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Theorem List for Metamath Proof Explorer - 28101-28200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremusgrav 28101 The classes of vertices and edges of an undirected simple graph without loops are sets. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( V  e.  _V  /\  E  e.  _V ) )
 
Theoremisuslgra 28102* The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USLGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremisusgra 28103* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 ) )
 
Theoremuslgraf 28104* The edge function of an undirected simple graph with loops is a one to one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremusgraf 28105* The edge function of an undirected simple graph without loops is a one to one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  =  2 }
 )
 
Theoremisusgra0 28106* The property of being an undirected simple graph without loops. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V USGrph  E  <->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 ) )
 
Theoremusgraf0 28107* The edge function of an undirected simple graph without loops is a one to one function into unordered pairs of vertices. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  ( V USGrph  E  ->  E : dom  E -1-1-> { x  e.  ~P V  |  ( # `  x )  =  2 }
 )
 
Theoremusgrafun 28108 The edge function of an undirected simple graph without loops is a function. (Contributed by Alexander van der Vekens, 18-Aug-2017.)
 |-  ( V USGrph  E  ->  Fun  E )
 
Theoremusgraedgop 28109 An edge of an undirected simple graph as second component of an ordered pair. (Contributed by Alexander van der Vekens, 17-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( ( E `  X )  =  { M ,  N }  <->  <. X ,  { M ,  N } >.  e.  E ) )
 
Theoremusgrass 28110 An edge is a subset of vertices, analogous to umgrass 23871. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  F  e.  dom  E ) 
 ->  ( E `  F )  C_  V )
 
Theoremusgraeq12d 28111 Equality of simple graphs without loops. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  (
 ( ( V  e.  X  /\  E  e.  Y )  /\  ( V  =  W  /\  E  =  F ) )  ->  ( V USGrph  E 
 <->  W USGrph  F ) )
 
Theoremuslisumgra 28112 An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USLGrph  E  ->  V UMGrph  E )
 
Theoremusisuslgra 28113 An undirected simple graph without loops is an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USLGrph  E )
 
Theoremusisumgra 28114 An undirected simple graph without loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 20-Aug-2017.)
 |-  ( V USGrph  E  ->  V UMGrph  E )
 
Theoremusgrares 28115 A subgraph of a graph (formed by removing some edges from the original graph) is a graph, analogous to umgrares 23876. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V USGrph  E  ->  V USGrph  ( E  |`  A ) )
 
Theoremusgra0 28116 The empty graph, with vertices but no edges, is a graph, analogous to umgra0 23877. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( V  e.  W  ->  V USGrph  (/) )
 
Theoremusgra0v 28117 The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  ( (/) USGrph  E 
 <->  E  =  (/) )
 
Theoremuslgra1 28118 The graph with one edge, analogous to umgra1 23878. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  V USLGrph  { <. A ,  { B ,  C } >. } )
 
Theoremusgra1 28119 The graph with one edge, analogous to umgra1 23878 ( with additional assumption that  B  =/=  C since otherwise the edge is a loop!). (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  ( B  =/=  C  ->  V USGrph  {
 <. A ,  { B ,  C } >. } )
 )
 
Theoremuslgraun 28120 If  <. V ,  E >. and  <. V ,  F >. are (simple) graphs (with loops), then  <. V ,  E  u.  F >. is a multigraph (the vertex set stays the same, but the edges from both graphs are kept, maybe resulting incident two edges between two vertices), analogous to umgraun 23879. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V USLGrph  E )   &    |-  ( ph  ->  V USLGrph  F )   =>    |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
 
Theoremusgraedg2 28121 The value of the "edge function" of a graph is a set containing two elements (the vertices the corresponding edge is connecting), analogous to umgrale 23873. (Contributed by Alexander van der Vekens, 11-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( # `  ( E `  X ) )  =  2 )
 
Theoremusgraedgprv 28122 In an undirected graph, an edge is an unordered pair of vertices. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  (
 ( V USGrph  E  /\  X  e.  dom  E ) 
 ->  ( ( E `  X )  =  { M ,  N }  ->  ( M  e.  V  /\  N  e.  V ) ) )
 
Theoremusgraedgrnv 28123 An edge of an undirected simple graph always connects to vertices. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  ( M  e.  V  /\  N  e.  V ) )
 
Theoremusgranloop 28124* In an undirected simple graph without loops, there is no edge connecting a vertex with itself. (Contributed by Alexander van der Vekens, 19-Aug-2017.)
 |-  ( V USGrph  E  ->  ( E. x  e.  dom  E ( E `  x )  =  { M ,  N }  ->  M  =/=  N ) )
 
Theoremusgraedgrn 28125 An edge of an undirected simple graph without loops always connects two different vertices. (Contributed by Alexander van der Vekens, 2-Sep-2017.)
 |-  (
 ( V USGrph  E  /\  { M ,  N }  e.  ran  E )  ->  M  =/=  N )
 
18.23.3.8  Undirected simple graphs (examples)
 
Theoremusgra1v 28126 A class with one (or no) vertex is a graph if and only if it has no edges. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( { A } USGrph  E  <->  E  =  (/) )
 
Theoremusgraexvlem 28127 Lemma for usgraexmpl 28133. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  V  =  ( {
 0 ,  1 ,  2 }  u.  {
 3 ,  4 } )
 
Theoremusgraex0elv 28128 Lemma 0 for usgraexmpl 28133. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  0  e.  V
 
Theoremusgraex1elv 28129 Lemma 1 for usgraexmpl 28133. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  1  e.  V
 
Theoremusgraex2elv 28130 Lemma 2 for usgraexmpl 28133. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  2  e.  V
 
Theoremusgraex3elv 28131 Lemma 3 for usgraexmpl 28133. (Contributed by Alexander van der Vekens, 13-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   =>    |-  3  e.  V
 
Theoremusgraexmpldifpr 28132 Lemma for usgraexmpl 28133: all "edges" are different. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  (
 ( { 0 ,  1 }  =/=  {
 1 ,  2 } 
 /\  { 0 ,  1 }  =/=  { 2 ,  0 }  /\  { 0 ,  1 }  =/=  { 0 ,  3 } )  /\  ( { 1 ,  2 }  =/=  { 2 ,  0 }  /\  { 1 ,  2 }  =/=  { 0 ,  3 }  /\  {
 2 ,  0 }  =/=  { 0 ,  3 } ) )
 
Theoremusgraexmpl 28133  <. V ,  E >. is a graph of five vertices  0 ,  1 , 
2 ,  3 ,  4, with edges  { 0 ,  1 } ,  {
1 ,  2 } ,  { 2 ,  0 } ,  {
0 ,  3 }. (Contributed by Alexander van der Vekens, 15-Aug-2017.)
 |-  V  =  ( 0 ... 4
 )   &    |-  E  =  <" {
 0 ,  1 }  { 1 ,  2 }  { 2 ,  0 }  { 0 ,  3 } ">   =>    |-  V USGrph  E
 
18.23.3.9  Neighbors, complete graphs and universal vertices
 
Syntaxcnbgra 28134 Extend class notation with Neighbors (of a vertex in a graph).
 class Neighbors
 
Syntaxccusgra 28135 Extend class notation with complete (undirected simple) graphs.
 class ComplUSGrph
 
Syntaxcuvtx 28136 Extend class notation with the universal vertices (in a graph).
 class UnivVertex
 
Definitiondf-nbgra 28137* Define the class of all Neighbors of a vertex in a graph. The neighbors of a vertex are all vertices which are connected with this vertex by an edge. (Contributed by Alexander van der Vekens and Mario Carneiro, 7-Oct-2017.)
 |- Neighbors  =  ( g  e.  _V ,  k  e.  ( 1st `  g )  |->  { n  e.  ( 1st `  g
 )  |  { k ,  n }  e.  ran  ( 2nd `  g ) } )
 
Definitiondf-cusgra 28138* Define the class of all complete (undirected simple) graphs. A undirected simple graph is called complete if every pair of distinct vertices is connected by a (unique) edge. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- ComplUSGrph  =  { <. v ,  e >.  |  ( v USGrph  e  /\  A. k  e.  v  A. n  e.  ( v  \  { k } ) { n ,  k }  e.  ran  e ) }
 
Definitiondf-uvtx 28139* Define the class of all universal vertices (in a graphs). A vertex is called universal if it is adjacent, i.e. connected by an edge, to all other vertices (of the graph). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |- UnivVertex  =  ( v  e.  _V ,  e  e.  _V  |->  { n  e.  v  |  A. k  e.  ( v  \  { n } ) { k ,  n }  e.  ran  e } )
 
Theoremnbgraop 28140* The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens, 7-Oct-2017.)
 |-  (
 ( ( V  e.  Y  /\  E  e.  Z )  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgrael 28141 The set of neighbors of an element of the first component of an ordered pair, especially of a vertex in a graph. (Contributed by Alexander van der Vekens and Mario Carneiro, 9-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( N  e.  ( <. V ,  E >. Neighbors  K ) 
 <->  ( K  e.  V  /\  N  e.  V  /\  { K ,  N }  e.  ran  E ) ) )
 
Theoremnbgranv0 28142 There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e/  V  ->  ( <. V ,  E >. Neighbors  N )  =  (/) )
 
Theoremnbusgra 28143* The set of neighbors of a vertex in a graph. (Contributed by Alexander van der Vekens, 9-Oct-2017.)
 |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N )  =  { n  e.  V  |  { N ,  n }  e.  ran  E } )
 
Theoremnbgra0nb 28144* A vertex which is not endpoint of an edge has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( A. x  e.  ran  E  N  e/  x  ->  ( <. V ,  E >. Neighbors  N )  =  (/) ) )
 
Theoremnbgraeledg 28145 A class/vertex is a neighbor of another class/vertex if and only if it is an endpoint of an edge. (Contributed by Alexander van der Vekens, 11-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  { N ,  K }  e.  ran  E ) )
 
Theoremnbgraisvtx 28146 Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  ->  N  e.  V ) )
 
Theoremnbgra0edg 28147 In a graph with no edges, every vertex has no neighbor. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph 
 (/)  ->  ( <. V ,  (/)
 >. Neighbors  K )  =  (/) )
 
Theoremnbgrassvt 28148 The neighbors of a node in a graph are a subset of all nodes of the graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( <. V ,  E >. Neighbors  N ) 
 C_  V )
 
Theoremnbgranself 28149* A node in a graph (without loops!) is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  A. v  e.  V  v  e/  ( <. V ,  E >. Neighbors  v
 ) )
 
Theoremnbgrassovt 28150 The neighbors of a vertex are a subset of the other vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  V  ->  ( <. V ,  E >. Neighbors  N ) 
 C_  ( V  \  { N } ) ) )
 
Theoremnbgranself2 28151 A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  N  e/  ( <. V ,  E >. Neighbors  N ) )
 
Theoremnbgrasym 28152 A vertex in a graph is a neighbor of a second vertex if and only if the second vertex is a neighbor of the first vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( V USGrph  E  ->  ( N  e.  ( <. V ,  E >. Neighbors  K )  <->  K  e.  ( <. V ,  E >. Neighbors  N ) ) )
 
Theoremiscusgra 28153* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V ComplUSGrph  E  <->  ( V USGrph  E  /\  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  ran  E ) ) )
 
Theoremiscusgra0 28154* The property of being a complete (undirected simple) graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  ( V USGrph  E 
 /\  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) { n ,  k }  e.  ran  E ) )
 
Theoremcusisusgra 28155 A complete (undirected simple) graph is an undirected simple graph. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  V USGrph  E )
 
Theoremcusgra0v 28156 A graph with no vertices (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  (/) ComplUSGrph  (/)
 
Theoremcusgra1v 28157 A graph with one vertex (and therefore no edges) is complete. (Contributed by Alexander van der Vekens, 13-Oct-2017.)
 |-  { A } ComplUSGrph  (/)
 
Theoremcusgra2v 28158 A graph with two (different) vertices is complete if and only if there is an edge between these two vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( A  e.  V  /\  B  e.  W  /\  A  =/=  B )  ->  ( { A ,  B } USGrph  E  ->  ( { A ,  B } ComplUSGrph  E  <->  { A ,  B }  e.  ran  E ) ) )
 
Theoremnbcusgra 28159 In a complete (undirected simple) graph, each vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V ComplUSGrph  E  /\  N  e.  V )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )
 )
 
Theoremisuvtx 28160* The set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( V UnivVertex  E )  =  { n  e.  V  |  A. k  e.  ( V  \  { n }
 ) { k ,  n }  e.  ran  E } )
 
Theoremuvtxel 28161* An element of the set of all universal vertices. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V  e.  X  /\  E  e.  Y ) 
 ->  ( N  e.  ( V UnivVertex  E )  <->  ( N  e.  V  /\  A. k  e.  ( V  \  { N } ) { k ,  N }  e.  ran  E ) ) )
 
Theoremuvtxisvtx 28162 A universal vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( N  e.  ( V UnivVertex  E )  ->  N  e.  V )
 
Theoremuvtx0 28163 There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  ( (/) UnivVertex  E )  =  (/)
 
Theoremuvtx01vtx 28164* If a graph/class has no edges, it has universal vertices if and only if it has at most one vertex. This theorem could have been stated  ( ( V UnivVertex  (/) )  =/=  (/)  <->  ( # `  V
)  =  1 ), but a lot of auxiliary theorems would have been needed. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
 |-  (
 ( V UnivVertex  (/) )  =/=  (/)  <->  E. x  V  =  { x } )
 
Theoremuvtxnbgra 28165 A universal vertex has all other vertices as neighbors. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  ( <. V ,  E >. Neighbors  N )  =  ( V  \  { N } )
 )
 
Theoremuvtxnm1nbgra 28166 A universal vertex has  n  -  1 neighbors in a graph with  n vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  V  e.  Fin )  ->  ( N  e.  ( V UnivVertex  E )  ->  ( # `
  ( <. V ,  E >. Neighbors  N ) )  =  ( ( # `  V )  -  1 ) ) )
 
Theoremuvtxnbgravtx 28167* A universal vertex is neighbor of all other vertices. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  (
 ( V USGrph  E  /\  N  e.  ( V UnivVertex  E ) )  ->  A. v  e.  ( V  \  { N } ) N  e.  ( <. V ,  E >. Neighbors  v ) )
 
Theoremcusgrauvtx 28168 In a complete (undirected simple) graph, each vertex is a universal vertex. (Contributed by Alexander van der Vekens, 14-Oct-2017.)
 |-  ( V ComplUSGrph  E  ->  ( V UnivVertex  E )  =  V )
 
18.23.3.10  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 28170 as special undirected simple graphs without loops (see frisusgra 28173) and the proofs of the friendship theorem for small graphs (with up to 3 vertices), see 1to3vfriendship 28186. 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 28170, a graph without vertices is a friendship graph (see frgra0 28175), 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 3468). Further results of this sections are: Any graph with exactly one vertex is a friendship graph, see frgra1v 28176, any graph with exactly 2 (different) vertices is not a friendship graph, see frgra2v 28177, 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 28180, and every friendship graph (with 1 or 3 vertices) is a windmill graph, see 1to3vfriswmgra 28185 (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" which was proven by Mertzios and Unger, see "The friendship problem on graphs", ROGICS'08, 12-17 May 2008, Mahida, Tunisia, pp 152-158).

 
Syntaxcfrgra 28169 Extend class notation with Friendship Graphs.
 class FriendGrph
 
Definitiondf-frgra 28170* 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 28171* 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 28172* 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 28173 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 28174 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 28175 Any empty graph (graph without vertices) is a friendship graph. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
 |-  (/) FriendGrph  (/)
 
Theoremfrgra1v 28176 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 28177 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 28178* Lemma 1 for frgra3v 28180. (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 28179* Lemma 2 for frgra3v 28180. (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 28180 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 28181* 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 28182* Lemma for 3vfriswmgra 28183. (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 28183* 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 28184* 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 28185* 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 28186* 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 ) )
 
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 28187 If a wff is true, it is true for at least one instance. Deductive form of 19.8a 1718. (Contributed by DAW, 13-Feb-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theoremsbidd 28188 An identity theorem for substitution. See sbid 1863. 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 28189 An identity theorem for substitution. See sbid 1863. 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 28190 Extend wff notation to include the 'greater than or equal to' relation, see df-gte 28192.
 class  >_
 
Syntaxcgt 28191 Extend wff notation to include the 'greater than' relation, see df-gt 28193.
 class  >
 
Definitiondf-gte 28192 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 8873.

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 28193), 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 28194 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 28193 The 'greater than' relation is merely the converse of the 'less than or equal to' relation defined by df-lt 8750. Defined in ISO 80000-2:2009(E) operation 2-7.12. See df-gte 28192 for a discussion on why this approach is used for the definition. See gt-lt 28195 and gt-lth 28197 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 28194 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 28195 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 28196 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 28197 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 28198 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 28199 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 28200 Extend class notation to include the hyperbolic sine function, see df-sinh 28203.
 class sinh
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