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Theorem List for Metamath Proof Explorer - 28401-28500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremconstr3pthlem1 28401 Lemma for constr3pth 28406. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( B  e.  V  /\  C  e.  W )  ->  ( P  |`  ( 1..^ ( # `  F ) ) )  =  { <. 1 ,  B >. ,  <. 2 ,  C >. } )
 
Theoremconstr3pthlem2 28402 Lemma for constr3pth 28406. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  B  =/=  C ) 
 ->  Fun  `' ( P  |`  ( 1..^ ( # `  F ) ) ) )
 
Theoremconstr3pthlem3 28403 Lemma for constr3pth 28406. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) ) 
 ->  ( ( P " { 0 ,  ( # `
  F ) }
 )  i^i  ( P " ( 1..^ ( # `  F ) ) ) )  =  (/) )
 
Theoremconstr3cycllem1 28404 Lemma for constr3cycl 28407. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  ->  ( P `
  0 )  =  ( P `  ( # `
  F ) ) )
 
Theoremconstr3trl 28405 Construction of a trail from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F ( V Trails  E ) P )
 
Theoremconstr3pth 28406 Construction of a path from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F ( V Paths  E ) P )
 
Theoremconstr3cycl 28407 Construction of a 3-cycle from three given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( F ( V Cycles  E ) P  /\  ( # `  F )  =  3 ) )
 
Theoremconstr3cyclp 28408 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.)
 |-  F  =  { <. 0 ,  ( `' E `  { A ,  B } ) >. , 
 <. 1 ,  ( `' E `  { B ,  C } ) >. , 
 <. 2 ,  ( `' E `  { C ,  A } ) >. }   &    |-  P  =  ( { <. 0 ,  A >. , 
 <. 1 ,  B >. }  u.  { <. 2 ,  C >. ,  <. 3 ,  A >. } )   =>    |-  ( ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V ) 
 /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( F ( V Cycles  E ) P  /\  ( # `  F )  =  3  /\  ( P `  0 )  =  A ) )
 
Theoremconstr3cyclpe 28409* If there are three (different) vertices in a graph 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.)
 |-  (
 ( V USGrph  E  /\  ( A  e.  V  /\  B  e.  V  /\  C  e.  V )  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3  /\  ( p `  0 )  =  A ) )
 
Theorem3v3e3cycl2 28410* If there are three (different) vertices in a graph which are mutually connected by edges, there is a 3-cycle in the graph. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
 |-  ( V USGrph  E  ->  ( E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E )  ->  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3 ) ) )
 
Theorem3v3e3cycl 28411* If and only if there is a 3-cycle in a graph, there are three (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 14-Nov-2017.)
 |-  ( V USGrph  E  ->  ( E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f
 )  =  3 )  <->  E. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  {
 c ,  a }  e.  ran  E ) ) )
 
Theorem4cycl4v4e 28412* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( Fun  E  /\  F ( V Cycles  E ) P  /\  ( # `  F )  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( {
 a ,  b }  e.  ran  E  /\  {
 b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) ) )
 
Theorem4cycl4dv 28413 In a simple graph, the vertices of a 4-cycle are mutually different. (Contributed by Alexander van der Vekens, 18-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  ( F  e. Word  dom  E  /\  Fun  `' F  /\  ( # `  F )  =  4 ) ) 
 ->  ( ( ( ( E `  ( F `
  0 ) )  =  { A ,  B }  /\  ( E `
  ( F `  1 ) )  =  { B ,  C } )  /\  ( ( E `  ( F `
  2 ) )  =  { C ,  D }  /\  ( E `
  ( F `  3 ) )  =  { D ,  A } ) )  ->  ( ( ( { A ,  B }  e.  ran  E  /\  { B ,  C }  e.  ran  E )  /\  ( { C ,  D }  e.  ran  E  /\  { D ,  A }  e.  ran  E ) ) 
 /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D ) ) ) ) )
 
Theorem4cycl4dv4e 28414* If there is a cycle of length 4 in a graph, there are four (different) vertices in the graph which are mutually connected by edges. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  (
 ( V USGrph  E  /\  F ( V Cycles  E ) P  /\  ( # `  F )  =  4 )  ->  E. a  e.  V  E. b  e.  V  E. c  e.  V  E. d  e.  V  ( ( ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E )  /\  ( { c ,  d }  e.  ran  E  /\  { d ,  a }  e.  ran  E ) ) 
 /\  ( ( a  =/=  b  /\  a  =/=  c  /\  a  =/=  d )  /\  (
 b  =/=  c  /\  b  =/=  d  /\  c  =/=  d ) ) ) )
 
18.23.3.15  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 28416 as special undirected simple graphs without loops (see frisusgra 28419) and the proofs of the friendship theorem for small graphs (with up to 3 vertices), see 1to3vfriendship 28432. 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 28416, a graph without vertices is a friendship graph (see frgra0 28421), 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 3481).

Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 28422, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 28423, 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 28426, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 28431 (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 28434 and n4cyclfrgra 28440 (these theorems correspond to Proposition 1 of [MertziosUnger] p. 153.).

 
Syntaxcfrgra 28415 Extend class notation with Friendship Graphs.
 class FriendGrph
 
Definitiondf-frgra 28416* 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 28417* 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 28418* 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 28419 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 28420 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 28421 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (/) FriendGrph  (/)
 
Theoremfrgra1v 28422 Any graph with only one vertex is a friendship graph. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  (
 ( V  e.  X  /\  { V } USGrph  E ) 
 ->  { V } FriendGrph  E )
 
Theoremfrgra2v 28423 Any graph with two (different) vertices is not a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B )  ->  -.  { A ,  B } FriendGrph  E )
 
Theoremfrgra3vlem1 28424* Lemma 1 for frgra3v 28426. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  (
 ( ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z )  /\  ( A  =/=  B  /\  A  =/=  C  /\  B  =/=  C ) )  /\  { A ,  B ,  C } USGrph  E )  ->  A. x A. y ( ( ( x  e. 
 { A ,  B ,  C }  /\  { { x ,  A } ,  { x ,  B } }  C_  ran  E )  /\  ( y  e. 
 { A ,  B ,  C }  /\  { { y ,  A } ,  { y ,  B } }  C_  ran 
 E ) )  ->  x  =  y )
 )
 
Theoremfrgra3vlem2 28425* Lemma 2 for frgra3v 28426. (Contributed by Alexander van der Vekens, 4-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( E! x  e.  { A ,  B ,  C }  { { x ,  A } ,  { x ,  B } }  C_  ran 
 E 
 <->  ( { C ,  A }  e.  ran  E 
 /\  { C ,  B }  e.  ran  E ) ) ) )
 
Theoremfrgra3v 28426 Any graph with three vertices which are completely connected with each other is a friendship graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) )  ->  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  B ,  C } FriendGrph  E  <->  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) ) )
 
Theorem1vwmgra 28427* Every graph with one vertex is a windmill graph. (Contributed by Alexander van der Vekens, 5-Oct-2017.)
 |-  (
 ( A  e.  X  /\  V  =  { A } )  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) )
 
Theorem3vfriswmgralem 28428* Lemma for 3vfriswmgra 28429. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E 
 ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
 
Theorem3vfriswmgra 28429* Every friendship graph with three (different) vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  (
 ( ( A  e.  X  /\  B  e.  Y  /\  C  e.  Z ) 
 /\  ( A  =/=  B 
 /\  A  =/=  C  /\  B  =/=  C ) 
 /\  V  =  { A ,  B ,  C } )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to2vfriswmgra 28430* Every friendship graph with one or two vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  (
 ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B } ) )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to3vfriswmgra 28431* Every friendship graph with one, two or three vertices is a windmill graph. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  (
 ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  ->  ( V FriendGrph  E  ->  E. h  e.  V  A. v  e.  ( V  \  { h } ) ( {
 v ,  h }  e.  ran  E  /\  E! w  e.  ( V  \  { h } ) { v ,  w }  e.  ran  E ) ) )
 
Theorem1to3vfriendship 28432* The friendship theorem for small graphs: In every friendship graph with one, two or three vertices, there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
 |-  (
 ( A  e.  X  /\  ( V  =  { A }  \/  V  =  { A ,  B }  \/  V  =  { A ,  B ,  C } ) )  ->  ( V FriendGrph  E  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E ) )
 
Theorem2pthfrgrarn 28433* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 15-Nov-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  {
 a } ) E. b  e.  V  ( { a ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E ) )
 
Theorem2pthfrgrarn2 28434* Between any two (different) vertices in a friendship graph is a 2-path (path of length 2), see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G ..., as well as the distance between any two nodes in G is at most two". (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  ( V FriendGrph  E  ->  A. a  e.  V  A. c  e.  ( V  \  {
 a } ) E. b  e.  V  (
 ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E ) 
 /\  ( a  =/=  b  /\  b  =/=  c ) ) )
 
Theorem3cyclfrgrarn1 28435* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  ( A  e.  V  /\  C  e.  V )  /\  A  =/=  C ) 
 ->  E. b  e.  V  E. c  e.  V  ( { A ,  b }  e.  ran  E  /\  { b ,  c }  e.  ran  E  /\  {
 c ,  A }  e.  ran  E ) )
 
Theorem3cyclfrgrarn 28436* Every vertex in a friendship graph ( with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 16-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. a  e.  V  E. b  e.  V  E. c  e.  V  ( { a ,  b }  e.  ran  E 
 /\  { b ,  c }  e.  ran  E  /\  { c ,  a }  e.  ran  E ) )
 
Theorem3cyclfrgra 28437* Every vertex in a friendship graph (with more than 1 vertex) is part of a 3-cycle. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  1  <  ( # `  V ) )  ->  A. v  e.  V  E. f E. p ( f ( V Cycles  E ) p  /\  ( # `  f )  =  3  /\  ( p `  0 )  =  v ) )
 
Theorem4cycl2v2nb 28438 In a (maybe degenerated) 4-cycle, two vertices have two (maybe not different) common neighbors. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 /\  ( { C ,  D }  e.  ran  E 
 /\  { D ,  A }  e.  ran  E ) )  ->  ( { { A ,  B } ,  { B ,  C } }  C_  ran  E  /\  { { A ,  D } ,  { D ,  C } }  C_  ran 
 E ) )
 
Theorem4cycl2vnunb 28439* In a 4-cycle, two distinct vertices have not a unique common neighbor. (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E ) 
 /\  ( { C ,  D }  e.  ran  E 
 /\  { D ,  A }  e.  ran  E ) 
 /\  ( B  e.  V  /\  D  e.  V  /\  B  =/=  D ) )  ->  -.  E! x  e.  V  { { A ,  x } ,  { x ,  C } }  C_  ran  E )
 
Theoremn4cyclfrgra 28440 There is no 4-cycle in a friendship graph, see Proposition 1 of [MertziosUnger] p. 153 : "A friendship graph G contains no C4 as a subgraph ...". (Contributed by Alexander van der Vekens, 19-Nov-2017.)
 |-  (
 ( V FriendGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `  F )  =/=  4 )
 
18.24  Mathbox for David A. Wheeler

This is the mathbox of David A. Wheeler, dwheeler at dwheeler dot com. Among other things, I have added a number of formal definitions for widely-used functions, e.g., those defined in ISO 80000-2:2009(E) Quantities and units - Part 2: Mathematical signs and symbols used in the natural sciences and technology and the NIST Digital Library of Mathematical Functions http://dlmf.nist.gov/.

 
18.24.1  Natural deduction
 
Theorem19.8ad 28441 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1730. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 28442 An identity theorem for substitution. See sbid 1875. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  ( ph  ->  [ x  /  x ] ps )   =>    |-  ( ph  ->  ps )
 
Theoremsbidd-misc 28443 An identity theorem for substitution. See sbid 1875. See Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by DAW, 18-Feb-2017.)
 |-  (
 ( ph  ->  [ x  /  x ] ps )  <->  (
 ph  ->  ps ) )
 
18.24.2  Greater than, greater than or equal to.

As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Here we formally define the widely-used relations 'greater than' and 'greater than or equal to', so that we have formal definitions of them, as well as a few related theorems.

 
Syntaxcge-real 28444 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 28446.
 class  >_
 
Syntaxcgt 28445 Extend wff notation to include the 'greater than' relation, see df-gt 28447.
 class  >
 
Definitiondf-gte 28446 Define the 'greater than or equal' predicate over the reals. Defined in ISO 80000-2:2009(E) operation 2-7.10. It is used as a primitive in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. This relation is merely the converse of the 'less than or equal to' relation defined by df-le 8889.

We do not write this as  ( x  >_  y  <->  y  <_  x ), and similarly we do not write ` > ` as  ( x  >  y  <->  y  <  x ), because these are not definitional axioms as understood by mmj2 (those definitions will be flagged as being "potentially non-conservative"). We could write them this way:  |-  >  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <  x ) } and  |-  >_  =  { <. x ,  y
>.  |  ( (
x  e.  RR*  /\  y  e.  RR* )  /\  y  <_  x ) } but these are very complicated. This definition of  >_, and the similar one for  > (df-gt 28447), are a bit strange when you see them for the first time, but these definitions are much simpler for us to process and are clearly conservative definitions. (My thanks to Mario Carneiro for pointing out this simpler approach.) See gte-lte 28448 for a more conventional expression of the relationship between  < and  >. As a stylistic issue, set.mm prefers 'less than' instead of 'greater than' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)

 |-  >_  =  `'  <_
 
Definitiondf-gt 28447 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 8766. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 28446 for a discussion on why this approach is used for the definition. See gt-lt 28449 and gt-lth 28451 for more conventional expression of the relationship between  < and  >.

As a stylistic issue, set.mm prefers 'less than or equal' instead of 'greater than or equal' to reduce the number of conversion steps. Thus, we discourage its use, but include its definition so that there is a formal definition of this symbol.

(Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)

 |-  >  =  `'  <
 
Theoremgte-lte 28448 Simple relationship between  <_ and  >_. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >_  B  <->  B 
 <_  A ) )
 
Theoremgt-lt 28449 Simple relationship between  < and  >. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  (
 ( A  e.  _V  /\  B  e.  _V )  ->  ( A  >  B  <->  B  <  A ) )
 
Theoremgte-lteh 28450 Relationship between  <_ and  >_ using hypotheses. (Contributed by David A. Wheeler, 10-May-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >_  B  <->  B  <_  A )
 
Theoremgt-lth 28451 Relationship between  < and  > using hypotheses. (Contributed by David A. Wheeler, 19-Apr-2015.) (New usage is discouraged.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  ( A  >  B  <->  B  <  A )
 
Theoremex-gt 28452 Simple example of  >, in this case, 0 is not greater than 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  -.  0  >  0
 
Theoremex-gte 28453 Simple example of  >_, in this case, 0 is greater than or equal to 0. This is useful as an example, and helps us gain confidence that we've correctly defined the symbol. (Contributed by David A. Wheeler, 1-Jan-2017.) (New usage is discouraged.)
 |-  0  >_  0
 
18.24.3  Hyperbolic trig functions

It is a convention of set.mm to not use sinh and so on directly, and instead of use expansions such as  ( cos `  ( _i  x.  x ) ). However, I believe it's important to give formal definitions for these conventional functions as they are typically used, so here they are. A few related identities are also proved.

 
Syntaxcsinh 28454 Extend class notation to include the hyperbolic sine function, see df-sinh 28457.
 class sinh
 
Syntaxccosh 28455 Extend class notation to include the hyperbolic cosine function. see df-cosh 28458.
 class cosh
 
Syntaxctanh 28456 Extend class notation to include the hyperbolic tangent function, see df-tanh 28459.
 class tanh
 
Definitiondf-sinh 28457 Define the hyperbolic sine function (sinh). We define it this way for cmpt 4093, which requires the form  (
x  e.  A  |->  B ). See sinhval-named 28460 for a simple way to evaluate it. We define this function by dividing by  _i, which uses fewer operations than many conventional definitions (and thus is more convenient to use in metamath). See sinh-conventional 28463 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |- sinh  =  ( x  e.  CC  |->  ( ( sin `  ( _i  x.  x ) ) 
 /  _i ) )
 
Definitiondf-cosh 28458 Define the hyperbolic cosine function (cosh). We define it this way for cmpt 4093, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- cosh  =  ( x  e.  CC  |->  ( cos `  ( _i  x.  x ) ) )
 
Definitiondf-tanh 28459 Define the hyperbolic tangent function (tanh). We define it this way for cmpt 4093, which requires the form  (
x  e.  A  |->  B ). (Contributed by David A. Wheeler, 10-May-2015.)
 |- tanh  =  ( x  e.  ( `'cosh " ( CC  \  { 0 } )
 )  |->  ( ( tan `  ( _i  x.  x ) )  /  _i ) )
 
Theoremsinhval-named 28460 Value of the named sinh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-sinh 28457. See sinhval 12450 for a theorem to convert this further. See sinh-conventional 28463 for a justification that our definition is the same as the conventional definition of sinh used in other sources. (Contributed by David A. Wheeler, 20-Apr-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  ( ( sin `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremcoshval-named 28461 Value of the named cosh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-cosh 28458. See coshval 12451 for a theorem to convert this further. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (cosh `  A )  =  ( cos `  ( _i  x.  A ) ) )
 
Theoremtanhval-named 28462 Value of the named tanh function. Here we show the simple conversion to the conventional form used in set.mm, using the definition given by df-tanh 28459. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  ( `'cosh " ( CC  \  {
 0 } ) ) 
 ->  (tanh `  A )  =  ( ( tan `  ( _i  x.  A ) ) 
 /  _i ) )
 
Theoremsinh-conventional 28463 Conventional definition of sinh. Here we show that the sinh definition we're using has the same meaning as the conventional definition used in some other sources. We choose a slightly different definition of sinh because it has fewer operations, and thus is more convenient to manipulate using metamath. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  ( A  e.  CC  ->  (sinh `  A )  =  (
 -u _i  x.  ( sin `  ( _i  x.  A ) ) ) )
 
Theoremsinhpcosh 28464 Prove that  (sinh `  A
)  +  (cosh `  A )  =  ( exp `  A ) using the conventional hyperbolic trig functions. (Contributed by David A. Wheeler, 27-May-2015.)
 |-  ( A  e.  CC  ->  ( (sinh `  A )  +  (cosh `  A )
 )  =  ( exp `  A ) )
 
18.24.4  Reciprocal trig functions (sec, csc, cot)

Define the traditional reciprocal trigonometric functions secant (sec), cosecant (csc), and cotangent (cos), along with various identities involving them.

 
Syntaxcsec 28465 Extend class notation to include the secant function, see df-sec 28468.
 class  sec
 
Syntaxccsc 28466 Extend class notation to include the cosecant function, see df-csc 28469.
 class  csc
 
Syntaxccot 28467 Extend class notation to include the cotangent function, see df-cot 28470.
 class  cot
 
Definitiondf-sec 28468* Define the secant function. We define it this way for cmpt 4093, which requires the form  ( x  e.  A  |->  B ). The sec function is defined in ISO 80000-2:2009(E) operation 2-13.6 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  sec  =  ( x  e.  {
 y  e.  CC  |  ( cos `  y )  =/=  0 }  |->  ( 1 
 /  ( cos `  x ) ) )
 
Definitiondf-csc 28469* Define the cosecant function. We define it this way for cmpt 4093, which requires the form  ( x  e.  A  |->  B ). The csc function is defined in ISO 80000-2:2009(E) operation 2-13.7 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  csc  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( 1 
 /  ( sin `  x ) ) )
 
Definitiondf-cot 28470* Define the cotangent function. We define it this way for cmpt 4093, which requires the form  ( x  e.  A  |->  B ). The cot function is defined in ISO 80000-2:2009(E) operation 2-13.5 and "NIST Digital Library of Mathematical Functions" section on "Trigonometric Functions" http://dlmf.nist.gov/4.14 (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  cot  =  ( x  e.  {
 y  e.  CC  |  ( sin `  y )  =/=  0 }  |->  ( ( cos `  x )  /  ( sin `  x ) ) )
 
Theoremsecval 28471 Value of the secant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  =  ( 1  /  ( cos `  A ) ) )
 
Theoremcscval 28472 Value of the cosecant function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  =  ( 1  /  ( sin `  A ) ) )
 
Theoremcotval 28473 Value of the cotangent function. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  =  ( ( cos `  A )  /  ( sin `  A ) ) )
 
Theoremseccl 28474 The closure of the secant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  CC )
 
Theoremcsccl 28475 The closure of the cosecant function with a complex argument. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  CC )
 
Theoremcotcl 28476 The closure of the cotangent function with a complex argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  CC )
 
Theoremreseccl 28477 The closure of the secant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( cos `  A )  =/=  0 )  ->  ( sec `  A )  e.  RR )
 
Theoremrecsccl 28478 The closure of the cosecant function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( csc `  A )  e.  RR )
 
Theoremrecotcl 28479 The closure of the cotangent function with a real argument. (Contributed by David A. Wheeler, 15-Mar-2014.)
 |-  (
 ( A  e.  RR  /\  ( sin `  A )  =/=  0 )  ->  ( cot `  A )  e.  RR )
 
Theoremrecsec 28480 The reciprocal of secant is cosine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( cos `  A )  =  ( 1  /  ( sec `  A ) ) )
 
Theoremreccsc 28481 The reciprocal of cosecant is sine. (Contributed by David A. Wheeler, 14-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( sin `  A )  =  ( 1  /  ( csc `  A ) ) )
 
Theoremreccot 28482 The reciprocal of cotangent is tangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( tan `  A )  =  ( 1  /  ( cot `  A ) ) )
 
Theoremrectan 28483 The reciprocal of tangent is cotangent. (Contributed by David A. Wheeler, 21-Mar-2014.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0  /\  ( cos `  A )  =/=  0 )  ->  ( cot `  A )  =  ( 1  /  ( tan `  A ) ) )
 
Theoremsec0 28484 The value of the secant function at zero is one. (Contributed by David A. Wheeler, 16-Mar-2014.)
 |-  ( sec `  0 )  =  1
 
Theoremonetansqsecsq 28485 Prove the tangent squared secant squared identity  ( 1  +  ( ( tan A ) ^ 2 ) ) = ( ( sec  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 25-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( cos `  A )  =/=  0 )  ->  ( 1  +  (
 ( tan `  A ) ^ 2 ) )  =  ( ( sec `  A ) ^ 2
 ) )
 
Theoremcotsqcscsq 28486 Prove the tangent squared cosecant squared identity  ( 1  +  ( ( cot A ) ^ 2 ) ) = ( ( csc  A ) ^ 2 ) ). (Contributed by David A. Wheeler, 27-May-2015.)
 |-  (
 ( A  e.  CC  /\  ( sin `  A )  =/=  0 )  ->  ( 1  +  (
 ( cot `  A ) ^ 2 ) )  =  ( ( csc `  A ) ^ 2
 ) )
 
18.24.5  Identities for "if"

Utility theorems for "if".

 
Theoremifnmfalse 28487 If A is not a member of B, but an "if" condition requires it, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3585 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e/  B  ->  if ( A  e.  B ,  C ,  D )  =  D )
 
18.24.6  Not-member-of
 
TheoremAnelBC 28488 If an element doesn't match the items in an unordered pair, it is not in the unordered pair, using 
e/. (Contributed by David A. Wheeler, 10-May-2015.)
 |-  A  =/=  B   &    |-  A  =/=  C   =>    |-  A  e/  { B ,  C }
 
18.24.7  Decimal point

Define the decimal point operator and the decimal fraction constructor. This can model traditional decimal point notation, and serve as a convenient way to write some fractional numbers. See df-dp 28492 and df-dp2 28491 for more information; ~? dfpval provides a more convenient way to obtain a value. This is intentionally similar to df-dec 10141.

TODO: Fix non-existent label dfpval.

 
Syntaxcdp2 28489 Constant used for decimal fraction constructor. See df-dp2 28491.
 class _ A B
 
Syntaxcdp 28490 Decimal point operator. See df-dp 28492.
 class  period
 
Definitiondf-dp2 28491 Define the "decimal fraction constructor", which is used to build up "decimal fractions" in base 10. This is intentionally similar to df-dec 10141. (Contributed by David A. Wheeler, 15-May-2015.)
 |- _ A B  =  ( A  +  ( B 
 /  10 ) )
 
Definitiondf-dp 28492* Define the  period (decimal point) operator. For example,  ( 1 period 5 )  =  ( 3  /  2 ), and  -u (; 3 2 period_ 7_ 1 8 )  = 
-u (;;;; 3 2 7 1 8  / ;;; 1 0 0 0 ) Unary minus, if applied, should normally be applied in front of the parentheses.

Metamath intentionally does not have a built-in construct for numbers, so it can show that numbers are something you can build based on set theory. However, that means that metamath has no built-in way to handle decimal numbers as traditionally written, e.g., "2.54", and its parsing system intentionally does not include the complexities necessary to define such a parsing system. Here we create a system for modeling traditional decimal point notation; it is not syntactically identical, but it is sufficiently similar so it is a reasonable model of decimal point notation. It should also serve as a convenient way to write some fractional numbers.

The RHS is  RR, not  QQ; this should simplify some proofs. The LHS is  NN0, since that is what is used in practice. The definition intentionally does not allow negative numbers on the LHS; if it did, nonzero fractions would produce the wrong results. (It would be possible to define the decimal point to do this, but using it would be more complicated, and the expression  -u ( A period B ) is just as convenient.) (Contributed by David A. Wheeler, 15-May-2015.)

 |-  period  =  ( x  e.  NN0 ,  y  e.  RR  |-> _ x y )
 
Theoremdp2cl 28493 Define the closure for the decimal fraction constructor if both values are reals. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  -> _ A B  e.  RR )
 
Theoremdpval 28494 Define the value of the decimal point operator. See df-dp 28492. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  = _ A B )
 
Theoremdpcl 28495 Prove that the closure of the decimal point is  RR as we have defined it. See df-dp 28492. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  e.  RR )
 
Theoremdpfrac1 28496 Prove a simple equivalence involving the decimal point. See df-dp 28492 and dpcl 28495. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (
 ( A  e.  NN0  /\  B  e.  RR )  ->  ( A period B )  =  (; A B  /  10 ) )
 
18.24.8  Signum (sgn or sign) function
 
Syntaxcsgn 28497 Extend class notation to include the Signum function.
 class sgn
 
Definitiondf-sgn 28498 Signum function. Pronounced "signum" , otherwise it might be confused with "sine". Defined as "sgn" in ISO 80000-2:2009(E) operation 2-9.13. It is named "sign" (with the same definition) in the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4. We define this over  RR* (df-xr 8887) instead of  RR so that it can accept  +oo and  -oo. Note that df-psgn 27518 defines the sign of a permutation, which is different. Value shown in sgnval 28499. (Contributed by David A. Wheeler, 15-May-2015.)
 |- sgn  =  ( x  e.  RR*  |->  if ( x  =  0 , 
 0 ,  if ( x  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgnval 28499 Value of Signum function. Pronounced "signum" . See df-sgn 28498. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  e.  RR*  ->  (sgn `  A )  =  if ( A  =  0 ,  0 ,  if ( A  <  0 ,  -u 1 ,  1 ) ) )
 
Theoremsgn0 28500 Proof that signum of 0 is 0. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  (sgn `  0 )  =  0
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