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Theorem List for Metamath Proof Explorer - 28001-28100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremswrdnd 28001 The value of the subword extractor is the empty set (undefined) if the range is not valid. (Contributed by Alexander van der Vekens, 16-Mar-2018.)
 |-  (
 ( W  e. Word  V  /\  F  e.  ZZ  /\  L  e.  ZZ )  ->  ( ( F  <  0  \/  L  <_  F  \/  ( # `  W )  <  L )  ->  ( W substr  <. F ,  L >. )  =  (/) ) )
 
Theoremswrd0 28002 A subword of an empty set is always the empty set. REMARK: The antecedent  ( F  e.  ZZ  /\  L  e.  ZZ ) should not be necessary! (Contributed by Alexander van der Vekens, 31-Mar-2018.)
 |-  (
 ( F  e.  ZZ  /\  L  e.  ZZ )  ->  ( (/) substr  <. F ,  L >. )  =  (/) )
 
Theoremswrdrlen 28003 Length of a right-anchored subword. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
 |-  (
 ( S  e. Word  A  /\  F  e.  ( 0
 ... ( # `  S ) ) )  ->  ( # `  ( S substr  <. F ,  ( # `  S ) >. ) )  =  ( ( # `  S )  -  F ) )
 
Theoremaddlenrevswrd 28004 The sum of the lengths of two parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( # `  ( W substr 
 <. M ,  ( # `  W ) >. ) )  +  ( # `  ( W substr 
 <. 0 ,  M >. ) ) )  =  ( # `  W ) )
 
Theoremlenrevcctswrd 28005 The length of two reversely concatenated parts of a word is the length of the word. (Contributed by Alexander van der Vekens, 1-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  M  e.  ( 0
 ... ( # `  W ) ) )  ->  ( # `  ( ( W substr  <. M ,  ( # `
  W ) >. ) concat 
 ( W substr  <. 0 ,  M >. ) ) )  =  ( # `  W ) )
 
Theoremelfzelfzccat 28006 An element of a finite set of sequential integers up to the length of a word is an element of an extended finite set of sequential integers up to the length of a concatenation of this word with another word. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V )  ->  ( N  e.  (
 0 ... ( # `  A ) )  ->  N  e.  ( 0 ... ( # `
  ( A concat  B ) ) ) ) )
 
Theoremswrdvalfn 28007 Value of the subword extractor as function with domain. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( S  e. Word  A  /\  F  e.  ( 0
 ... L )  /\  L  e.  ( 0 ... ( # `  S ) ) )  ->  ( S substr  <. F ,  L >. )  Fn  (
 0..^ ( L  -  F ) ) )
 
Theoremccatvalfn 28008 The concatenation of two words is a function over the half-open interval of integers having the sum of the lengths of the word as length. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V )  ->  ( A concat  B )  Fn  ( 0..^ ( ( # `  A )  +  ( # `  B ) ) ) )
 
19.22.3.17  Words over a set - extension (subwords of subwords)
 
Theoremswrd0swrd 28009 A prefix of a subword. (Contributed by Alexander van der Vekens, 2-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( L  e.  ( 0 ... ( N  -  M ) )  ->  ( ( W substr  <. M ,  N >. ) substr  <. 0 ,  L >. )  =  ( W substr  <. M ,  ( M  +  L ) >. ) ) )
 
Theoremswrdswrdlem 28010 Lemma for swrdswrd 28011. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
 |-  (
 ( ( W  e. Word  V 
 /\  N  e.  (
 0 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  /\  ( K  e.  ( 0 ... ( N  -  M ) )  /\  L  e.  ( K ... ( N  -  M ) ) ) )  ->  ( W  e. Word  V  /\  ( M  +  K )  e.  ( 0 ... ( M  +  L )
 )  /\  ( M  +  L )  e.  (
 0 ... ( # `  W ) ) ) )
 
Theoremswrdswrd 28011 A subword of a subword. (Contributed by Alexander van der Vekens, 4-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  M  e.  ( 0 ... N ) )  ->  ( ( K  e.  ( 0
 ... ( N  -  M ) )  /\  L  e.  ( K ... ( N  -  M ) ) )  ->  ( ( W substr  <. M ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. ( M  +  K ) ,  ( M  +  L ) >. ) ) )
 
Theoremswrd0swrdid 28012 A prefix of a prefix with the same length is the prefix. (Contributed by Alexander van der Vekens, 5-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. 0 ,  N >. )  =  ( W substr  <. 0 ,  N >. ) )
 
Theoremswrdswrd0 28013 A subword of a prefix. (Contributed by Alexander van der Vekens, 6-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) ) )  ->  ( ( K  e.  ( 0 ... N )  /\  L  e.  ( K ... N ) ) 
 ->  ( ( W substr  <. 0 ,  N >. ) substr  <. K ,  L >. )  =  ( W substr  <. K ,  L >. ) ) )
 
Theoremswrd0swrd0 28014 A prefix of a prefix. (Contributed by Alexander van der Vekens, 7-Apr-2018.)
 |-  (
 ( W  e. Word  V  /\  N  e.  ( 0
 ... ( # `  W ) )  /\  L  e.  ( 0 ... N ) )  ->  ( ( W substr  <. 0 ,  N >. ) substr  <. 0 ,  L >. )  =  ( W substr  <. 0 ,  L >. ) )
 
19.22.3.18  Words over a set - extension (subwords of concatenations)
 
Theoremswrdccat3a0 28015 The subword of a concatenation of an empty word with another word is the subword of the second concatenated word. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
 |-  (
 ( A  e.  Y  /\  B  e. Word  V )  ->  ( ( # `  A )  =  0  ->  ( ( A concat  B ) substr  <. M ,  N >. )  =  ( B substr  <. M ,  N >. ) ) )
 
Theoremswrdccatin1 28016 The subword of a concatenation of two words within the first of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  (
 ( A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... N )  /\  N  e.  (
 0 ... ( # `  A ) ) )  ->  ( ( A concat  B ) substr 
 <. M ,  N >. )  =  ( A substr  <. M ,  N >. ) ) )
 
Theoremswrdccatin2lem1 28017 Lemma for swrdccatin2 28018. (Contributed by Alexander van der Vekens, 29-Mar-2018.)
 |-  (
 ( ( A  e.  NN0  /\  B  e.  NN0 )  /\  ( M  e.  ( A ... N )  /\  N  e.  ( ( A  +  1 ) ... ( A  +  B ) )  /\  K  e.  ( 0..^ ( N  -  M ) ) ) )  ->  ( K  +  M )  e.  ( A..^ ( A  +  B ) ) )
 
Theoremswrdccatin2 28018 The subword of a concatenation of two words within the second of the concatenated words. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( L ... N ) 
 /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A concat  B ) substr 
 <. M ,  N >. )  =  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ) )
 
Theoremswrdccatin12lem1 28019 Lemma 1 for swrdccatin12 28026. (Contributed by Alexander van der Vekens, 28-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0..^ L )  /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) )  ->  ( A  e. Word  V  /\  M  e.  ( 0 ... L )  /\  L  e.  (
 0 ... ( # `  A ) ) ) )
 
Theoremswrdccatin12lem2 28020 Lemma 2 for swrdccatin12 28026. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( L  e.  NN0  /\  M  e.  NN0  /\  N  e.  ZZ )  ->  (
 ( K  e.  (
 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  K  e.  ( ( L  -  M )..^ ( ( L  -  M )  +  ( N  -  L ) ) ) ) )
 
Theoremswrdccatin12lem3a 28021 Lemma 1 for swrdccatin12lem3 28024. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( B  e.  NN0  /\  M  e.  ( 0..^ L )  /\  N  e.  ( ( L  +  1 ) ... ( L  +  B )
 ) )  ->  (
 ( K  e.  (
 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  +  M )  e.  ( L..^ ( L  +  B ) ) ) )
 
Theoremswrdccatin12lem3b 28022 Lemma 2 for swrdccatin12lem3 28024. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( B  e.  NN0  /\  M  e.  ( 0..^ L )  /\  N  e.  ( ( L  +  1 ) ... ( L  +  B )
 ) )  ->  (
 ( K  e.  (
 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( K  -  ( L  -  M ) )  e.  ( 0..^ ( ( N  -  L )  -  0 ) ) ) )
 
Theoremswrdccatin12lem3c 28023 Lemma for swrdccatin12lem3 28024 and swrdccatin12lem4 28025. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0..^ L )  /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( A concat  B )  e. Word  V  /\  M  e.  ( 0
 ... N )  /\  N  e.  ( 0 ... ( # `  ( A concat  B ) ) ) ) )
 
Theoremswrdccatin12lem3 28024 Lemma 3 for swrdccatin12 28026. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0..^ L )  /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  -.  K  e.  ( 0..^ ( L  -  M ) ) )  ->  ( (
 ( A concat  B ) substr  <. M ,  N >. ) `
  K )  =  ( ( B substr  <. 0 ,  ( N  -  L ) >. ) `  ( K  -  ( # `  ( A substr 
 <. M ,  L >. ) ) ) ) ) )
 
Theoremswrdccatin12lem4 28025 Lemma 4 for swrdccatin12 28026. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  /\  ( M  e.  ( 0..^ L )  /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) )  ->  ( ( K  e.  ( 0..^ ( N  -  M ) )  /\  K  e.  ( 0..^ ( L  -  M ) ) ) 
 ->  ( ( ( A concat  B ) substr  <. M ,  N >. ) `  K )  =  ( ( A substr 
 <. M ,  L >. ) `
  K ) ) )
 
Theoremswrdccatin12 28026 The subword of a concatenation of two words within both of the concatenated words. REMARK: If swrdccatin12c 28028 is proven directly, this theorem would be special case of swrdccatin12c 28028). (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0..^ L )  /\  N  e.  ( ( L  +  1 ) ... ( L  +  ( # `
  B ) ) ) )  ->  (
 ( A concat  B ) substr  <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat 
 ( B substr  <. 0 ,  ( N  -  L ) >. ) ) ) )
 
Theoremswrdccatin12b 28027 The subword of a concatenation of two words within both of the concatenated words. REMARK: If swrdccatin12c 28028 is proven directly, this theorem would be obsolete then). (Contributed by Alexander van der Vekens, 5-Apr-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... L )  /\  N  e.  (
 ( L  +  1 ) ... ( L  +  ( # `  B ) ) ) ) 
 ->  ( ( A concat  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat 
 ( B substr  <. 0 ,  ( N  -  L ) >. ) ) ) )
 
Theoremswrdccatin12c 28028 The subword of a concatenation of two words within both of the concatenated words. REMARK: Maybe this can be proven directly, without using swrdccatin12 28026 (this would be a special case of the current theorem) and swrdccatin12b 28027 (this would be obsolete then). (Contributed by Alexander van der Vekens, 5-Apr-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... L )  /\  N  e.  ( L ... ( L  +  ( # `  B ) ) ) )  ->  ( ( A concat  B ) substr 
 <. M ,  N >. )  =  ( ( A substr  <. M ,  L >. ) concat 
 ( B substr  <. 0 ,  ( N  -  L ) >. ) ) ) )
 
Theoremswrdccat3 28029 The subword of a concatenation is either a subword of the first concatenated word or a subword of the second concatenated word or a concatenation of a suffix of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 30-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( ( M  e.  ( 0 ... N )  /\  N  e.  (
 0 ... ( L  +  ( # `  B ) ) ) )  ->  ( ( A concat  B ) substr 
 <. M ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. M ,  N >. ) ,  if ( L 
 <_  M ,  ( B substr  <. ( M  -  L ) ,  ( N  -  L ) >. ) ,  ( ( A substr  <. M ,  L >. ) concat  ( B substr  <.
 0 ,  ( N  -  L ) >. ) ) ) ) ) )
 
Theoremswrdccat3a 28030 A prefix of a concatenation is either a prefix of the first concatenated word or a concatenation of the first word with a prefix of the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( N  e.  (
 0 ... ( L  +  ( # `  B ) ) )  ->  (
 ( A concat  B ) substr  <.
 0 ,  N >. )  =  if ( N 
 <_  L ,  ( A substr  <. 0 ,  N >. ) ,  ( A concat  ( B substr 
 <. 0 ,  ( N  -  L ) >. ) ) ) ) )
 
Theoremswrdccat3b 28031 A suffix of a concatenation is either a suffix of the second concatenated word or a concatenation of a suffix of the first word with the second word. (Contributed by Alexander van der Vekens, 31-Mar-2018.)
 |-  L  =  ( # `  A )   =>    |-  ( ( V  e.  X  /\  A  e. Word  V  /\  B  e. Word  V )  ->  ( M  e.  (
 0 ... ( L  +  ( # `  B ) ) )  ->  (
 ( A concat  B ) substr  <. M ,  ( L  +  ( # `  B ) ) >. )  =  if ( L  <_  M ,  ( B substr  <. ( M  -  L ) ,  ( # `  B ) >. ) ,  (
 ( A substr  <. M ,  L >. ) concat  B )
 ) ) )
 
19.22.4  Graph theory
 
19.22.4.1  Undirected hypergraphs
 
Theoremuhgraedgrnv 28032 An edge of an undirected hypergraph contains only vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V UHGrph  E  /\  F  e.  ran  E  /\  N  e.  F )  ->  N  e.  V )
 
19.22.4.2  Undirected simple graphs
 
Theoremusisuhgra 28033 An undirected simple graph without loops is an undirected hypergraph. (Contributed by Alexander van der Vekens, 9-Feb-2018.)
 |-  ( V USGrph  E  ->  V UHGrph  E )
 
19.22.4.3  Neighbors, complete graphs and universal vertices
 
Theoremnbfiusgrafi 28034 The class of neighbors of a vertex in a finite graph is a finite set. (Contributed by Alexander van der Vekens, 7-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  e.  Fin )
 
19.22.4.4  Walks, Paths and Cycles
 
Theoremusgra2pthspth 28035 In a undirected simple graph, any path of length 2 is a simple path. (Contributed by Alexander van der Vekens, 25-Jan-2018.)
 |-  (
 ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  ( F ( V Paths  E ) P  <->  F ( V SPaths  E ) P ) )
 
Theoremusgra2wlkspthlem1 28036* Lemma 1 for usgra2wlkspth 28038. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  (
 ( F  e. Word  dom  E 
 /\  E : dom  E
 -1-1-> ran  E  /\  ( # `
  F )  =  2 )  ->  (
 ( ( ( P `
  0 )  =  A  /\  ( P `
  ( # `  F ) )  =  B  /\  A  =/=  B ) 
 /\  A. i  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  ->  Fun  `' F ) )
 
Theoremusgra2wlkspthlem2 28037* Lemma 2 for usgra2wlkspth 28038. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  (
 ( ( F  e. Word  dom 
 E  /\  ( # `  F )  =  2 )  /\  ( V USGrph  E  /\  P : ( 0 ... ( # `  F ) ) --> V ) )  ->  ( (
 ( ( P `  0 )  =  A  /\  ( P `  ( # `
  F ) )  =  B  /\  A  =/=  B )  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  (
 i  +  1 ) ) } )  ->  Fun  `' P ) )
 
Theoremusgra2wlkspth 28038 In a undirected simple graph, any walk of length 2 between two different vertices is a simple path. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  ( # `  F )  =  2  /\  A  =/=  B )  ->  ( F ( A ( V WalkOn  E ) B ) P  <->  F ( A ( V SPathOn  E ) B ) P ) )
 
Theoremspthdifv 28039 The vertices of a simple path are distinct, so the vertex function is one-to-one. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
 |-  ( F ( V SPaths  E ) P  ->  P :
 ( 0 ... ( # `
  F ) )
 -1-1-> V )
 
Theoremusgra2pthlem1 28040* Lemma for usgra2pth 28041. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  (
 ( F : ( 0..^ ( # `  F ) ) -1-1-> dom  E  /\  P : ( 0
 ... ( # `  F ) ) --> V  /\  A. i  e.  ( 0..^ ( # `  F ) ) ( E `
  ( F `  i ) )  =  { ( P `  i ) ,  ( P `  ( i  +  1 ) ) }
 )  ->  ( ( V USGrph  E  /\  ( # `  F )  =  2 )  ->  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y }
 ) ( ( ( P `  0 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `
  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) )
 
Theoremusgra2pth 28041* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  y  /\  ( P `  2 )  =  z )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  y }  /\  ( E `
  ( F `  1 ) )  =  { y ,  z } ) ) ) ) )
 
Theoremusgra2pth0 28042* In a undirected simply graph, there is a path of length 2 if and only if there are three distinct vertices so that one of them is connected to each of the two others by an edge. (Contributed by Alexander van der Vekens, 27-Jan-2018.)
 |-  ( V USGrph  E  ->  ( ( F ( V Paths  E ) P  /\  ( # `  F )  =  2 )  <->  ( F :
 ( 0..^ 2 )
 -1-1-> dom  E  /\  P : ( 0 ... 2 ) -1-1-> V  /\  E. x  e.  V  E. y  e.  ( V  \  { x } ) E. z  e.  ( V  \  { x ,  y } ) ( ( ( P `  0
 )  =  x  /\  ( P `  1 )  =  z  /\  ( P `  2 )  =  y )  /\  (
 ( E `  ( F `  0 ) )  =  { x ,  z }  /\  ( E `
  ( F `  1 ) )  =  { z ,  y } ) ) ) ) )
 
Theoremusgra2adedgspthlem1 28043 Lemma 1 for usgra2adedgspth 28045. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  (
 ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) 
 ->  ( ( E `  ( `' E `  { A ,  B } ) )  =  { A ,  B }  /\  ( E `
  ( `' E ` 
 { B ,  C } ) )  =  { B ,  C } ) )
 
Theoremusgra2adedgspthlem2 28044 Lemma for usgra2adedgspth 28045. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  (
 ( ( V USGrph  E  /\  A  =/=  C ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) )  ->  ( ( `' E `  { A ,  B } )  =/=  ( `' E `  { B ,  C }
 )  /\  ( E `  ( `' E `  { A ,  B }
 ) )  =  { A ,  B }  /\  ( E `  ( `' E `  { B ,  C } ) )  =  { B ,  C } ) )
 
Theoremusgra2adedgspth 28045 In an undirected simple graph, two adjacent edges form a simple path of length 2. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( ( V USGrph  E  /\  A  =/=  C ) 
 ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  F ( V SPaths  E ) P ) )
 
Theoremusgra2adedgwlk 28046 In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  ( F ( V Walks  E ) P  /\  ( # `  F )  =  2  /\  ( A  =  ( P `  0 )  /\  B  =  ( P `  1
 )  /\  C  =  ( P `  2 ) ) ) ) )
 
Theoremusgra2adedgwlkon 28047 In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  F ( A ( V WalkOn  E ) C ) P ) )
 
Theoremusgra2adedglem1 28048 In an undirected simple graph, two adjacent edges are an unordered pair of unordered pairs. (Contributed by Alexander van der Vekens, 1-Feb-2018.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. }   &    |-  P  =  { <. 0 ,  A >. ,  <. 1 ,  B >. ,  <. 2 ,  C >. }   =>    |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 ->  { { A ,  B } ,  { B ,  C } }  =  ( E " ran  F ) ) )
 
Theoremusg2wlk 28049* In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V USGrph  E  /\  { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) )
 
Theoremusg2wlkon 28050* In an undirected simple graph, two adjacent edges form a walk between two (different) vertices. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  ( V USGrph  E  ->  ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  ->  E. f E. p  f ( A ( V WalkOn  E ) C ) p ) )
 
19.22.4.5  Walks/paths of length 2 as ordered triples
 
Syntaxc2wlkot 28051 Extend class notation with walks (of a graph) of length 2 as ordered triple.
 class 2WalksOt
 
Syntaxc2wlkonot 28052 Extend class notation with walks between two vertices (within a graph) of length 2 as ordered triple.
 class 2WalksOnOt
 
Syntaxc2spthot 28053 Extend class notation with paths (of a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Syntaxc2pthonot 28054 Extend class notation with simple paths between two vertices (within a graph) of length 2 as ordered triple.
 class 2SPathOnOt
 
Definitiondf-2wlkonot 28055* Define the collection of walks of length 2 with particular endpoints as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |- 2WalksOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { t  e.  ( ( v  X.  v )  X.  v
 )  |  E. f E. p ( f ( a ( v WalkOn  e
 ) b ) p 
 /\  ( # `  f
 )  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Definitiondf-2wlksot 28056* Define the collection of all walks of length 2 as ordered triple (in a graph). (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |- 2WalksOt  =  ( v  e.  _V ,  e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v
 )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2WalksOnOt  e ) b ) } )
 
Definitiondf-2spthonot 28057* Define the collection of simple paths of length 2 with particular endpoints as ordered triple (in a graph) . (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |- 2SPathOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  ( a  e.  v ,  b  e.  v  |->  { t  e.  ( ( v  X.  v )  X.  v
 )  |  E. f E. p ( f ( a ( v SPathOn  e
 ) b ) p 
 /\  ( # `  f
 )  =  2  /\  ( ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Definitiondf-2spthsot 28058* Define the collection of all simple paths of length 2 as ordered triple. (in a graph) (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |- 2SPathOnOt  =  ( v  e.  _V ,  e  e.  _V  |->  { t  e.  ( ( v  X.  v )  X.  v
 )  |  E. a  e.  v  E. b  e.  v  t  e.  ( a ( v 2SPathOnOt  e ) b ) } )
 
Theoremel2wlkonotlem 28059 Lemma for el2wlkonot 28066. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 )  ->  ( p `  1 )  e.  V )
 
Theoremis2wlkonot 28060* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V 2WalksOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( a ( V WalkOn  E )
 b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Theoremis2spthonot 28061* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V 2SPathOnOt  E )  =  ( a  e.  V ,  b  e.  V  |->  { t  e.  ( ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( a ( V SPathOn  E )
 b ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  a  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  b ) ) } ) )
 
Theorem2wlkonot 28062* The set of walks of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V 2WalksOnOt  E ) B )  =  {
 t  e.  ( ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( A ( V WalkOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  B ) ) } )
 
Theorem2spthonot 28063* The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V 2SPathOnOt  E ) B )  =  {
 t  e.  ( ( V  X.  V )  X.  V )  | 
 E. f E. p ( f ( A ( V SPathOn  E ) B ) p  /\  ( # `  f )  =  2  /\  (
 ( 1st `  ( 1st `  t ) )  =  A  /\  ( 2nd `  ( 1st `  t
 ) )  =  ( p `  1 ) 
 /\  ( 2nd `  t
 )  =  B ) ) } )
 
Theorem2wlksot 28064* The set of walks of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V 2WalksOt  E )  =  { t  e.  (
 ( V  X.  V )  X.  V )  | 
 E. a  e.  V  E. b  e.  V  t  e.  ( a
 ( V 2WalksOnOt  E ) b ) } )
 
Theorem2spthsot 28065* The set of simple paths of length 2 (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V 2SPathOnOt  E )  =  { t  e.  (
 ( V  X.  V )  X.  V )  | 
 E. a  e.  V  E. b  e.  V  t  e.  ( a
 ( V 2SPathOnOt  E ) b ) } )
 
Theoremel2wlkonot 28066* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V ) )  ->  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V Walks  E ) p 
 /\  ( # `  f
 )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  C  =  ( p `  2
 ) ) ) ) ) )
 
Theoremel2spthonot 28067* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V ) )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  b  =  ( p `  1
 )  /\  C  =  ( p `  2 ) ) ) ) ) )
 
Theoremel2spthonot0 28068* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V ) )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( T  =  <. A ,  b ,  C >.  /\  <. A ,  b ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C ) ) ) )
 
Theoremel2wlkonotot0 28069* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V ) )  ->  ( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <-> 
 ( A  =  R  /\  C  =  S  /\  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) ) ) )
 
Theoremel2wlkonotot 28070* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 15-Feb-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  C  e.  V ) )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) ) )
 
Theoremel2wlkonotot1 28071 A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( R  e.  V  /\  S  e.  V ) )  ->  ( <. A ,  B ,  C >.  e.  ( R ( V 2WalksOnOt  E ) S )  <-> 
 ( A  =  R  /\  C  =  S  /\  <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C ) ) ) )
 
Theorem2wlkonot3v 28072 If an ordered triple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  ( T  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  ( ( V  X.  V )  X.  V ) ) )
 
Theorem2spthonot3v 28073 If an ordered triple represents a simple path of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  C  e.  V )  /\  T  e.  ( ( V  X.  V )  X.  V ) ) )
 
Theorem2wlkonotv 28074 If an ordered tripple represents a walk of length 2, its components are vertices. (Contributed by Alexander van der Vekens, 19-Feb-2018.)
 |-  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  ->  ( ( V  e.  _V  /\  E  e.  _V )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 ) )
 
Theoremel2wlksoton 28075* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( T  e.  ( V 2WalksOt  E )  <->  E. a  e.  V  E. b  e.  V  T  e.  ( a
 ( V 2WalksOnOt  E ) b ) ) )
 
Theoremel2spthsoton 28076* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( T  e.  ( V 2SPathOnOt  E )  <->  E. a  e.  V  E. b  e.  V  T  e.  ( a
 ( V 2SPathOnOt  E ) b ) ) )
 
Theoremel2wlksot 28077* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 21-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( T  e.  ( V 2WalksOt  E )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V Walks  E ) p 
 /\  ( # `  f
 )  =  2  /\  ( a  =  ( p `  0 )  /\  b  =  ( p `  1 )  /\  c  =  ( p `  2
 ) ) ) ) ) )
 
Theoremel2pthsot 28078* A simple path of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 28-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( T  e.  ( V 2SPathOnOt  E )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( T  =  <. a ,  b ,  c >.  /\  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  (
 a  =  ( p `
  0 )  /\  b  =  ( p `  1 )  /\  c  =  ( p `  2
 ) ) ) ) ) )
 
Theoremel2wlksotot 28079* A walk of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 26-Feb-2018.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) )  ->  ( <. A ,  B ,  C >.  e.  ( V 2WalksOt  E )  <->  E. f E. p ( f ( V Walks  E ) p  /\  ( # `  f )  =  2 
 /\  ( A  =  ( p `  0 ) 
 /\  B  =  ( p `  1 ) 
 /\  C  =  ( p `  2 ) ) ) ) )
 
Theoremusg2wlkonot 28080 A walk of length 2 between two vertices as ordered triple in an undirected simple graph. This theorem would also hold for undirected multigraphs, but to proof this the cases  A  =  B and/or  B  =  C must be considered separately. (Contributed by Alexander van der Vekens, 18-Feb-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <-> 
 ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) ) )
 
Theoremusg2wotspth 28081* A walk of length 2 between two different vertices as ordered triple corresponds to a simple path of length 2 in an undirected simple graph. (Contributed by Alexander van der Vekens, 16-Feb-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  A  =/=  C ) 
 ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2WalksOnOt  E ) C )  <->  E. f E. p ( f ( V SPaths  E ) p  /\  ( # `  f )  =  2  /\  ( A  =  ( p `  0 )  /\  B  =  ( p `  1
 )  /\  C  =  ( p `  2 ) ) ) ) )
 
Theorem2pthwlkonot 28082 For two different vertices, a walk of length 2 between these vertices as ordered triple is a simple path of length 2 between these vertices as ordered triple in an undirected simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V ) 
 /\  A  =/=  B )  ->  ( A ( V 2SPathOnOt  E ) B )  =  ( A ( V 2WalksOnOt  E ) B ) )
 
Theorem2wot2wont 28083* The set of (simple) paths of length 2 (in a graph) is the set of (simple) paths of length 2 between any two different vertices. (Contributed by Alexander van der Vekens, 27-Feb-2018.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V 2WalksOt  E )  =  U_ x  e.  V  U_ y  e.  V  ( x ( V 2WalksOnOt  E ) y ) )
 
Theorem2spontn0vne 28084 If the set of simple paths of length 2 between two vertices (in a graph) is not empty, the two vertices must be not equal. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
 |-  (
 ( X ( V 2SPathOnOt  E ) Y )  =/=  (/)  ->  X  =/=  Y )
 
Theoremusg2spthonot 28085 A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( <. A ,  B ,  C >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
 ( A  =/=  C  /\  { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E ) ) )
 
Theoremusg2spthonot0 28086 A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 8-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )
 )  ->  ( <. S ,  B ,  T >.  e.  ( A ( V 2SPathOnOt  E ) C )  <-> 
 ( ( S  =  A  /\  T  =  C  /\  A  =/=  C ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) ) ) )
 
Theoremusg2spthonot1 28087* A simple path of length 2 between two vertices as ordered triple corresponds to two adjacent edges in an undirected simple graph. (Contributed by Alexander van der Vekens, 9-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  C  e.  V ) )  ->  ( T  e.  ( A ( V 2SPathOnOt  E ) C )  <->  E. b  e.  V  ( ( T  =  <. A ,  b ,  C >.  /\  A  =/=  C )  /\  ( { A ,  b }  e.  ran  E  /\  {
 b ,  C }  e.  ran  E ) ) ) )
 
Theorem2spot2iun2spont 28088* The set of simple paths of length 2 (in a graph) is the double union of the simple paths of length 2 between different vertices. (Contributed by Alexander van der Vekens, 3-Mar-2018.)
 |-  (
 ( V  e.  _V  /\  E  e.  _V )  ->  ( V 2SPathOnOt  E )  =  U_ x  e.  V  U_ y  e.  ( V 
 \  { x }
 ) ( x ( V 2SPathOnOt  E ) y ) )
 
Theorem2spotfi 28089 In a finite graph, the set of simple paths of length 2 between two vertices (as ordered triples) is finite. (Contributed by Alexander van der Vekens, 4-Mar-2018.)
 |-  (
 ( ( V  e.  Fin  /\  E  e.  X ) 
 /\  ( A  e.  V  /\  B  e.  V ) )  ->  ( A ( V 2SPathOnOt  E ) B )  e.  Fin )
 
19.22.4.6  Vertex Degree
 
Theoremusgfidegfi 28090* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin )  ->  A. v  e.  V  ( ( V VDeg  E ) `  v )  e. 
 NN0 )
 
Theoremusgfiregdegfi 28091* In a finite graph, the degree of each vertex is finite. (Contributed by Alexander van der Vekens, 6-Mar-2018.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin  /\  V  =/= 
 (/) )  ->  ( A. v  e.  V  ( ( V VDeg  E ) `  v )  =  K  ->  K  e.  NN0 ) )
 
19.22.4.7  Friendship graphs

In this section, the basics for the friendship theorem, which is one from the "100 theorem list" (#83), are provided, including the definition of friendship graphs df-frgra 28093 as special undirected simple graphs without loops (see frisusgra 28096) and the proofs of the friendship theorem for small graphs (with up to 3 vertices), see 1to3vfriendship 28112. The general friendship theorem, which should be called "friendship", but which is still to be proven, would be  |-  ( V  =/=  (/)  ->  ( V FriendGrph  E  ->  E. v  e.  V A. w  e.  ( V  \  { v } ) { v ,  w }  e.  ran  E ) ). The case  V  =  (/) (a graph without vertices) must be excluded either from the definition of a friendship graph, or from the theorem. If it is not excluded from the definition, which is the case with df-frgra 28093, a graph without vertices is a friendship graph (see frgra0 28098), but the friendship condition  E. v  e.  V A. w  e.  ( V  \  { v } ) { v ,  w }  e.  ran  E does not hold (because of  -.  E. x  e.  (/) ph, see rex0 3601).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 28102, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 28103, a graph with exactly 3 (different) vertices is a friendship graph if and only if it is a complete graph (every two vertices are connected by an edge), see frgra3v 28106, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 28111 (The generalization of this theorem "Every friendship graph (with at least one vertex) is a windmill graph" is a stronger result than the "friendship theorem". This generalization was proven by Mertzios and Unger, see Theorem 1 of [MertziosUnger] p. 152.).

The first steps to prove the friendship theorem following the approach of Mertzios and Unger are already made, see 2pthfrgrarn2 28114 and n4cyclfrgra 28122 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.). In addition, the first three Lemmas ("claims") in the proof of [Huneke] p. 1 are proven, see frgrancvvdgeq 28146, frgraregorufr 28156 and frgregordn0 28173.

 
Syntaxcfrgra 28092 Extend class notation with Friendship Graphs.
 class FriendGrph
 
Definitiondf-frgra 28093* Define the class of all Friendship Graphs. A graph is called a friendship graph if every pair of its vertices has exactly one common neighbor. (Contributed by Alexander van der Vekens and Mario Carneiro, 2-Oct-2017.)
 |- FriendGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v  A. l  e.  ( v  \  { k } ) E! x  e.  v  { { x ,  k } ,  { x ,  l } }  C_  ran  e ) }
 
Theoremisfrgra 28094* The property of being a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V FriendGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. l  e.  ( V 
 \  { k }
 ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E )
 ) )
 
Theoremfrisusgrapr 28095* A friendship graph is an undirected simple graph without loops with special properties. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( V FriendGrph  E  ->  ( V USGrph  E 
 /\  A. k  e.  V  A. l  e.  ( V 
 \  { k }
 ) E! x  e.  V  { { x ,  k } ,  { x ,  l } }  C_  ran  E )
 )
 
Theoremfrisusgra 28096 A friendship graph is an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( V FriendGrph  E  ->  V USGrph  E )
 
Theoremfrgra0v 28097 Any graph with no vertex is a friendship graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  ( (/) FriendGrph  E  <->  E  =  (/) )
 
Theoremfrgra0 28098 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (/) FriendGrph  (/)
 
Theoremfrgraunss 28099* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b  e.  V  { { A ,  b } ,  { b ,  C } }  C_  ran 
 E ) )
 
Theoremfrgraun 28100* Any two (different) vertices in a friendship graph have a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Dec-2017.)
 |-  ( V FriendGrph  E  ->  ( ( A  e.  V  /\  C  e.  V  /\  A  =/=  C )  ->  E! b  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  C }  e.  ran  E ) ) )
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