment of upgrspthswlk 26690. (Contributed by Alexander van der Vekens, 27-Oct-2017.) (Revised by AV, 17-Jan-2021.)
((𝐺 ∈ UPGraph ∧ 𝐹(Walks‘𝐺)𝑃 ∧ Fun 𝑃) → 𝐹(SPaths‘𝐺)𝑃)

Theorempthsonfval 26692* The set of paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(PathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(Paths‘𝐺)𝑝)})

Theoremspthson 26693* The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       ((𝐴𝑉𝐵𝑉) → (𝐴(SPathsOn‘𝐺)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(TrailsOn‘𝐺)𝐵)𝑝𝑓(SPaths‘𝐺)𝑝)})

Theoremispthson 26694 Properties of a pair of functions to be a path between two given vertices. (Contributed by Alexander van der Vekens, 8-Nov-2017.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))

Theoremisspthson 26695 Properties of a pair of functions to be a simple path between two given vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.) (Revised by AV, 21-Mar-2021.)
𝑉 = (Vtx‘𝐺)       (((𝐴𝑉𝐵𝑉) ∧ (𝐹𝑈𝑃𝑍)) → (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 ↔ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)))

Theorempthsonprop 26696 Properties of a path between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)))

Theoremspthonprop 26697 Properties of a simple path between two vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.) (Revised by AV, 16-Jan-2021.)
𝑉 = (Vtx‘𝐺)       (𝐹(𝐴(SPathsOn‘𝐺)𝐵)𝑃 → ((𝐺 ∈ V ∧ 𝐴𝑉𝐵𝑉) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃𝐹(SPaths‘𝐺)𝑃)))

Theorempthonispth 26698 A path between two vertices is a path. (Contributed by Alexander van der Vekens, 12-Dec-2017.) (Revised by AV, 17-Jan-2021.)
(𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹(Paths‘𝐺)𝑃)

Theorempthontrlon 26699 A path between two vertices is a trail between these vertices. (Contributed by AV, 24-Jan-2021.)
(𝐹(𝐴(PathsOn‘𝐺)𝐵)𝑃𝐹(𝐴(TrailsOn‘𝐺)𝐵)𝑃)

Theorempthonpth 26700 A path is a path between its endpoints. (Contributed by AV, 31-Jan-2021.)
(𝐹(Paths‘𝐺)𝑃𝐹((𝑃‘0)(PathsOn‘𝐺)(𝑃‘(#‘𝐹)))𝑃)

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