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Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcusgrasizeindslem2 24601* Lemma 2 for cusgrasizeinds 24603. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( dom  F  i^i  { x  e.  dom  E  |  N  e.  ( E `  x ) }
 )  =  (/)
 
Theoremcusgrasizeindslem3 24602* Lemma 3 for cusgrasizeinds 24603. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  F  =  ( E  |`  { x  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # `
  { x  e. 
 dom  E  |  N  e.  ( E `  x ) } )  =  ( ( # `  V )  -  1 ) )
 
Theoremcusgrasizeinds 24603* Part 1 of induction step in cusgrasize 24605. 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  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( # `
  E )  =  ( ( ( # `  V )  -  1
 )  +  ( # `  F ) ) )
 
Theoremcusgrasize2inds 24604* Induction step in cusgrasize 24605. 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  e.  dom  E  |  N  e/  ( E `  x ) }
 )   =>    |-  ( Y  e.  NN0  ->  ( ( V ComplUSGrph  E  /\  ( # `  V )  =  Y  /\  N  e.  V )  ->  (
 ( # `  F )  =  ( ( # `  ( V  \  { N } ) )  _C  2 )  ->  ( # `  E )  =  ( ( # `  V )  _C  2 ) ) ) )
 
Theoremcusgrasize 24605 The size of a finite complete simple graph with  n vertices ( n  e.  NN0) is  ( n  _C  2 ) ("
n choose 2") resp.  ( (
( n  -  1 ) * n )  /  2 ). (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  ( ( V ComplUSGrph  E  /\  V  e.  Fin )  ->  ( # `  E )  =  ( ( # `  V )  _C  2
 ) )
 
Theoremcusgrafilem1 24606* Lemma 1 for cusgrafi 24609. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   =>    |-  (
 ( V ComplUSGrph  E  /\  N  e.  V )  ->  P  C_ 
 ran  E )
 
Theoremcusgrafilem2 24607* Lemma 2 for cusgrafi 24609. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( ( V  e.  W  /\  N  e.  V )  ->  F : ( V  \  { N } ) -1-1-onto-> P )
 
Theoremcusgrafilem3 24608* Lemma 3 for cusgrafi 24609. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  P  =  { x  e.  ~P V  |  E. a  e.  V  (
 a  =/=  N  /\  x  =  { a ,  N } ) }   &    |-  F  =  ( x  e.  ( V  \  { N }
 )  |->  { x ,  N } )   =>    |-  ( ( V  e.  W  /\  N  e.  V )  ->  ( -.  V  e.  Fin  ->  -.  P  e.  Fin ) )
 
Theoremcusgrafi 24609 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  E  /\  E  e.  Fin )  ->  V  e.  Fin )
 
Theoremusgrasscusgra 24610* An undirected simple graph is a subgraph of a complete simple graph. (Contributed by Alexander van der Vekens, 11-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  A. e  e.  ran  E E. f  e.  ran  F  e  =  f )
 
Theoremsizeusglecusglem1 24611 Lemma 1 for sizeusglecusg 24613. (Contributed by Alexander van der Vekens, 12-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  (  _I  |`  ran  E ) : ran  E -1-1-> ran  F )
 
Theoremsizeusglecusglem2 24612 Lemma 2 for sizeusglecusg 24613. (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F  /\  F  e.  Fin )  ->  E  e.  Fin )
 
Theoremsizeusglecusg 24613 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.)
 |-  ( ( V USGrph  E  /\  V ComplUSGrph  F )  ->  ( # `
  E )  <_  ( # `  F ) )
 
Theoremusgramaxsize 24614 The maximum size of an undirected simple graph with  n vertices ( n  e.  NN0) is  ( ( ( n  - 
1 ) * n )  /  2 ). (Contributed by Alexander van der Vekens, 13-Jan-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( # `  E )  <_  ( ( # `  V )  _C  2
 ) )
 
16.1.4.3  Universal vertices
 
Theoremisuvtx 24615* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n } ) { k ,  n }  e.  ran  E } )
 
Theoremuvtxel 24616* A universal vertex, i.e. an element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
 
Theoremuvtxisvtx 24617 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e.  ( V UnivVertex  E )  ->  N  e.  V )
 
Theoremuvtx0 24618 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( (/) UnivVertex  E )  =  (/)
 
Theoremuvtx01vtx 24619* 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  (/) )  =/=  (/) 
 <-> 
 E. x  V  =  { x } )
 
Theoremuvtxnbgra 24620 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N }
 ) )
 
Theoremuvtxnm1nbgra 24621 A universal vertex has  n  -  1 neighbors in a graph with  n vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin )  ->  ( N  e.  ( V UnivVertex  E )  ->  ( # `
  ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremuvtxnbgravtx 24622* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  ( ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  A. v  e.  ( V  \  { N }
 ) N  e.  ( <. V ,  E >. Neighbors  v
 ) )
 
Theoremcusgrauvtxb 24623 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 24624 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  E  /\  N  e.  V ) 
 ->  ( N  e.  ( V UnivVertex  E )  <->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } ) ) )
 
16.1.5  Walks, paths and cycles
 
Syntaxcwalk 24625 Extend class notation with Walks (of a graph).
 class Walks
 
Syntaxctrail 24626 Extend class notation with Trails (of a graph).
 class Trails
 
Syntaxcpath 24627 Extend class notation with Paths (of a graph).
 class Paths
 
Syntaxcspath 24628 Extend class notation with Simple Paths (of a graph).
 class SPaths
 
Syntaxcwlkon 24629 Extend class notation with Walks between two vertices (within a graph).
 class WalkOn
 
Syntaxctrlon 24630 Extend class notation with Trails between two vertices (within a graph).
 class TrailOn
 
Syntaxcpthon 24631 Extend class notation with Paths between two vertices (within a graph).
 class PathOn
 
Syntaxcspthon 24632 Extend class notation with simple paths between two vertices (within a graph).
 class SPathOn
 
Syntaxccrct 24633 Extend class notation with Circuits (of a graph).
 class Circuits
 
Syntaxccycl 24634 Extend class notation with Cycles (of a graph).
 class Cycles
 
Definitiondf-wlk 24635* 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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f  e. Word  dom  e  /\  p : ( 0 ... ( # `  f ) ) --> v  /\  A. k  e.  ( 0..^ ( # `  f
 ) ) ( e `
  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }
 ) } )
 
Definitiondf-trail 24636* 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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v Walks 
 e ) p  /\  Fun  `' f ) } )
 
Definitiondf-pth 24637* 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 24737).

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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v Trails 
 e ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Definitiondf-spth 24638* 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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v Trails 
 e ) p  /\  Fun  `' p ) } )
 
Definitiondf-crct 24639* 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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v Trails 
 e ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
Definitiondf-cycl 24640* 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  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( f ( v Paths 
 e ) p  /\  ( p `  0 )  =  ( p `  ( # `  f ) ) ) } )
 
Definitiondf-wlkon 24641* 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  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( v Walks  e ) p 
 /\  ( p `  0 )  =  a  /\  ( p `  ( # `
  f ) )  =  b ) }
 ) )
 
Definitiondf-trlon 24642* 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  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
 ) b ) p 
 /\  f ( v Trails 
 e ) p ) } ) )
 
Definitiondf-pthon 24643* 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  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
 ) b ) p 
 /\  f ( v Paths 
 e ) p ) } ) )
 
Definitiondf-spthon 24644* Define the collection of simple paths with particular endpoints (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |- SPathOn  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { <. f ,  p >.  |  ( f ( a ( v WalkOn  e
 ) b ) p 
 /\  f ( v SPaths 
 e ) p ) } ) )
 
16.1.5.1  Walks and trails
 
Theoremrelwlk 24645 The walks (in an undirected simple graph) are (ordered) pairs. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
 |- 
 Rel  ( V Walks  E )
 
Theoremwlks 24646* The set of walks (in an undirected graph). (Contributed by Alexander van der Vekens, 19-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Walks  E )  =  { <. f ,  p >.  |  (
 f  e. Word  dom  E  /\  p : ( 0 ... ( # `  f
 ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f
 ) ) ( E `
  ( f `  k ) )  =  { ( p `  k ) ,  ( p `  ( k  +  1 ) ) }
 ) } )
 
Theoremiswlk 24647* Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( F ( V Walks  E ) P  <-> 
 ( F  e. Word  dom  E 
 /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theorem2mwlk 24648 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  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V ) )
 
Theoremwlkres 24649* Restrictions of walks (i.e. special kinds of walks, for example paths - see pths 24695) are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.)
 |-  ( f ( V W E ) p 
 ->  f ( V Walks  E ) p )   =>    |-  ( ( V  e.  _V 
 /\  E  e.  _V )  ->  { <. f ,  p >.  |  (
 f ( V W E ) p  /\  ph ) }  e.  _V )
 
Theoremwlkbprop 24650 Basic properties of a walk. (Contributed by Alexander van der Vekens, 31-Oct-2017.)
 |-  ( F ( V Walks  E ) P  ->  ( ( # `  F )  e.  NN0  /\  ( V  e.  _V  /\  E  e.  _V )  /\  ( F  e.  _V  /\  P  e.  _V ) ) )
 
Theoremiswlkg 24651* Generalisation of iswlk 24647: Properties of a pair of functions to be a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( F ( V Walks  E ) P  <-> 
 ( F  e. Word  dom  E 
 /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremwlkcomp 24652* A walk expressed by properties of its components. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
 |-  F  =  ( 1st `  W )   &    |-  P  =  ( 2nd `  W )   =>    |-  (
 ( V  e.  X  /\  E  e.  Y  /\  W  e.  ( S  X.  T ) )  ->  ( W  e.  ( V Walks  E )  <->  ( F  e. Word  dom 
 E  /\  P :
 ( 0 ... ( # `
  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `
  k ) )  =  { ( P `
  k ) ,  ( P `  (
 k  +  1 ) ) } ) ) )
 
Theoremwlkcompim 24653* Implications for the properties of the components of a walk. (Contributed by Alexander van der Vekens, 23-Jun-2018.)
 |-  F  =  ( 1st `  W )   &    |-  P  =  ( 2nd `  W )   =>    |-  ( W  e.  ( V Walks  E )  ->  ( F  e. Word  dom  E  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) )
 
Theoremwlkn0 24654 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 ) P  ->  P  =/=  (/) )
 
Theoremwlkop 24655 A walk (in an undirected simple graph) is an ordered pair. (Contributed by Alexander van der Vekens, 30-Jun-2018.)
 |-  ( W  e.  ( V Walks  E )  ->  W  =  <. ( 1st `  W ) ,  ( 2nd `  W ) >. )
 
Theoremwlkcpr 24656 A walk as class with two components. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( W  e.  ( V Walks  E )  <->  ( 1st `  W ) ( V Walks  E ) ( 2nd `  W ) )
 
Theoremwlkelwrd 24657 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  e.  ( V Walks  E )  ->  (
 ( 1st `  W )  e. Word  dom  E  /\  ( 2nd `  W ) : ( 0 ... ( # `
  ( 1st `  W ) ) ) --> V ) )
 
Theoremedgwlk 24658* The (connected) edges of a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 22-Jul-2018.)
 |-  ( F ( V Walks  E ) P  ->  A. k  e.  ( 0..^ ( # `  F ) ) { ( P `  k ) ,  ( P `  (
 k  +  1 ) ) }  e.  ran  E )
 
Theoremwlklenvm1 24659 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 24660* The set of walks between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )
 )  ->  ( A ( V WalkOn  E ) B )  =  { <. f ,  p >.  |  ( f ( V Walks  E ) p  /\  ( p `
  0 )  =  A  /\  ( p `
  ( # `  f
 ) )  =  B ) } )
 
Theoremiswlkon 24661 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  ( F ( V Walks  E ) P 
 /\  ( P `  0 )  =  A  /\  ( P `  ( # `
  F ) )  =  B ) ) )
 
Theoremwlkonprop 24662 Properties of a walk between two vertices. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V WalkOn  E ) B ) P  ->  ( ( ( V  e.  _V 
 /\  E  e.  _V )  /\  ( F  e.  _V 
 /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V ) )  /\  ( F ( V Walks  E ) P  /\  ( P `
  0 )  =  A  /\  ( P `
  ( # `  F ) )  =  B ) ) )
 
Theoremwlkoniswlk 24663 A walk between to vertices is a walk. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V WalkOn  E ) B ) P  ->  F ( V Walks  E ) P )
 
Theoremwlkonwlk 24664 A walk is a walk between its endpoints. (Contributed by Alexander van der Vekens, 2-Nov-2017.)
 |-  ( F ( V Walks  E ) P  ->  F ( ( P `  0 ) ( V WalkOn  E ) ( P `
  ( # `  F ) ) ) P )
 
Theoremtrls 24665* The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Trails  E )  =  { <. f ,  p >.  |  (
 ( f  e. Word  dom  E 
 /\  Fun  `' f
 )  /\  p :
 ( 0 ... ( # `
  f ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  f ) ) ( E `  ( f `
  k ) )  =  { ( p `
  k ) ,  ( p `  (
 k  +  1 ) ) } ) }
 )
 
Theoremistrl 24666* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( F ( V Trails  E ) P  <-> 
 ( ( F  e. Word  dom 
 E  /\  Fun  `' F )  /\  P : ( 0 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremistrl2 24667* Properties of a pair of functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( F ( V Trails  E ) P  <-> 
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 ) ) )
 
Theoremtrliswlk 24668 A trail is a walk (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( F ( V Trails  E ) P  ->  F ( V Walks  E ) P )
 
Theoremtrlon 24669* The set of trails between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 4-Nov-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V )
 )  ->  ( A ( V TrailOn  E ) B )  =  { <. f ,  p >.  |  ( f ( A ( V WalkOn  E ) B ) p  /\  f ( V Trails  E ) p ) } )
 
Theoremistrlon 24670 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( F ( A ( V TrailOn  E ) B ) P  <->  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonprop 24671 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  e.  _V 
 /\  E  e.  _V )  /\  ( F  e.  _V 
 /\  P  e.  _V )  /\  ( A  e.  V  /\  B  e.  V ) )  /\  ( F ( A ( V WalkOn  E ) B ) P  /\  F ( V Trails  E ) P ) ) )
 
Theoremtrlonistrl 24672 A trail between to vertices is a trail. (Contributed by Alexander van der Vekens, 12-Dec-2017.)
 |-  ( F ( A ( V TrailOn  E ) B ) P  ->  F ( V Trails  E ) P )
 
Theoremtrlonwlkon 24673 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 ) P  ->  F ( A ( V WalkOn  E ) B ) P )
 
Theorem0wlk 24674 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Walks  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0trl 24675 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Trails  E ) P 
 <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0wlkon 24676 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  (/) ( N ( V WalkOn  E ) N ) P ) )
 
Theorem0trlon 24677 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  N  e.  V )  ->  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  (/) ( N ( V TrailOn  E ) N ) P ) )
 
Theorem2trllemF 24678 Lemma 5 for constr2trl 24728. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( ( ( E `
  I )  =  { X ,  Y }  /\  Y  e.  V )  ->  I  e.  dom  E )
 
Theorem2trllemA 24679 Lemma 1 for constr2trl 24728. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( # `  F )  =  2
 
Theorem2trllemB 24680 Lemma 2 for constr2trl 24728. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( 0..^ ( # `  F ) )  =  { 0 ,  1 }
 
Theorem2trllemH 24681 Lemma 3 for constr2trl 24728. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `
  J )  =  { B ,  C } ) )  ->  F : ( 0..^ ( # `  F ) ) --> dom  E )
 
Theorem2trllemE 24682 Lemma 4 for constr2trl 24728. (Contributed by Alexander van der Vekens, 31-Jan-2018.)
 |-  ( I  e.  U  /\  J  e.  W )   &    |-  F  =  { <. 0 ,  I >. ,  <. 1 ,  J >. }   =>    |-  ( ( ( V  e.  X  /\  E  e.  Y  /\  B  e.  V )  /\  I  =/= 
 J  /\  ( ( E `  I )  =  { A ,  B }  /\  ( E `  J )  =  { B ,  C }
 ) )  ->  F : ( 0..^ ( # `  F ) )
 -1-1-> dom  E )
 
Theorem2wlklemA 24683 Lemma for constr2wlk 24727. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( A  e.  V  ->  ( P `  0 )  =  A )
 
Theorem2wlklemB 24684 Lemma for constr2wlk 24727. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( B  e.  V  ->  ( P `  1 )  =  B )
 
Theorem2wlklemC 24685 Lemma for constr2wlk 24727. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( C  e.  V  ->  ( P `  2 )  =  C )
 
Theorem2trllemD 24686 Lemma 4 for constr2trl 24728. (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 >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P  Fn  { 0 ,  1 ,  2 } )
 
Theorem2trllemG 24687 Lemma 7 for constr2trl 24728. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  P :
 ( 0 ... 2
 ) --> V )
 
Theoremwlkntrllem1 24688 Lemma 1 for wlkntrl 24691: 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  e. Word  dom  E
 
Theoremwlkntrllem2 24689* Lemma 2 for wlkntrl 24691: 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 >. }   =>    |-  A. k  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 
Theoremwlkntrllem3 24690* Lemma 3 for wlkntrl 24691: 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 24691* 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  x  =/=  y. (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 )
 
Theoremusgrnloop 24692* In an undirected simple graph, each walk has no loops! (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  ( ( V USGrph  E  /\  F ( V Walks  E ) P )  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `  k )  =/=  ( P `  ( k  +  1
 ) ) )
 
Theorem2wlklem 24693* Lemma for is2wlk 24694 and 2wlklemA 24683. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  ( A. k  e. 
 { 0 ,  1 }  ( E `  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  (
 k  +  1 ) ) }  <->  ( ( E `
  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P `  1 ) }  /\  ( E `  ( F `  1 ) )  =  { ( P `
  1 ) ,  ( P `  2
 ) } ) )
 
Theoremis2wlk 24694 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( ( F ( V Walks  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 ) --> dom  E  /\  P : ( 0 ... 2 ) --> V  /\  ( ( E `  ( F `  0 ) )  =  { ( P `  0 ) ,  ( P `  1
 ) }  /\  ( E `  ( F `  1 ) )  =  { ( P `  1 ) ,  ( P `  2 ) }
 ) ) ) )
 
16.1.5.2  Paths and simple paths
 
Theorempths 24695* The set of paths (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V Paths  E )  =  { <. f ,  p >.  |  (
 f ( V Trails  E ) p  /\  Fun  `' ( p  |`  ( 1..^ ( # `  f
 ) ) )  /\  ( ( p " { 0 ,  ( # `
  f ) }
 )  i^i  ( p " ( 1..^ ( # `  f ) ) ) )  =  (/) ) }
 )
 
Theoremspths 24696* The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V SPaths  E )  =  { <. f ,  p >.  |  (
 f ( V Trails  E ) p  /\  Fun  `' p ) } )
 
Theoremispth 24697 Properties of a pair of functions to be a path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( F ( V Paths  E ) P  <-> 
 ( F ( V Trails  E ) P  /\  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )  /\  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) ) ) )
 
Theoremisspth 24698 Properties of a pair of functions to be a simple path (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  ( F  e.  W  /\  P  e.  Z )
 )  ->  ( F ( V SPaths  E ) P  <-> 
 ( F ( V Trails  E ) P  /\  Fun  `' P ) ) )
 
Theorem0pth 24699 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V Paths  E ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0spth 24700 A pair of an empty set (of edges) and a second set (of vertices) is a simple 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.)
 |-  ( ( ( V  e.  X  /\  E  e.  Y )  /\  P  e.  Z )  ->  ( (/) ( V SPaths  E ) P 
 <->  P : ( 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 38201-38213
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