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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcrctcshtrl 40001* Cyclically shifting the indices of a circuit  <. F ,  P >. results in a trail  <. H ,  Q >.. (Contributed by AV, 14-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ph  ->  H (TrailS `  G ) Q )
 
Theoremcrctcsh 40002* Cyclically shifting the indices of a circuit  <. F ,  P >. results in a circuit  <. H ,  Q >.. (Contributed by AV, 10-Mar-2021.)
 |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  F (CircuitS `  G ) P )   &    |-  N  =  ( # `  F )   &    |-  ( ph  ->  S  e.  (
 0..^ N ) )   &    |-  H  =  ( F cyclShift  S )   &    |-  Q  =  ( x  e.  ( 0
 ... N )  |->  if ( x  <_  ( N  -  S ) ,  ( P `  ( x  +  S )
 ) ,  ( P `
  ( ( x  +  S )  -  N ) ) ) )   =>    |-  ( ph  ->  H (CircuitS `  G ) Q )
 
21.33.8.20  Examples for walks, trails and paths
 
Theorem0ewlk 40003 The empty set (empty sequence of edges) is an s-walk of edges for all s. (Contributed by AV, 4-Jan-2021.)
 |-  (
 ( G  e.  _V  /\  S  e. NN0* )  ->  (/) 
 e.  ( G EdgWalks  S ) )
 
Theorem1ewlk 40004 A sequence of 1 edge is an s-walk of edges for all s. (Contributed by AV, 5-Jan-2021.)
 |-  (
 ( G  e.  _V  /\  S  e. NN0*  /\  I  e. 
 dom  (iEdg `  G )
 )  ->  <" I ">  e.  ( G EdgWalks  S ) )
 
Theorem01wlk 40005 A pair of an empty set (of edges) and a second set (of vertices) is a walk iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 3-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  U  /\  P  e.  Z ) 
 ->  ( (/) (1Walks `  G ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theoremis01wlklem 40006 Lemma for is01wlk 40007 and is0Trl 40012. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P  =  { <. 0 ,  N >. } 
 /\  N  e.  V )  ->  P : ( 0 ... 0 ) --> V )
 
Theoremis01wlk 40007 A pair of an empty set (of edges) and a sequence of one vertex is a walk (of length 0). (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P  =  { <. 0 ,  N >. } 
 /\  N  e.  V )  ->  (/) (1Walks `  G ) P )
 
Theorem0wlkOnlem1 40008 Lemma 1 for 0wlkOn 40010 and 0TrlOn 40013. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  ( N  e.  V  /\  N  e.  V )
 )
 
Theorem0wlkOnlem2 40009 Lemma 2 for 0wlkOn 40010 and 0TrlOn 40013. (Contributed by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  P  e.  ( V  ^pm  (
 0 ... 0 ) ) )
 
Theorem0wlkOn 40010 A walk of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 3-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  (/) ( N (WalksOn `  G ) N ) P )
 
Theorem0Trl 40011 A pair of an empty set (of edges) and a second set (of vertices) is a trail iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 7-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  U  /\  P  e.  Z ) 
 ->  ( (/) (TrailS `  G ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theoremis0Trl 40012 A pair of an empty set (of edges) and a sequence of one vertex is a trail (of length 0). (Contributed by AV, 7-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P  =  { <. 0 ,  N >. } 
 /\  N  e.  V )  ->  (/) (TrailS `  G ) P )
 
Theorem0TrlOn 40013 A trail of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 2-Dec-2017.) (Revised by AV, 8-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  (/) ( N (TrailsOn `  G ) N ) P )
 
Theorem0pth-av 40014 A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 19-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  P  e.  Z ) 
 ->  ( (/) (PathS `  G ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0spth-av 40015 A pair of an empty set (of edges) and a second set (of vertices) is a simple path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 18-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  W  /\  P  e.  Z ) 
 ->  ( (/) (SPathS `  G ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0pthon-av 40016 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  V  ->  ( ( P : ( 0 ... 0 ) --> V  /\  ( P `
  0 )  =  N )  ->  (/) ( N (PathsOn `  G ) N ) P ) )
 
Theorem0pthon1-av 40017 A path of length 0 from a vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 20-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  V  ->  (/) ( N (PathsOn `  G ) N ) { <. 0 ,  N >. } )
 
Theorem0pthonv-av 40018* For each vertex there is a path of length 0 from the vertex to itself. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 21-Jan-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  ( N  e.  V  ->  E. f E. p  f ( N (PathsOn `  G ) N ) p )
 
Theorem0clWlk 40019 A pair of an empty set (of edges) and a second set (of vertices) is a closed walk if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 15-Mar-2018.) (Revised by AV, 17-Feb-2021.)
 |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e.  X  /\  P  e.  Z ) 
 ->  ( (/) (ClWalkS `  G ) P  <->  P : ( 0
 ... 0 ) --> V ) )
 
Theorem0clwlk0 40020 There is no closed walk in the empty set (i.e. the null graph). (Contributed by Alexander van der Vekens, 2-Sep-2018.) (Revised by AV, 5-Mar-2021.)
 |-  (ClWalkS `  (/) )  =  (/)
 
Theorem0Crct 40021 A pair of an empty set (of edges) and a second set (of vertices) is a circuit if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  P  e.  Z ) 
 ->  ( (/) (CircuitS `  G ) P 
 <->  P : ( 0
 ... 0 ) --> (Vtx `  G ) ) )
 
Theorem0Cycl 40022 A pair of an empty set (of edges) and a second set (of vertices) is a cycle if and only if the second set contains exactly one vertex (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017.) (Revised by AV, 31-Jan-2021.)
 |-  (
 ( G  e.  W  /\  P  e.  Z ) 
 ->  ( (/) (CycleS `  G ) P  <->  P : ( 0
 ... 0 ) --> (Vtx `  G ) ) )
 
Theorem1pthdlem1 40023 Lemma 1 for 1pthd 40031. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   =>    |- 
 Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) )
 
Theorem1pthdlem2 40024 Lemma 2 for 1pthd 40031. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   =>    |-  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/)
 
Theorem11wlkdlem1 40025 Lemma 1 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   =>    |-  ( ph  ->  P : ( 0 ... ( # `  F ) ) --> V )
 
Theorem11wlkdlem2 40026 Lemma 2 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   =>    |-  ( ph  ->  X  e.  ( I `  J ) )
 
Theorem11wlkdlem3 40027 Lemma 3 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   =>    |-  ( ph  ->  F  e. Word  dom 
 I )
 
Theorem11wlkdlem4 40028* Lemma 4 for 11wlkd 40029. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F ) )if- ( ( P `  k
 )  =  ( P `
  ( k  +  1 ) ) ,  ( I `  ( F `  k ) )  =  { ( P `
  k ) } ,  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) ) )
 
Theorem11wlkd 40029 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F (1Walks `  G ) P )
 
Theorem1trld 40030 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F (TrailS `  G ) P )
 
Theorem1pthd 40031 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 3-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F (PathS `  G ) P )
 
Theorem1pthond 40032 In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Revised by AV, 23-Mar-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  V )   &    |-  ( ph  ->  Y  e.  V )   &    |-  (
 ( ph  /\  X  =  Y )  ->  ( I `
  J )  =  { X } )   &    |-  (
 ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  ( I `  J ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F ( X (PathsOn `  G ) Y ) P )
 
Theoremupgr11wlkdlem1 40033 Lemma 1 for upgr11wlkd 40035. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   =>    |-  ( ( ph  /\  X  =  Y )  ->  (
 (iEdg `  G ) `  J )  =  { X } )
 
Theoremupgr11wlkdlem2 40034 Lemma 2 for upgr11wlkd 40035. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   =>    |-  ( ( ph  /\  X  =/=  Y )  ->  { X ,  Y }  C_  (
 (iEdg `  G ) `  J ) )
 
Theoremupgr11wlkd 40035 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a walk. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   &    |-  ( ph  ->  G  e. UPGraph  )   =>    |-  ( ph  ->  F (1Walks `  G ) P )
 
Theoremupgr1trld 40036 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   &    |-  ( ph  ->  G  e. UPGraph  )   =>    |-  ( ph  ->  F (TrailS `  G ) P )
 
Theoremupgr1pthd 40037 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   &    |-  ( ph  ->  G  e. UPGraph  )   =>    |-  ( ph  ->  F (PathS `  G ) P )
 
Theoremupgr1pthond 40038 In a pseudograph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a path from one of these vertices to the other vertex. The two vertices need not be distinct (in the case of a loop) - in this case, however, the path is not a simple path. (Contributed by AV, 22-Jan-2021.)
 |-  P  =  <" X Y ">   &    |-  F  =  <" J ">   &    |-  ( ph  ->  X  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  Y  e.  (Vtx `  G )
 )   &    |-  ( ph  ->  (
 (iEdg `  G ) `  J )  =  { X ,  Y }
 )   &    |-  ( ph  ->  G  e. UPGraph  )   =>    |-  ( ph  ->  F ( X (PathsOn `  G ) Y ) P )
 
Theoremlppthon 40039 A loop (which is an edge at index 
J) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  J  e.  dom  I  /\  ( I `  J )  =  { A } )  -> 
 <" J "> ( A (PathsOn `  G ) A ) <" A A "> )
 
Theoremlp1cycl 40040 A loop (which is an edge at index 
J) induces a cycle of length 1 in a hypergraph. (Contributed by AV, 2-Feb-2021.)
 |-  I  =  (iEdg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  J  e.  dom  I  /\  ( I `  J )  =  { A } )  -> 
 <" J "> (CycleS `  G ) <" A A "> )
 
Theorem1pthon2v-av 40041* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V )  /\  E. e  e.  E  { A ,  B }  C_  e )  ->  E. f E. p  f ( A (PathsOn `  G ) B ) p )
 
Theorem1pthon2ve 40042* For each pair of adjacent vertices there is a path of length 1 from one vertex to the other in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 22-Jan-2021.) (Proof shortened by AV, 15-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V )  /\  { A ,  B }  e.  E )  ->  E. f E. p  f ( A (PathsOn `  G ) B ) p )
 
Theorem1wlk2v2elem1 40043 Lemma 1 for 1wlk2v2e 40045: 
F is a length 2 word of over  { 0 }, the domain of the singleton word  I. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
 |-  I  =  <" { X ,  Y } ">   &    |-  F  =  <" 0 0 ">   =>    |-  F  e. Word  dom  I
 
Theorem1wlk2v2elem2 40044* Lemma 2 for 1wlk2v2e 40045: The values of  I after 
F are edges between two vertices enumerated by  P. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 9-Jan-2021.)
 |-  I  =  <" { X ,  Y } ">   &    |-  F  =  <" 0 0 ">   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  P  =  <" X Y X ">   =>    |-  A. k  e.  ( 0..^ ( # `  F ) ) ( I `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 
Theorem1wlk2v2e 40045 In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk. Notice that  G is a simple graph (without loops) only if  X  =/=  Y. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
 |-  I  =  <" { X ,  Y } ">   &    |-  F  =  <" 0 0 ">   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  P  =  <" X Y X ">   &    |-  G  =  <. { X ,  Y } ,  I >.   =>    |-  F (1Walks `  G ) P
 
Theoremntrl2v2e 40046 A walk which is not a trail: In a graph with two vertices and one edge connecting these two vertices, to go from one vertex to the other and back to the first vertex via the same/only edge is a walk, see 1wlk2v2e 40045, but not a trail. Notice that  G is a simple graph (without loops) only if  X  =/=  Y. (Contributed by Alexander van der Vekens, 22-Oct-2017.) (Revised by AV, 8-Jan-2021.)
 |-  I  =  <" { X ,  Y } ">   &    |-  F  =  <" 0 0 ">   &    |-  X  e.  _V   &    |-  Y  e.  _V   &    |-  P  =  <" X Y X ">   &    |-  G  =  <. { X ,  Y } ,  I >.   =>    |-  -.  F (TrailS `  G ) P
 
Theorem21wlkdlem1 40047 Lemma 1 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   =>    |-  ( # `  P )  =  ( ( # `
  F )  +  1 )
 
Theorem21wlkdlem2 40048 Lemma 2 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   =>    |-  ( 0..^ ( # `  F ) )  =  { 0 ,  1 }
 
Theorem21wlkdlem3 40049 Lemma 3 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   =>    |-  ( ph  ->  (
 ( P `  0
 )  =  A  /\  ( P `  1 )  =  B  /\  ( P `  2 )  =  C ) )
 
Theorem21wlkdlem4 40050* Lemma 4 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   =>    |-  ( ph  ->  A. k  e.  ( 0 ... ( # `
  F ) ) ( P `  k
 )  e.  V )
 
Theorem21wlkdlem5 40051* Lemma 5 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `
  k )  =/=  ( P `  (
 k  +  1 ) ) )
 
Theorem2pthdlem1 40052* Lemma 1 for 2pthd 40062. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ ( # `  F ) ) ( k  =/=  j  ->  ( P `  k )  =/=  ( P `  j ) ) )
 
Theorem21wlkdlem6 40053 Lemma 6 for 21wlkd 40058. (Contributed by AV, 23-Jan-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   =>    |-  ( ph  ->  ( B  e.  ( I `
  J )  /\  B  e.  ( I `  K ) ) )
 
Theorem21wlkdlem7 40054 Lemma 7 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   =>    |-  ( ph  ->  ( J  e.  _V  /\  K  e.  _V )
 )
 
Theorem21wlkdlem8 40055 Lemma 8 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   =>    |-  ( ph  ->  ( ( F `  0
 )  =  J  /\  ( F `  1 )  =  K ) )
 
Theorem21wlkdlem9 40056 Lemma 9 for 21wlkd 40058. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   =>    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  ( F `  0 ) )  /\  { B ,  C }  C_  ( I `  ( F `  1 ) ) ) )
 
Theorem21wlkdlem10 40057* Lemma 10 for 31wlkd 40084. (Contributed by AV, 14-Feb-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F ) ) { ( P `  k ) ,  ( P `  (
 k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) )
 
Theorem21wlkd 40058 Construction of a walk from two given edges in a graph. (Contributed by Alexander van der Vekens, 5-Feb-2018.) (Revised by AV, 23-Jan-2021.) (Proof shortened by AV, 14-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F (1Walks `  G ) P )
 
Theorem21wlkond 40059 A 1-walk of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F ( A (WalksOn `  G ) C ) P )
 
Theorem2trld 40060 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  J  =/=  K )   =>    |-  ( ph  ->  F (TrailS `  G ) P )
 
Theorem2trlond 40061 A trail of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  J  =/=  K )   =>    |-  ( ph  ->  F ( A (TrailsOn `  G ) C ) P )
 
Theorem2pthd 40062 A path of length 2 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  J  =/=  K )   =>    |-  ( ph  ->  F (PathS `  G ) P )
 
Theorem2spthd 40063 A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 24-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  J  =/=  K )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  F (SPathS `  G ) P )
 
Theorem2pthond 40064 A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 24-Jan-2021.) (Proof shortened by AV, 30-Jan-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C ">   &    |-  F  =  <" J K ">   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )   &    |-  ( ph  ->  ( A  =/=  B  /\  B  =/=  C ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  J  =/=  K )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  F ( A (SPathsOn `  G ) C ) P )
 
Theorem2pthon3v-av 40065* For a vertex adjacent to two other vertices there is a simple path of length 2 between these other vertices in a hypergraph. (Contributed by Alexander van der Vekens, 4-Dec-2017.) (Revised by AV, 24-Jan-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( ( G  e. UHGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  /\  ( A  =/=  B  /\  A  =/=  C 
 /\  B  =/=  C )  /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E )
 )  ->  E. f E. p ( f ( A (SPathsOn `  G ) C ) p  /\  ( # `  f )  =  2 ) )
 
Theoremumgr2adedgwlklem 40066 Lemma for umgr2adedgwlk 40067, umgr2adedgspth 40070, etc. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  ->  ( ( A  =/=  B  /\  B  =/=  C )  /\  ( A  e.  (Vtx `  G )  /\  B  e.  (Vtx `  G )  /\  C  e.  (Vtx `  G )
 ) ) )
 
Theoremumgr2adedgwlk 40067 In a multigraph, two adjacent edges form a walk of length 2. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 29-Jan-2021.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  <" J K ">   &    |-  P  =  <" A B C ">   &    |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E ) )   &    |-  ( ph  ->  ( I `  J )  =  { A ,  B }
 )   &    |-  ( ph  ->  ( I `  K )  =  { B ,  C } )   =>    |-  ( ph  ->  ( F (1Walks `  G ) P  /\  ( # `  F )  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1 )  /\  C  =  ( P `  2
 ) ) ) )
 
Theoremumgr2adedgwlkon 40068 In a multigraph, two adjacent edges form a walk between two vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  <" J K ">   &    |-  P  =  <" A B C ">   &    |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E ) )   &    |-  ( ph  ->  ( I `  J )  =  { A ,  B }
 )   &    |-  ( ph  ->  ( I `  K )  =  { B ,  C } )   =>    |-  ( ph  ->  F ( A (WalksOn `  G ) C ) P )
 
Theoremumgr2adedgwlkonALT 40069 Alternate proof for umgr2adedgwlkon 40068, using umgr2adedgwlk 40067, but with a much longer proof! In a multigraph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  <" J K ">   &    |-  P  =  <" A B C ">   &    |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E ) )   &    |-  ( ph  ->  ( I `  J )  =  { A ,  B }
 )   &    |-  ( ph  ->  ( I `  K )  =  { B ,  C } )   =>    |-  ( ph  ->  F ( A (WalksOn `  G ) C ) P )
 
Theoremumgr2adedgspth 40070 In a multigraph, two adjacent edges with different endvertices form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.) (Revised by AV, 29-Jan-2021.)
 |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   &    |-  F  =  <" J K ">   &    |-  P  =  <" A B C ">   &    |-  ( ph  ->  G  e. UMGraph  )   &    |-  ( ph  ->  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E ) )   &    |-  ( ph  ->  ( I `  J )  =  { A ,  B }
 )   &    |-  ( ph  ->  ( I `  K )  =  { B ,  C } )   &    |-  ( ph  ->  A  =/=  C )   =>    |-  ( ph  ->  F (SPathS `  G ) P )
 
Theoremumgr2wlk 40071* In a multigraph, there is a 1-walk of length 2 for each pair of adjacent edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  ->  E. f E. p ( f (1Walks `  G ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) )
 
Theoremumgr2wlkon 40072* For each pair of adjacent edges in a multigraph, there is a 1-walk of length 2 between the not common vertices of the edges. (Contributed by Alexander van der Vekens, 18-Feb-2018.) (Revised by AV, 30-Jan-2021.)
 |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  { A ,  B }  e.  E  /\  { B ,  C }  e.  E )  ->  E. f E. p  f ( A (WalksOn `  G ) C ) p )
 
Theorem31wlkdlem1 40073 Lemma 1 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   =>    |-  ( # `  P )  =  ( ( # `
  F )  +  1 )
 
Theorem31wlkdlem2 40074 Lemma 2 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   =>    |-  ( 0..^ ( # `  F ) )  =  { 0 ,  1 ,  2 }
 
Theorem31wlkdlem3 40075 Lemma 3 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Revised by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   =>    |-  ( ph  ->  ( ( ( P `  0 )  =  A  /\  ( P `  1
 )  =  B ) 
 /\  ( ( P `
  2 )  =  C  /\  ( P `
  3 )  =  D ) ) )
 
Theorem31wlkdlem4 40076* Lemma 4 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   =>    |-  ( ph  ->  A. k  e.  ( 0
 ... ( # `  F ) ) ( P `
  k )  e.  V )
 
Theorem31wlkdlem5 40077* Lemma 5 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 11-Nov-2017.) (Revised by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F ) ) ( P `
  k )  =/=  ( P `  (
 k  +  1 ) ) )
 
Theorem3pthdlem1 40078* Lemma 1 for 3pthd 40088. (Contributed by AV, 9-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  P ) ) A. j  e.  ( 1..^ ( # `  F ) ) ( k  =/=  j  ->  ( P `  k )  =/=  ( P `  j ) ) )
 
Theorem31wlkdlem6 40079 Lemma 6 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   =>    |-  ( ph  ->  ( A  e.  ( I `  J )  /\  B  e.  ( I `  K )  /\  C  e.  ( I `  L ) ) )
 
Theorem31wlkdlem7 40080 Lemma 7 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   =>    |-  ( ph  ->  ( J  e.  _V  /\  K  e.  _V  /\  L  e.  _V ) )
 
Theorem31wlkdlem8 40081 Lemma 8 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   =>    |-  ( ph  ->  (
 ( F `  0
 )  =  J  /\  ( F `  1 )  =  K  /\  ( F `  2 )  =  L ) )
 
Theorem31wlkdlem9 40082 Lemma 9 for 31wlkd 40084. (Contributed by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   =>    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  ( F `  0 ) ) 
 /\  { B ,  C }  C_  ( I `  ( F `  1 ) )  /\  { C ,  D }  C_  ( I `  ( F `  2 ) ) ) )
 
Theorem31wlkdlem10 40083* Lemma 10 for 31wlkd 40084. (Contributed by Alexander van der Vekens, 12-Nov-2017.) (Revised by AV, 7-Feb-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   =>    |-  ( ph  ->  A. k  e.  ( 0..^ ( # `  F ) ) {
 ( P `  k
 ) ,  ( P `
  ( k  +  1 ) ) }  C_  ( I `  ( F `  k ) ) )
 
Theorem31wlkd 40084 Construction of a walk from two given edges in a graph. (Contributed by AV, 7-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F (1Walks `  G ) P )
 
Theorem31wlkond 40085 A 1-walk of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  ( ph  ->  F ( A (WalksOn `  G ) D ) P )
 
Theorem3trld 40086 Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   =>    |-  ( ph  ->  F (TrailS `  G ) P )
 
Theorem3trlond 40087 A trail of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 8-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   =>    |-  ( ph  ->  F ( A (TrailsOn `  G ) D ) P )
 
Theorem3pthd 40088 A path of length 3 from one vertex to another vertex via a third vertex. (Contributed by Alexander van der Vekens, 6-Dec-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   =>    |-  ( ph  ->  F (PathS `  G ) P )
 
Theorem3pthond 40089 A path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   =>    |-  ( ph  ->  F ( A (PathsOn `  G ) D ) P )
 
Theorem3spthd 40090 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   &    |-  ( ph  ->  A  =/=  D )   =>    |-  ( ph  ->  F (SPathS `  G ) P )
 
Theorem3spthond 40091 A simple path of length 3 from one vertex to another, different vertex via a third vertex. (Contributed by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   &    |-  ( ph  ->  A  =/=  D )   =>    |-  ( ph  ->  F ( A (SPathsOn `  G ) D ) P )
 
Theorem3cycld 40092 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   &    |-  ( ph  ->  A  =  D )   =>    |-  ( ph  ->  F (CycleS `  G ) P )
 
Theorem3cyclpd 40093 Construction of a 3-cycle from three given edges in a graph, containing an endpoint of one of these edges. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 10-Feb-2021.) (Revised by AV, 24-Mar-2021.)
 |-  P  =  <" A B C D ">   &    |-  F  =  <" J K L ">   &    |-  ( ph  ->  (
 ( A  e.  V  /\  B  e.  V ) 
 /\  ( C  e.  V  /\  D  e.  V ) ) )   &    |-  ( ph  ->  ( ( A  =/=  B  /\  A  =/=  C )  /\  ( B  =/=  C  /\  B  =/=  D )  /\  C  =/=  D ) )   &    |-  ( ph  ->  ( { A ,  B }  C_  ( I `  J )  /\  { B ,  C }  C_  ( I `  K )  /\  { C ,  D }  C_  ( I `
  L ) ) )   &    |-  V  =  (Vtx `  G )   &    |-  I  =  (iEdg `  G )   &    |-  ( ph  ->  ( J  =/=  K  /\  J  =/=  L  /\  K  =/=  L ) )   &    |-  ( ph  ->  A  =  D )   =>    |-  ( ph  ->  ( F (CycleS `  G ) P  /\  ( # `  F )  =  3  /\  ( P `  0 )  =  A ) )
 
Theoremupgr3v3e3cycl 40094* If there is a cycle of length 3 in a pseudograph, there are three distinct vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  E  =  (Edg `  G )   &    |-  V  =  (Vtx `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (CycleS `  G ) P 
 /\  ( # `  F )  =  3 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E )  /\  (
 a  =/=  b  /\  b  =/=  c  /\  c  =/=  a ) ) )
 
Theoremuhgr3cyclexlem 40095 Lemma for uhgr3cyclex 40096. (Contributed by AV, 12-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   &    |-  I  =  (iEdg `  G )   =>    |-  (
 ( ( ( A  e.  V  /\  B  e.  V )  /\  A  =/=  B )  /\  (
 ( J  e.  dom  I 
 /\  { B ,  C }  =  ( I `  J ) )  /\  ( K  e.  dom  I 
 /\  { C ,  A }  =  ( I `  K ) ) ) )  ->  J  =/=  K )
 
Theoremuhgr3cyclex 40096* If there are three different vertices in a hypergraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UHGraph  /\  (
 ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) ) 
 /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E  /\  { C ,  A }  e.  E ) )  ->  E. f E. p ( f (CycleS `  G ) p  /\  ( # `  f )  =  3 
 /\  ( p `  0 )  =  A ) )
 
Theoremumgr3cyclex 40097* If there are three (different) vertices in a multigraph which are mutually connected by edges, there is a 3-cycle in the graph containing one of these vertices. (Contributed by Alexander van der Vekens, 17-Nov-2017.) (Revised by AV, 12-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UMGraph  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  E  /\  { B ,  C }  e.  E  /\  { C ,  A }  e.  E ) )  ->  E. f E. p ( f (CycleS `  G ) p  /\  ( # `  f )  =  3 
 /\  ( p `  0 )  =  A ) )
 
Theoremumgr3v3e3cycl 40098* If and only if there is a 3-cycle in a multigraph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.) (Revised by AV, 12-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  ( G  e. UMGraph  ->  ( E. f E. p ( f (CycleS `  G ) p  /\  ( # `  f
 )  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  E  /\  { b ,  c }  e.  E  /\  { c ,  a }  e.  E ) ) )
 
Theoremupgr4cycl4dv4e 40099* If there is a cycle of length 4 in a pseudograph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.) (Revised by AV, 13-Feb-2021.)
 |-  V  =  (Vtx `  G )   &    |-  E  =  (Edg `  G )   =>    |-  (
 ( G  e. UPGraph  /\  F (CycleS `  G ) P 
 /\  ( # `  F )  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( {
 a ,  b }  e.  E  /\  { b ,  c }  e.  E )  /\  ( { c ,  d }  e.  E  /\  { d ,  a }  e.  E )
 )  /\  ( (
 a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
 b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
 
21.33.8.21  Connected graphs
 
Syntaxcconngr 40100 Extend class notation with connected graphs.
 class ConnGraph
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