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Theorem List for Metamath Proof Explorer - 26001-26100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcusgrares 26001* Restricting a complete simple graph. (Contributed by Alexander van der Vekens, 2-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → (𝑉 ∖ {𝑁}) ComplUSGrph 𝐹)

Theoremcusgrasizeindslem1 26002* Lemma 1 for cusgrasizeinds 26004. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       (dom 𝐹 ∩ {𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ∅

Theoremcusgrasizeindslem2 26003* Lemma 2 for cusgrasizeinds 26004. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘{𝑥 ∈ dom 𝐸𝑁 ∈ (𝐸𝑥)}) = ((#‘𝑉) − 1))

Theoremcusgrasizeinds 26004* Part 1 of induction step in cusgrasize 26006. The size of a complete simple graph with 𝑛 vertices is (𝑛 − 1) plus the size of the complete graph reduced by one vertex. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       ((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin ∧ 𝑁𝑉) → (#‘𝐸) = (((#‘𝑉) − 1) + (#‘𝐹)))

Theoremcusgrasize2inds 26005* Induction step in cusgrasize 26006. If the size of the complete graph with 𝑛 vertices reduced by one vertex is "(𝑛 − 1) choose 2", the size of the complete graph with 𝑛 vertices is "𝑛 choose 2". (Contributed by Alexander van der Vekens, 11-Jan-2018.)
𝐹 = (𝐸 ↾ {𝑥 ∈ dom 𝐸𝑁 ∉ (𝐸𝑥)})       (𝑌 ∈ ℕ0 → ((𝑉 ComplUSGrph 𝐸 ∧ (#‘𝑉) = 𝑌𝑁𝑉) → ((#‘𝐹) = ((#‘(𝑉 ∖ {𝑁}))C2) → (#‘𝐸) = ((#‘𝑉)C2))))

Theoremcusgrasize 26006 The size of a finite complete simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (𝑛C2) ("𝑛 choose 2") resp. (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 11-Jan-2018.)
((𝑉 ComplUSGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) = ((#‘𝑉)C2))

Theoremcusgrafilem1 26007* Lemma 1 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}       ((𝑉 ComplUSGrph 𝐸𝑁𝑉) → 𝑃 ⊆ ran 𝐸)

Theoremcusgrafilem2 26008* Lemma 2 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       ((𝑉𝑊𝑁𝑉) → 𝐹:(𝑉 ∖ {𝑁})–1-1-onto𝑃)

Theoremcusgrafilem3 26009* Lemma 3 for cusgrafi 26010. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
𝑃 = {𝑥 ∈ 𝒫 𝑉 ∣ ∃𝑎𝑉 (𝑎𝑁𝑥 = {𝑎, 𝑁})}    &   𝐹 = (𝑥 ∈ (𝑉 ∖ {𝑁}) ↦ {𝑥, 𝑁})       ((𝑉𝑊𝑁𝑉) → (¬ 𝑉 ∈ Fin → ¬ 𝑃 ∈ Fin))

Theoremcusgrafi 26010 If the size of a complete simple graph is finite, then also its order is finite. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 ComplUSGrph 𝐸𝐸 ∈ Fin) → 𝑉 ∈ Fin)

Theoremusgrasscusgra 26011* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ∀𝑒 ∈ ran 𝐸𝑓 ∈ ran 𝐹 𝑒 = 𝑓)

Theoremsizeusglecusglem1 26012 Lemma 1 for sizeusglecusg 26014. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → ( I ↾ ran 𝐸):ran 𝐸1-1→ran 𝐹)

Theoremsizeusglecusglem2 26013 Lemma 2 for sizeusglecusg 26014. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹𝐹 ∈ Fin) → 𝐸 ∈ Fin)

Theoremsizeusglecusg 26014 The size of an undirected simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Proof shortened by AV, 4-May-2021.)
((𝑉 USGrph 𝐸𝑉 ComplUSGrph 𝐹) → (#‘𝐸) ≤ (#‘𝐹))

Theoremusgramaxsize 26015 The maximum size of an undirected simple graph with 𝑛 vertices (𝑛 ∈ ℕ0) is (((𝑛 − 1)∗𝑛) / 2). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (#‘𝐸) ≤ ((#‘𝑉)C2))

17.1.4.3  Universal vertices

Theoremisuvtx 26016* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 UnivVertex 𝐸) = {𝑛𝑉 ∣ ∀𝑘 ∈ (𝑉 ∖ {𝑛}){𝑘, 𝑛} ∈ ran 𝐸})

Theoremuvtxel 26017* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (𝑁𝑉 ∧ ∀𝑘 ∈ (𝑉 ∖ {𝑁}){𝑘, 𝑁} ∈ ran 𝐸)))

Theoremuvtxisvtx 26018 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(𝑁 ∈ (𝑉 UnivVertex 𝐸) → 𝑁𝑉)

Theoremuvtx0 26019 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
(∅ UnivVertex 𝐸) = ∅

Theoremuvtx01vtx 26020* If a graph/class has no edges, it has universal vertices if and only if it has exactly one vertex. This theorem could have been stated ((𝑉 UnivVertex ∅) ≠ ∅ ↔ (#‘𝑉) = 1), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
((𝑉 UnivVertex ∅) ≠ ∅ ↔ ∃𝑥 𝑉 = {𝑥})

Theoremuvtxnbgra 26021 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (𝑉 UnivVertex 𝐸)) → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁}))

Theoremuvtxnm1nbgra 26022 A universal vertex has 𝑛 − 1 neighbors in a graph with 𝑛 vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑉 ∈ Fin) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) → (#‘(⟨𝑉, 𝐸⟩ Neighbors 𝑁)) = ((#‘𝑉) − 1)))

Theoremuvtxnbgravtx 26023* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
((𝑉 USGrph 𝐸𝑁 ∈ (𝑉 UnivVertex 𝐸)) → ∀𝑣 ∈ (𝑉 ∖ {𝑁})𝑁 ∈ (⟨𝑉, 𝐸⟩ Neighbors 𝑣))

Theoremcusgrauvtxb 26024 An undirected simple graph is complete if and only if each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.) (Revised by Alexander van der Vekens, 18-Jan-2018.)
(𝑉 USGrph 𝐸 → (𝑉 ComplUSGrph 𝐸 ↔ (𝑉 UnivVertex 𝐸) = 𝑉))

Theoremuvtxnb 26025 A vertex in a undirected simple graph is universal iff all the other vertices are its neighbors. (Contributed by Alexander van der Vekens, 13-Jul-2018.)
((𝑉 USGrph 𝐸𝑁𝑉) → (𝑁 ∈ (𝑉 UnivVertex 𝐸) ↔ (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = (𝑉 ∖ {𝑁})))

17.1.5  Walks, paths and cycles

Syntaxcwalk 26026 Extend class notation with Walks (of a graph).
class Walks

Syntaxctrail 26027 Extend class notation with Trails (of a graph).
class Trails

Syntaxcpath 26028 Extend class notation with Paths (of a graph).
class Paths

Syntaxcspath 26029 Extend class notation with Simple Paths (of a graph).
class SPaths

Syntaxcwlkon 26030 Extend class notation with Walks between two vertices (within a graph).
class WalkOn

Syntaxctrlon 26031 Extend class notation with Trails between two vertices (within a graph).
class TrailOn

Syntaxcpthon 26032 Extend class notation with Paths between two vertices (within a graph).
class PathOn

Syntaxcspthon 26033 Extend class notation with simple paths between two vertices (within a graph).
class SPathOn

Syntaxccrct 26034 Extend class notation with Circuits (of a graph).
class Circuits

Syntaxccycl 26035 Extend class notation with Cycles (of a graph).
class Cycles

Definitiondf-wlk 26036* Define the set of all Walks (in an undirected graph).

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). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)

Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝑒𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Definitiondf-trail 26037* 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.)

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

Definitiondf-pth 26038* 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 wlkdvspth 26138).

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.)

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

Definitiondf-spth 26039* 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.)

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

Definitiondf-crct 26040* Define the set of all circuits (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A circuit can be a closed walk allowing repetitions of vertices but not edges;"; according to Wikipedia ("Glossary of graph theory terms", https://en.wikipedia.org/wiki/Glossary_of_graph_theory_terms, 3-Oct-2017): "A circuit may refer to ... a trail (a closed tour without repeated edges), ...".

Following Bollobas ("A trail whose endvertices coincide (a closed trail) is called a circuit.", see Definition of [Bollobas] p. 5.), a circuit is a closed trail without repeated edges. So the circuit 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)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Circuits = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Definitiondf-cycl 26041* Define the set of all (simple) cycles (in an undirected graph).

According to Wikipedia ("Cycle (graph theory)", https://en.wikipedia.org/wiki/Cycle_(graph_theory), 3-Oct-2017): "A simple cycle may be defined either as a closed walk with no repetitions of vertices and edges allowed, other than the repetition of the starting and ending vertex,"

According to Bollobas: "If a walk W = x0 x1 ... x(l) is such that l >= 3, x0=x(l), and the vertices x(i), 0 < i < l, are distinct from each other and x0, then W is said to be a cycle.", see Definition of [Bollobas] p. 5.

However, since a walk consisting of distinct vertices (except the first and the last vertex) is a path, a cycle can be defined as path whose first and last vertices coincide. So a cycle is represented by the following sequence: p(0) e(f(1)) p(1) ... p(n-1) e(f(n)) p(n)=p(0). (Contributed by Alexander van der Vekens, 3-Oct-2017.)

Cycles = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Paths 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))})

Definitiondf-wlkon 26042* Define the collection of walks with particular endpoints (in an un- directed graph). 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.)
WalkOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = 𝑎 ∧ (𝑝‘(#‘𝑓)) = 𝑏)}))

Definitiondf-trlon 26043* Define the collection of trails with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
TrailOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Trails 𝑒)𝑝)}))

Definitiondf-pthon 26044* Define the collection of paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.)
PathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 Paths 𝑒)𝑝)}))

Definitiondf-spthon 26045* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
SPathOn = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑎𝑣, 𝑏𝑣 ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑎(𝑣 WalkOn 𝑒)𝑏)𝑝𝑓(𝑣 SPaths 𝑒)𝑝)}))

17.1.5.1  Walks and trails

Theoremrelwlk 26046 The walks (in an undirected simple graph) are (ordered) pairs. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
Rel (𝑉 Walks 𝐸)

Theoremwlks 26047* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Walks 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom 𝐸𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremiswlk 26048* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theorem2mwlk 26049 The two mappings determining a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉))

Theoremwlkres 26050* Restrictions of walks (i.e. special kinds of walks, for example paths - see pths 26096) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
(𝑓(𝑉𝑊𝐸)𝑝𝑓(𝑉 Walks 𝐸)𝑝)       ((𝑉 ∈ V ∧ 𝐸 ∈ V) → {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉𝑊𝐸)𝑝𝜑)} ∈ V)

Theoremwlkbprop 26051 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃 → ((#‘𝐹) ∈ ℕ0 ∧ (𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)))

Theoremiswlkg 26052* Generalisation of iswlk 26048: Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
((𝑉𝑋𝐸𝑌) → (𝐹(𝑉 Walks 𝐸)𝑃 ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremwlkcomp 26053* A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       ((𝑉𝑋𝐸𝑌𝑊 ∈ (𝑆 × 𝑇)) → (𝑊 ∈ (𝑉 Walks 𝐸) ↔ (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremwlkcompim 26054* Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
𝐹 = (1st𝑊)    &   𝑃 = (2nd𝑊)       (𝑊 ∈ (𝑉 Walks 𝐸) → (𝐹 ∈ Word dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}))

Theoremwlkn0 26055 The set 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.)
(𝐹(𝑉 Walks 𝐸)𝑃𝑃 ≠ ∅)

Theoremwlkop 26056 A walk (in an undirected simple graph) is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) → 𝑊 = ⟨(1st𝑊), (2nd𝑊)⟩)

Theoremwlkcpr 26057 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) ↔ (1st𝑊)(𝑉 Walks 𝐸)(2nd𝑊))

Theoremwlkelwrd 26058 The components of a walk are words/functions over a zero based range of integers. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
(𝑊 ∈ (𝑉 Walks 𝐸) → ((1st𝑊) ∈ Word dom 𝐸 ∧ (2nd𝑊):(0...(#‘(1st𝑊)))⟶𝑉))

Theoremedgwlk 26059* The (connected) edges of a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 22-Jul-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → ∀𝑘 ∈ (0..^(#‘𝐹)){(𝑃𝑘), (𝑃‘(𝑘 + 1))} ∈ ran 𝐸)

Theoremwlklenvm1 26060 The number of edges of a walk (in an undirected graph) is the number of its vertices minus 1. (Contributed by Alexander van der Vekens, 1-Jul-2018.)
(𝐹(𝑉 Walks 𝐸)𝑃 → (#‘𝐹) = ((#‘𝑃) − 1))

Theoremwlkon 26061* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 WalkOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Walks 𝐸)𝑝 ∧ (𝑝‘0) = 𝐴 ∧ (𝑝‘(#‘𝑓)) = 𝐵)})

Theoremiswlkon 26062 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.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

Theoremwlkonprop 26063 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝑉 Walks 𝐸)𝑃 ∧ (𝑃‘0) = 𝐴 ∧ (𝑃‘(#‘𝐹)) = 𝐵)))

Theoremwlkoniswlk 26064 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremwlkonwlk 26065 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
(𝐹(𝑉 Walks 𝐸)𝑃𝐹((𝑃‘0)(𝑉 WalkOn 𝐸)(𝑃‘(#‘𝐹)))𝑃)

Theoremtrls 26066* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Trails 𝐸) = {⟨𝑓, 𝑝⟩ ∣ ((𝑓 ∈ Word dom 𝐸 ∧ Fun 𝑓) ∧ 𝑝:(0...(#‘𝑓))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝐸‘(𝑓𝑘)) = {(𝑝𝑘), (𝑝‘(𝑘 + 1))})})

Theoremistrl 26067* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ ((𝐹 ∈ Word dom 𝐸 ∧ Fun 𝐹) ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremistrl2 26068* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))})))

Theoremtrliswlk 26069 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
(𝐹(𝑉 Trails 𝐸)𝑃𝐹(𝑉 Walks 𝐸)𝑃)

Theoremtrlon 26070* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐴𝑉𝐵𝑉)) → (𝐴(𝑉 TrailOn 𝐸)𝐵) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑝𝑓(𝑉 Trails 𝐸)𝑝)})

Theoremistrlon 26071 Properties of a pair of functions to be a trail between two given vertices(in an undirected graph). (Contributed by Alexander van der Vekens, 3-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍) ∧ (𝐴𝑉𝐵𝑉)) → (𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃 ↔ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)))

Theoremtrlonprop 26072 Properties of a trail between two vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃 → (((𝑉 ∈ V ∧ 𝐸 ∈ V) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V) ∧ (𝐴𝑉𝐵𝑉)) ∧ (𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)))

Theoremtrlonistrl 26073 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃𝐹(𝑉 Trails 𝐸)𝑃)

Theoremtrlonwlkon 26074 A trail between two vertices is a walk between these vertices. (Contributed by Alexander van der Vekens, 5-Nov-2017.)
(𝐹(𝐴(𝑉 TrailOn 𝐸)𝐵)𝑃𝐹(𝐴(𝑉 WalkOn 𝐸)𝐵)𝑃)

Theorem0wlk 26075 A pair of an empty set (of edges) and a second set (of vertices) is a walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Walks 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0trl 26076 A pair of an empty set (of edges) and a second set (of vertices) is a trail if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Proof shortened by AV, 7-Jan-2020.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Trails 𝐸)𝑃𝑃:(0...0)⟶𝑉))

Theorem0wlkon 26077 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(𝑉 WalkOn 𝐸)𝑁)𝑃))

Theorem0trlon 26078 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑁𝑉) → ((𝑃:(0...0)⟶𝑉 ∧ (𝑃‘0) = 𝑁) → ∅(𝑁(𝑉 TrailOn 𝐸)𝑁)𝑃))

Theorem2trllemF 26079 Lemma 5 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(((𝐸𝐼) = {𝑋, 𝑌} ∧ 𝑌𝑉) → 𝐼 ∈ dom 𝐸)

Theorem2trllemA 26080 Lemma 1 for constr2trl 26129. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (#‘𝐹) = 2

Theorem2trllemB 26081 Lemma 2 for constr2trl 26129. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (0..^(#‘𝐹)) = {0, 1}

Theorem2trllemH 26082 Lemma 3 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (((𝑉𝑋𝐸𝑌𝐵𝑉) ∧ ((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))⟶dom 𝐸)

Theorem2trllemE 26083 Lemma 4 for constr2trl 26129. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
(𝐼𝑈𝐽𝑊)    &   𝐹 = {⟨0, 𝐼⟩, ⟨1, 𝐽⟩}       (((𝑉𝑋𝐸𝑌𝐵𝑉) ∧ 𝐼𝐽 ∧ ((𝐸𝐼) = {𝐴, 𝐵} ∧ (𝐸𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)

Theorem2wlklemA 26084 Lemma for constr2wlk 26128. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐴𝑉 → (𝑃‘0) = 𝐴)

Theorem2wlklemB 26085 Lemma for constr2wlk 26128. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐵𝑉 → (𝑃‘1) = 𝐵)

Theorem2wlklemC 26086 Lemma for constr2wlk 26128. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       (𝐶𝑉 → (𝑃‘2) = 𝐶)

Theorem2trllemD 26087 Lemma 4 for constr2trl 26129. (Contributed by Alexander van der Vekens, 5-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃 Fn {0, 1, 2})

Theorem2trllemG 26088 Lemma 7 for constr2trl 26129. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
𝑃 = {⟨0, 𝐴⟩, ⟨1, 𝐵⟩, ⟨2, 𝐶⟩}       ((𝐴𝑉𝐵𝑉𝐶𝑉) → 𝑃:(0...2)⟶𝑉)

Theoremwlkntrllem1 26089 Lemma 1 for wlkntrl 26092: F is a word over {0}, the domain of E. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       𝐹 ∈ Word dom 𝐸

Theoremwlkntrllem2 26090* Lemma 2 for wlkntrl 26092: The values of E after F are edges between two vertices enumerated by P. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹𝑘)) = {(𝑃𝑘), (𝑃‘(𝑘 + 1))}

Theoremwlkntrllem3 26091* Lemma 3 for wlkntrl 26092: F is not injective. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}        ¬ Fun 𝐹

Theoremwlkntrl 26092* A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one edge to the other is a walk, but not a trail. Notice that 𝑉, 𝐸 is a simple graph (without loops) only if 𝑥𝑦. (Contributed by Alexander van der Vekens, 22-Oct-2017.)
𝑉 = {𝑥, 𝑦}    &   𝐸 = {⟨0, {𝑥, 𝑦}⟩}    &   𝐹 = {⟨0, 0⟩, ⟨1, 0⟩}    &   𝑃 = {⟨0, 𝑥⟩, ⟨1, 𝑦⟩, ⟨2, 𝑥⟩}       (𝐹(𝑉 Walks 𝐸)𝑃 ∧ ¬ 𝐹(𝑉 Trails 𝐸)𝑃)

Theoremusgrwlknloop 26093* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
((𝑉 USGrph 𝐸𝐹(𝑉 Walks 𝐸)𝑃) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝑃𝑘) ≠ (𝑃‘(𝑘 + 1)))

Theorem2wlklem 26094* Lemma for is2wlk 26095 and 2wlklemA 26084. (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)}))

Theoremis2wlk 26095 Properties of a pair of functions to be a walk of length 2 (in an undirected graph). (Contributed by Alexander van der Vekens, 16-Feb-2018.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → ((𝐹(𝑉 Walks 𝐸)𝑃 ∧ (#‘𝐹) = 2) ↔ (𝐹:(0..^2)⟶dom 𝐸𝑃:(0...2)⟶𝑉 ∧ ((𝐸‘(𝐹‘0)) = {(𝑃‘0), (𝑃‘1)} ∧ (𝐸‘(𝐹‘1)) = {(𝑃‘1), (𝑃‘2)}))))

17.1.5.2  Paths and simple paths

Theorempths 26096* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 Paths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun (𝑝 ↾ (1..^(#‘𝑓))) ∧ ((𝑝 “ {0, (#‘𝑓)}) ∩ (𝑝 “ (1..^(#‘𝑓)))) = ∅)})

Theoremspths 26097* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
((𝑉𝑋𝐸𝑌) → (𝑉 SPaths 𝐸) = {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑉 Trails 𝐸)𝑝 ∧ Fun 𝑝)})

Theoremispth 26098 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 Paths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun (𝑃 ↾ (1..^(#‘𝐹))) ∧ ((𝑃 “ {0, (#‘𝐹)}) ∩ (𝑃 “ (1..^(#‘𝐹)))) = ∅)))

Theoremisspth 26099 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ (𝐹𝑊𝑃𝑍)) → (𝐹(𝑉 SPaths 𝐸)𝑃 ↔ (𝐹(𝑉 Trails 𝐸)𝑃 ∧ Fun 𝑃)))

Theorem0pth 26100 A pair of an empty set (of edges) and a second set (of vertices) is a path if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.)
(((𝑉𝑋𝐸𝑌) ∧ 𝑃𝑍) → (∅(𝑉 Paths 𝐸)𝑃𝑃:(0...0)⟶𝑉))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42360
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