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Theorem List for Metamath Proof Explorer - 21601-21700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcyclnspth 21601 A (non trivial) cycle is not a simple path. (Contributed by Alexander van der Vekens, 30-Oct-2017.)
 |-  ( F  =/=  (/)  ->  ( F ( V Cycles  E ) P  ->  -.  F ( V SPaths  E ) P ) )
 
Theoremcycliswlk 21602 A cycle is a walk. (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( V Walks  E ) P )
 
Theoremcyclispthon 21603 A cycle is a path starting and ending at its first vertex. (Contributed by Alexander van der Vekens, 8-Nov-2017.)
 |-  ( F ( V Cycles  E ) P  ->  F ( ( P `  0 ) ( V PathOn  E ) ( P `
  0 ) ) P )
 
Theoremfargshiftlem 21604 If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  ( ( N  e.  NN0  /\  X  e.  ( 0..^ N ) )  ->  ( X  +  1
 )  e.  ( 1
 ... N ) )
 
Theoremfargshiftfv 21605* If a class is a function, then the values of the "shifted function" correspond to the function values of the class. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( X  e.  ( 0..^ N )  ->  ( G `  X )  =  ( F `  ( X  +  1
 ) ) ) )
 
Theoremfargshiftf 21606* If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  G : ( 0..^ ( # `  F ) ) --> dom  E )
 
Theoremfargshiftf1 21607* If a function is 1-1, then also the shifted function is 1-1. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -1-1-> dom  E )  ->  G :
 ( 0..^ ( # `  F ) ) -1-1-> dom  E )
 
Theoremfargshiftfo 21608* If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) -onto-> dom 
 E )  ->  G : ( 0..^ ( # `  F ) )
 -onto->
 dom  E )
 
Theoremfargshiftfva 21609* The values of a shifted function correspond to the value of the original function. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( ( N  e.  NN0  /\  F : ( 1
 ... N ) --> dom  E )  ->  ( A. k  e.  ( 1 ... N ) ( E `  ( F `  k ) )  =  [_ k  /  x ]_ P  ->  A. l  e.  ( 0..^ N ) ( E `
  ( G `  l ) )  = 
 [_ ( l  +  1 )  /  x ]_ P ) )
 
Theoremusgrcyclnl1 21610 In an undirected simple graph (with no loops!) there are no cycles with length 1 (consisting of one edge ). (Contributed by Alexander van der Vekens, 7-Nov-2017.)
 |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  1 )
 
Theoremusgrcyclnl2 21611 In an undirected simple graph (with no loops!) there are no cycles with length 2 (consisting of two edges ). (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  ( ( V USGrph  E  /\  F ( V Cycles  E ) P )  ->  ( # `
  F )  =/=  2 )
 
Theorem3cycl3dv 21612 In a simple graph, the vertices of a 3-cycle are mutually different. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
 |-  ( ( V USGrph  E  /\  ( { A ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  ( A  =/=  B  /\  B  =/=  C  /\  C  =/=  A ) )
 
Theoremnvnencycllem 21613 Lemma for 3v3e3cycl1 21614 and 4cycl4v4e 21636. (Contributed by Alexander van der Vekens, 9-Nov-2017.)
 |-  ( ( ( Fun 
 E  /\  F  e. Word  dom 
 E )  /\  ( X  e.  NN0  /\  X  <  ( # `  F ) ) )  ->  ( ( E `  ( F `  X ) )  =  { A ,  B }  ->  { A ,  B }  e.  ran  E ) )
 
Theorem3v3e3cycl1 21614* If there is a cycle of length 3 in a graph, there are three (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 )  =  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 ) )
 
Theoremconstr3lem1 21615 Lemma for constr3trl 21629 etc. (Contributed by Alexander van der Vekens, 10-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 >. } )   =>    |-  ( F  e.  _V  /\  P  e.  _V )
 
Theoremconstr3lem2 21616 Lemma for constr3trl 21629 etc. (Contributed by Alexander van der Vekens, 10-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 >. } )   =>    |-  ( # `  F )  =  3
 
Theoremconstr3lem4 21617 Lemma for constr3trl 21629 etc. (Contributed by Alexander van der Vekens, 10-Nov-2017.) (Proof shortened by Alexander van der Vekens, 16-Dec-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 )  =  A  /\  ( P `
  1 )  =  B )  /\  (
 ( P `  2
 )  =  C  /\  ( P `  3 )  =  A ) ) )
 
Theoremconstr3lem5 21618 Lemma for constr3trl 21629 etc. (Contributed by Alexander van der Vekens, 12-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 >. } )   =>    |-  ( ( F `  0 )  =  ( `' E `  { A ,  B } )  /\  ( F `  1 )  =  ( `' E ` 
 { B ,  C } )  /\  ( F `
  2 )  =  ( `' E `  { C ,  A }
 ) )
 
Theoremconstr3lem6 21619 Lemma for constr3pthlem3 21627. (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 ) ,  ( P `  3 ) }  i^i  { ( P `  1 ) ,  ( P `  2 ) }
 )  =  (/) )
 
Theoremconstr3trllem1 21620 Lemma for constr3trl 21629. (Contributed by Alexander van der Vekens, 10-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 ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  F  e. Word  dom  E )
 
Theoremconstr3trllem2 21621 Lemma for constr3trl 21629. (Contributed by Alexander van der Vekens, 12-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 ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  Fun  `' F )
 
Theoremconstr3trllem3 21622 Lemma for constr3trl 21629. (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
 ... ( # `  F ) ) --> V )
 
Theoremconstr3trllem4 21623 Lemma for constr3trl 21629. (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
 ... 3 ) --> V )
 
Theoremconstr3trllem5 21624* Lemma for constr3trl 21629. (Contributed by Alexander van der Vekens, 12-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 ,  B }  e.  ran  E 
 /\  { B ,  C }  e.  ran  E  /\  { C ,  A }  e.  ran  E ) ) 
 ->  A. k  e.  (
 0..^ ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  k ) ,  ( P `  ( k  +  1 ) ) }
 )
 
Theoremconstr3pthlem1 21625 Lemma for constr3pth 21630. (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 21626 Lemma for constr3pth 21630. (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 21627 Lemma for constr3pth 21630. (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 21628 Lemma for constr3cycl 21631. (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 21629 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 21630 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 21631 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 21632 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 21633* 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 21634* 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 21635* 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 21636* 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 21637 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 21638* 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 ) ) ) )
 
14.1.5.4  Connected graphs
 
Syntaxcconngra 21639 Extend class notation with connected graphs.
 class ConnGrph
 
Definitiondf-conngra 21640* Define the class of all connected graphs. A graph (or, more generally, any pair representing a structure consisting of "vertices" and "edges") is called connected if there is a path between every pair of (distinct) vertices. The distinctness of the vertices is not necessary for the definition, because there is always a path (of length 0) from a vertex to itself, see 0pthonv 21564 and dfconngra1 21641. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |- ConnGrph  =  { <. v ,  e >.  |  A. k  e.  v  A. n  e.  v  E. f E. p  f ( k ( v PathOn  e ) n ) p }
 
Theoremdfconngra1 21641* Alternative definition of the class of all connected graphs, requiring paths between distinct vertices. (Contributed by Alexander van der Vekens, 3-Dec-2017.)
 |- ConnGrph  =  { <. v ,  e >.  |  A. k  e.  v  A. n  e.  ( v  \  {
 k } ) E. f E. p  f ( k ( v PathOn  e
 ) n ) p }
 
Theoremisconngra 21642* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ConnGrph  E  <->  A. k  e.  V  A. n  e.  V  E. f E. p  f ( k ( V PathOn  E ) n ) p ) )
 
Theoremisconngra1 21643* The property of being a connected graph. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( ( V  e.  X  /\  E  e.  Y )  ->  ( V ConnGrph  E  <->  A. k  e.  V  A. n  e.  ( V 
 \  { k }
 ) E. f E. p  f ( k ( V PathOn  E ) n ) p ) )
 
Theorem0conngra 21644 A class/graph without vertices is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( E  e.  V  -> 
 (/) ConnGrph  E )
 
Theorem1conngra 21645 A class/graph with (at most) one vertex is connected. (Contributed by Alexander van der Vekens, 2-Dec-2017.)
 |-  ( E  e.  V  ->  { A } ConnGrph  E )
 
Theoremcusconngra 21646 A complete (undirected simple) graph is connected. (Contributed by Alexander van der Vekens, 4-Dec-2017.)
 |-  ( V ComplUSGrph  E  ->  V ConnGrph  E )
 
14.1.6  Vertex Degree
 
Syntaxcvdg 21647 Extend class notation with the vertex degree function.
 class VDeg
 
Definitiondf-vdgr 21648* Define the vertex degree function (for an undirected multigraph). To be appropriate for multigraphs, we have to double-count those edges that contain  u "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. Since the degree of a vertex can be (positive) infinity (if the graph containing the vertex is not of finite size), the extended addition  + e is used for the summation of the number of "ordinary" edges" and the number of "loops". (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
 |- VDeg  =  ( v  e.  _V ,  e  e.  _V  |->  ( u  e.  v  |->  ( ( # `  { x  e.  dom  e  |  u  e.  ( e `  x ) } ) + e
 ( # `  { x  e.  dom  e  |  ( e `  x )  =  { u } } ) ) ) )
 
Theoremvdgrfval 21649* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X ) 
 ->  ( V VDeg  E )  =  ( u  e.  V  |->  ( ( # ` 
 { x  e.  A  |  u  e.  ( E `  x ) }
 ) + e ( # `  { x  e.  A  |  ( E `
  x )  =  { u } }
 ) ) ) )
 
Theoremvdgrval 21650* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  /\  U  e.  V )  ->  ( ( V VDeg  E ) `  U )  =  (
 ( # `  { x  e.  A  |  U  e.  ( E `  x ) } ) + e
 ( # `  { x  e.  A  |  ( E `
  x )  =  { U } }
 ) ) )
 
Theoremvdgrfival 21651* The value of the vertex degree function (for graphs of finite size). (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
 |-  ( ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  /\  U  e.  V )  ->  ( ( V VDeg  E ) `  U )  =  (
 ( # `  { x  e.  A  |  U  e.  ( E `  x ) } )  +  ( # `
  { x  e.  A  |  ( E `
  x )  =  { U } }
 ) ) )
 
Theoremvdgrf 21652 The vertex degree function is a function from vertices to nonnegative integers or plus infinity. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X ) 
 ->  ( V VDeg  E ) : V --> ( NN0  u. 
 {  +oo } ) )
 
Theoremvdgrfif 21653 The vertex degree function on graphs of finite size is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )
 
Theoremvdgr0 21654 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V  e.  W  /\  U  e.  V )  ->  ( ( V VDeg  (/) ) `  U )  =  0 )
 
Theoremvdgrun 21655 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  B  e.  Y )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  ( E  u.  F ) ) `
  U )  =  ( ( ( V VDeg 
 E ) `  U ) + e ( ( V VDeg  F ) `  U ) ) )
 
Theoremvdgrfiun 21656 The degree of a vertex in the union of two graphs (of finite size) on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Jan-2018.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  ( E  u.  F ) ) `
  U )  =  ( ( ( V VDeg 
 E ) `  U )  +  ( ( V VDeg  F ) `  U ) ) )
 
Theoremvdgr1d 21657 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { U } >. } ) `  U )  =  2 )
 
Theoremvdgr1b 21658 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  ( ( V VDeg  { <. A ,  { U ,  B } >. } ) `  U )  =  1
 )
 
Theoremvdgr1c 21659 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  ( ( V VDeg  { <. A ,  { B ,  U } >. } ) `  U )  =  1
 )
 
Theoremvdgr1a 21660 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 22-Dec-2017.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  C  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { B ,  C } >. } ) `  U )  =  0 )
 
Theoremvdusgraval 21661* The value of the vertex degree function for simple undirected graphs. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  ( ( V VDeg  E ) `  U )  =  ( # `  { x  e.  dom  E  |  U  e.  ( E `  x ) } ) )
 
Theoremvdusgra0nedg 21662* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge. (Contributed by Alexander van der Vekens, 8-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V  /\  E  e.  Fin )  ->  ( ( ( V VDeg 
 E ) `  U )  =  0  ->  -. 
 E. v  e.  V  { U ,  v }  e.  ran  E ) )
 
Theoremvdgrnn0pnf 21663 The degree of a vertex is either a nonnegative integer or positive infinity. (Contributed by Alexander van der Vekens, 30-Dec-2017.)
 |-  ( ( V USGrph  E  /\  X  e.  V ) 
 ->  ( ( V VDeg  E ) `  X )  e.  ( NN0  u.  {  +oo } ) )
 
Theoremhashnbgravd 21664 The size of the set of the neighbors of a vertex is the vertex degree of this vertex. (Contributed by Alexander van der Vekens, 17-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V  /\  E  e.  Fin )  ->  ( # `  ( <. V ,  E >. Neighbors  U ) )  =  ( ( V VDeg  E ) `  U ) )
 
Theoremhashnbgravdg 21665 The size of the set of the neighbors of a vertex is the vertex degree of this vertex, analogous to hashnbgravd 21664. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  ( # `  ( <. V ,  E >. Neighbors  U ) )  =  (
 ( V VDeg  E ) `  U ) )
 
Theoremusgravd0nedg 21666* If a vertex in a simple graph has degree 0, the vertex is not adjacent to another vertex via an edge, analogous to vdusgra0nedg 21662. (Contributed by Alexander van der Vekens, 20-Dec-2017.)
 |-  ( ( V USGrph  E  /\  U  e.  V ) 
 ->  ( ( ( V VDeg 
 E ) `  U )  =  0  ->  -. 
 E. v  e.  V  { U ,  v }  e.  ran  E ) )
 
14.2  Eulerian paths and the Konigsberg Bridge problem
 
14.2.1  Eulerian paths
 
Syntaxceup 21667 Extend class notation with Eulerian paths.
 class EulPaths
 
Definitiondf-eupa 21668* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- EulPaths  =  ( v  e.  _V ,  e  e.  _V  |->  {
 <. f ,  p >.  |  ( v UMGrph  e  /\  E. n  e.  NN0  (
 f : ( 1
 ... n ) -1-1-onto-> dom  e  /\  p : ( 0
 ... n ) --> v  /\  A. k  e.  ( 1
 ... n ) ( e `  ( f `
  k ) )  =  { ( p `
  ( k  -  1 ) ) ,  ( p `  k
 ) } ) ) } )
 
Theoremreleupa 21669 The set  ( V EulPaths  E ) of all Eulerian paths on  <. V ,  E >. is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- 
 Rel  ( V EulPaths  E )
 
Theoremiseupa 21670* The property " <. F ,  P >. is an Eulerian path on the graph  <. V ,  E >.". An Eulerian path is defined as bijection  F from the edges to a set  1 ... N a function  P :
( 0 ... N
) --> V into the vertices such that for each 
1  <_  k  <_  N,  F ( k ) is an edge from  P ( k  -  1 ) to  P
( k ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( dom  E  =  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1
 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1
 ... n ) ( E `  ( F `
  k ) )  =  { ( P `
  ( k  -  1 ) ) ,  ( P `  k
 ) } ) ) ) )
 
Theoremeupagra 21671 If an eulerian path exists, then 
<. V ,  E >. is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  V UMGrph  E )
 
Theoremeupai 21672* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( ( ( # `  F )  e.  NN0  /\  F : ( 1
 ... ( # `  F ) ) -1-1-onto-> A  /\  P :
 ( 0 ... ( # `
  F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  ( k  -  1
 ) ) ,  ( P `  k ) }
 ) )
 
Theoremeupatrl 21673* An Eulerian path is a trail.

Unfortunately, the edge function  F of an Eulerian path has the domain  ( 1 ... ( # `  F
) ), whereas the edge functions of all kinds of walks defined here have the domain  ( 0..^ ( # `  F
) ) (i.e. the edge functions are "words of edge indices", see discussion and proposal of Mario Carneiro at https://groups.google.com/d/msg/metamath/KdVXdL3IH3k/2-BYcS_ACQAJ). Therefore, the arguments of the edge function of an Eulerian path must be shifted by 1 to obtain an edge function of a trail in this theorem, using the auxiliary theorems above (fargshiftlem 21604, fargshiftfv 21605, etc.). TODO: The definition of an Eulerian path and all related theorems should be modified to fit to the general definition of a trail. (Contributed by Alexander van der Vekens, 24-Nov-2017.)

 |-  G  =  ( x  e.  ( 0..^ ( # `  F ) ) 
 |->  ( F `  ( x  +  1 )
 ) )   =>    |-  ( F ( V EulPaths  E ) P  ->  G ( V Trails  E ) P )
 
Theoremeupacl 21674 An Eulerian path has length 
# ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  ( # `  F )  e. 
 NN0 )
 
Theoremeupaf1o 21675 The  F function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  F : ( 1 ... ( # `  F ) ) -1-1-onto-> A )
 
Theoremeupafi 21676 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  A  e.  Fin )
 
Theoremeupapf 21677 The  P function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  P : ( 0 ... ( # `  F ) ) --> V )
 
Theoremeupaseg 21678 The  N-th edge in an eulerian path is the edge from  P ( N  - 
1 ) to  P ( N ). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ( F ( V EulPaths  E ) P  /\  N  e.  ( 1 ... ( # `  F ) ) )  ->  ( E `  ( F `
  N ) )  =  { ( P `
  ( N  -  1 ) ) ,  ( P `  N ) } )
 
Theoremeupa0 21679 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( V  e.  W  /\  A  e.  V )  ->  (/) ( V EulPaths  (/) ) { <. 0 ,  A >. } )
 
Theoremeupares 21680 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  ( 0 ... ( # `  G ) ) )   &    |-  F  =  ( E  |`  ( G
 " ( 1 ...
 N ) ) )   &    |-  H  =  ( G  |`  ( 1 ... N ) )   &    |-  Q  =  ( P  |`  ( 0 ... N ) )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupap1 21681 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  =  ( # `  G ) )   &    |-  F  =  ( E  u.  { <. B ,  { ( P `
  N ) ,  C } >. } )   &    |-  H  =  ( G  u.  { <. ( N  +  1 ) ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupath2lem1 21682 Lemma for eupath2 21685. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupath2lem2 21683 Lemma for eupath2 21685. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  B  e.  _V   =>    |-  ( ( B  =/=  C  /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  U  e.  if ( A  =  C ,  (/) ,  { A ,  C } ) ) )
 
Theoremeupath2lem3 21684* Lemma for eupath2 21685. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( N  +  1 ) 
 <_  ( # `  F ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  ( E  |`  ( F "
 ( 1 ... N ) ) ) ) `
  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   =>    |-  ( ph  ->  ( -.  2  ||  ( ( V VDeg  ( E  |`  ( F
 " ( 1 ... ( N  +  1 ) ) ) ) ) `  U )  <->  U  e.  if (
 ( P `  0
 )  =  ( P `
  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupath2 21685* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( # `
  F ) ) ,  (/) ,  { ( P `  0 ) ,  ( P `  ( # `
  F ) ) } ) )
 
Theoremeupath 21686* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ( V EulPaths  E )  =/=  (/)  ->  ( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }
 )  e.  { 0 ,  2 } )
 
14.2.2  The Konigsberg Bridge problem
 
Theoremvdeg0i 21687 The base case for the induction for calculating the degree of a vertex. The degree of  U in the empty graph is  0. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  V  e.  _V   &    |-  U  e.  V   =>    |-  ( ( V VDeg  (/) ) `  U )  =  0
 
Theoremumgrabi 21688* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  X  e.  V   &    |-  Y  e.  V   =>    |-  ( ph  ->  { X ,  Y }  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremvdegp1ai 21689* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  Y } to the edge set, where  X  =/=  U  =/= 
Y, yields degree  P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  Y  e.  V   &    |-  Y  =/=  U   &    |-  F  =  ( E concat  <" { X ,  Y } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  P
 
Theoremvdegp1bi 21690* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { U ,  X } to the edge set, where 
X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { U ,  X } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremvdegp1ci 21691* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  U } to the edge set, where  X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { X ,  U } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremkonigsberg 21692 The Konigsberg Bridge problem. If  <. V ,  E >. is the graph on four vertices  0 ,  1 ,  2 ,  3, with edges  { 0 ,  1 } ,  { 0 ,  2 } ,  { 0 ,  3 } ,  {
1 ,  2 } ,  { 1 ,  2 } ,  {
2 ,  3 } ,  { 2 ,  3 }, then vertices  0 ,  1 ,  3 each have degree three, and  2 has degree five, so there are four vertices of odd degree and thus by eupath 21686 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  =  ( 0
 ... 3 )   &    |-  E  =  <" { 0 ,  1 }  {
 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  { 1 ,  2 }  {
 2 ,  3 }  { 2 ,  3 } ">   =>    |-  ( V EulPaths  E )  =  (/)
 
PART 15  GUIDES AND MISCELLANEA
 
15.1  Guides (conventions, explanations, and examples)
 
15.1.1  Conventions

This section describes the conventions we use. However, these conventions often refer to existing mathematical practices, which are discussed in more detail in other references. Logic and set theory provide a foundation for all of mathematics. To learn about them, you should study one or more of the references listed below. We indicate references using square brackets. The textbooks provide a motivation for what we are doing, whereas Metamath lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts include the following:

  • Axioms of propositional calculus - [Margaris].
  • Axioms of predicate calculus - [Megill] (System S3' in the article referenced).
  • Theorems of propositional calculus - [WhiteheadRussell].
  • Theorems of pure predicate calculus - [Margaris].
  • Theorems of equality and substitution - [Monk2], [Tarski], [Megill].
  • Axioms of set theory - [BellMachover].
  • Development of set theory - [TakeutiZaring]. (The first part of [Quine] has a good explanation of the powerful device of "virtual" or class abstractions, which is essential to our development.)
  • Construction of real and complex numbers - [Gleason]
  • Theorems about real numbers - [Apostol]
 
Theoremconventions 21693 Here are some of the conventions we use in the Metamath Proof Explorer (aka "set.mm"), and how they correspond to typical textbook language (skipping the many cases where they're identical):

  • Notation. Where possible, the notation attempts to conform to modern conventions, with variations due to our choice of the axiom system or to make proofs shorter. However, our notation is strictly sequential (left-to-right). For example, summation is written in the form  sum_ k  e.  A B (df-sum 12463) which denotes that index variable  k ranges over  A when evaluating  B. Thus,  sum_ k  e.  NN  ( 1  /  ( 2 ^ k ) )  =  1 means 1/2 + 1/4 + 1/8 + ... = 1 (geoihalfsum 12642). Also, the method of definition, the axioms for predicate calculus, and the development of substitution are somewhat different from those found in standard texts. For example, the expressions for the axioms were designed for direct derivation of standard results without excessive use of metatheorems. (See Theorem 9.7 of [Megill] p. 448 for a rigorous justification.) The notation is usually explained in more detail when first introduced.
  • Axiomatic assertions ($a). All axiomatic assertions ($a statements) starting with "  |-" have labels starting with "ax-" (axioms) or "df-" (definitions). A statement with a label starting with "ax-" corresponds to what is traditionally called an axiom. A statement with a label starting with "df-" introduces new symbols or a new relationship among symbols that can be eliminated; they always extend the definition of a wff or class. Metamath blindly treats $a statements as new given facts but does not try to justify them. The mmj2 program will justify the definitions as sound, except for 4 (df-bi, df-cleq, df-clel, df-clab) that require a more complex metalogical justification by hand.
  • Proven axioms. In some cases we wish to treat an expression as an axiom in later theorems, even though it can be proved. For example, we derive the postulates or axioms of complex arithmetic as theorems of ZFC set theory. For convenience, after deriving the postulates we re-introduce them as new axioms on top of set theory. This lets us easily identify which axioms are needed for a particular complex number proof, without the obfuscation of the set theory used to derive them. For more, see http://us.metamath.org/mpeuni/mmcomplex.html. When we wish to use a previously-proven assertion as an axiom, our convention is that we use the regular "ax-NAME" label naming convention to define the axiom, but we precede it with a proof of the same statement with the label "axNAME" . An example is complex arithmetic axiom ax-1cn 9032, proven by the preceding theorem ax1cn 9008. The metamath.exe program will warn if an axiom does not match the preceding theorem that justifies it if the names match in this way.
  • Definitions (df-...). We encourage definitions to include hypertext links to proven examples.
  • Statements with hypotheses. Many theorems and some axioms, such as ax-mp 8, have hypotheses that must be satisfied in order for the conclusion to hold, in this case min and maj. When presented in summarized form such as in the Theorem List (click on "Nearby theorems" on the ax-mp 8 page), the hypotheses are connected with an ampersand and separated from the conclusion with a big arrow, such as in "  |-  ph &  |-  ( ph  ->  ps ) =>  |-  ps". These symbols are not part of the Metamath language but are just informal notation meaning "and" and "implies".
  • Discouraged use and modification. If something should only be used in limited ways, it is marked with "(New usage is discouraged.)". This is used, for example, when something can be constructed in more than one way, and we do not want later theorems to depend on that specific construction. This marking is also used if we want later proofs to use proven axioms. For example, we want later proofs to use ax-1cn 9032 (not ax1cn 9008) and ax-1ne0 9043 (not ax1ne0 9019), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9008 and ax1ne0 9019 are marked as "(New usage is discouraged.)". In some cases a proof should not normally be changed, e.g., when it demonstrates some specific technique. These are marked with "(Proof modification is discouraged.)".
  • New definitions infrequent. Typically, we are minimalist when introducing new definitions; they are introduced only when a clear advantage becomes apparent for reducing the number of symbols, shortening proofs, etc. We generally avoid the introduction of gratuitous definitions because each one requires associated theorems and additional elimination steps in proofs. For example, we use  < and  <_ for inequality expressions, and use  ( ( sin `  ( _i  x.  A ) )  /  _i ) instead of  (sinh `  A ) for the hyperbolic sine.
  • Axiom of choice. The axiom of choice (df-ac 7981) is widely accepted, and ZFC is the most commonly-accepted fundamental set of axioms for mathematics. However, there have been and still are some lingering controversies about the Axiom of Choice. Therefore, where a proof does not require the axiom of choice, we prefer that proof instead. E.g., our proof of the Schroeder-Bernstein Theorem (sbth 7213) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8299) or axiom of dependent choice (ax-dc 8310) can be used instead.
  • Variables. Typically, Greek letters ( ph = phi,  ps = psi,  ch = chi, etc.),... are used for propositional (wff) variables;  x,  y,  z,... for individual (set) variables; and  A,  B,  C,... for class variables.
  • Turnstile. " |-", meaning "It is provable that," is the first token of all assertions and hypotheses that aren't syntax constructions. This is a standard convention in logic. For us, it also prevents any ambiguity with statements that are syntax constructions, such as "wff  -.  ph".
  • Biconditional ( <->). There are basically two ways to maximize the effectiveness of biconditionals ( <->): you can either have one-directional simplifications of all theorems that produce biconditionals, or you can have one-directional simplifications of theorems that consume biconditionals. Some tools (like Lean) follow the first approach, but set.mm follows the second approach. Practically, this means that in set.mm, for every theorem that uses an implication in the hypothesis, like ax-mp 8, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 200 or mpbir 201. We prefer this second approach because the number of duplications in the second approach is bounded by the size of the propositional calculus section, which is much smaller than the number of possible theorems in all later sections that produce biconditionals. So although theorems like biimpi 187 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 887, sylbir 205, or 3imtr4i 258.
  • Substitution. " [ y  /  x ] ph" should be read "the wff that results from the proper substitution of  y for  x in wff  ph." See df-sb 1659 and the related df-sbc 3149 and df-csb 3239.
  • Is-a set. " A  e.  _V" should be read "Class  A is a set (i.e. exists)." This is a convenient convention based on Definition 2.9 of [Quine] p. 19. See df-v 2945 and isset 2947.
  • Converse. " `' R" should be read "converse of (relation)  R" and is the same as the more standard notation R^{-1} (the standard notation is ambiguous). See df-cnv 4872. This can be used to define a subset, e.g., df-tan 12657 notates "the set of values whose cosine is a nonzero complex number" as  ( `' cos " ( CC  \  { 0 } ) ).
  • Function application. "( F `  x)" should be read "the value of function  F at  x" and has the same meaning as the more familiar but ambiguous notation F(x). For example,  ( cos `  0 )  =  1 (see cos0 12734). The left apostrophe notation originated with Peano and was adopted in Definition *30.01 of [WhiteheadRussell] p. 235, Definition 10.11 of [Quine] p. 68, and Definition 6.11 of [TakeutiZaring] p. 26. See df-fv 5448. In the ASCII (input) representation there are spaces around the grave accent; there is a single accent when it is used directly, and it is doubled within comments.
  • Infix and parentheses. When a function that takes two classes and produces a class is applied as part of an infix expression, the expression is always surrounded by parentheses (see df-ov 6070). For example, the  + in  ( 2  +  2 ); see 2p2e4 10082. Function application is itself an example of this. Similarly, predicate expressions in infix form that take two or three wffs and produce a wff are also always surrounded by parentheses, such as  ( ph  ->  ps ),  ( ph  \/  ps ),  ( ph  /\  ps ), and  ( ph  <->  ps ) (see wi 4, df-or 360, df-an 361, and df-bi 178 respectively). In contrast, a binary relation (which compares two classes and produces a wff) applied in an infix expression is not surrounded by parentheses. This includes set membership  A  e.  B (see wel 1726), equality  A  =  B (see df-cleq 2423), subset  A  C_  B (see df-ss 3321), and less-than  A  <  B (see df-lt 8987). For the general definition of a binary relation in the form  A R B, see df-br 4200. For example,  0  <  1 ( see 0lt1 9534) does not use parentheses.
  • Unary minus. The symbol  -u is used to indicate a unary minus, e.g.,  -u 1. It is specially defined because it is so commonly used. See cneg 9276.
  • Function definition. Functions are typically defined by first defining the constant symbol (using $c) and declaring that its symbol is a class with the label cNAME (e.g., ccos 12650). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 12656). Typically, there are other proofs such as its closure labelled NAMEcl (e.g., coscl 12711), its function application form labelled NAMEval (e.g., cosval 12707), and at least one simple value (e.g., cos0 12734).
  • Factorial. The factorial function is traditionally a postfix operation, but we treat it as a normal function applied in prefix form, e.g.,  ( ! `  4 )  = ; 2 4 (df-fac 11550 and fac4 11557).
  • Unambiguous symbols. A given symbol has a single unambiguous meaning in general. Thus, where the literature might use the same symbol with different meanings, here we use different (variant) symbols for different meanings. These variant symbols often have suffixes, subscripts, or underlines to distinguish them. For example, here " 0" always means the value zero (df-0 8981), while " 0g" is the group identity element (df-0g 13710), " 0." is the poset zero (df-p0 14451), " 0 p" is the zero polynomial (df-0p 19545), " 0vec" is the zero vector in a normed complex vector space (df-0v 22060), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 14453). There are other class variables used as connective symbols where traditional notation would use ambiguous symbols, including " .1.", " .+", " .*", and " .||". These symbols are very similar to traditional notation, but because they are different symbols they eliminate ambiguity.
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use  NN (df-nn 9985) for the integer numbers starting from 1, and  NN0 (df-n0 10206) for the set of nonnegative integers starting at zero.
  • Decimal numbers. Numbers larger than ten are often expressed in base 10 using the decimal constructor df-dec 10367, e.g., ;;; 4 0 0 1 (see 4001prm 13447 for a proof that 4001 is prime).
  • Theorem forms. We often call a theorem a "deduction" whenever the conclusion and all hypotheses are each prefixed with the same antecedent  ph  ->. Deductions are often the preferred form for theorems because they allow us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem would be used. See, for example, a1d 23. A deduction hypothesis can have an indirect antecedent via definitions, e.g., see lhop 19883. Deductions have a label suffix of "d" if there are other forms of the same theorem. By contrast, we tend to call the simpler version with no common antecedent an "inference" and suffix its label with "i"; compare theorem a1i 11. Finally, a "tautology" would be the form with no hypotheses, and its label would have no suffix. For example, see pm2.43 49, pm2.43i 45, and pm2.43d 46.
  • Deduction theorem. The Deduction Theorem is a metalogical theorem that provides an algorithm for constructing a proof of a theorem from the proof of its corresponding deduction. In ordinary mathematics, no one actually carries out the algorithm, because (in its most basic form) it involves an exponential explosion of the number of proof steps as more hypotheses are eliminated. Instead, in ordinary mathematics the Deduction Theorem is invoked simply to claim that something can be done in principle, without actually doing it. For more details, see http://us.metamath.org/mpeuni/mmdeduction.html. The Deduction Theorem is a metalogical theorem that cannot be applied directly in metamath, and the explosion of steps would be a problem anyway, so alternatives are used. One alternative we use sometimes is the "weak deduction theorem" dedth 3767, which works in certain cases in set theory. We also sometimes use dedhb 3091. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in the deduction theorem form (aka "deduction style") described earlier; the prefixed  ph  -> mimics the context in a deduction proof system. In practice this mechanism works very well. This approach is described in the deduction form and natural deduction page; a list of translations for common natural deduction rules is given in natded 21694.
  • Recursion. We define recursive functions using various "recursion constructors". These allow us to define, with compact direct definitions, functions that are usually defined in textbooks with indirect self-referencing recursive definitions. This produces compact definition and much simpler proofs, and greatly reduces the risk of creating unsound definitions. Examples of recursion constructors include recs ( F ) in df-recs 6619,  rec ( F ,  I ) in df-rdg 6654, seq𝜔 ( F ,  I ) in df-seqom 6691, and  seq  M (  .+  ,  F ) in df-seq 11307. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 13711 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 6619, but for the "average user" the most useful one is probably df-seq 11307- provided that a countable sequence is sufficient for the recursion.
  • Extensible structures. Mathematics includes many structures such as ring, group, poset, etc. We define an "extensible structure" which is then used to define group, ring, poset, etc. This allows theorems from more general structures (groups) to be reused for more specialized structures (rings) without having to reprove them. See df-struct 13454.
  • Organizing proofs. Humans have trouble understanding long proofs. It is often preferable to break longer proofs into smaller parts (just as with traditional proofs). In Metamath this is done by creating separate proofs of the separate parts. A proof with the sole purpose of supporting a final proof is a lemma; the naming convention for a lemma is the final proof's name followed by "lem", and a number if there is more than one. E.g., sbthlem1 7203 is the first lemma for sbth 7213. Also, consider proving reusable results separately, so that others will be able to easily reuse that part of your work.
  • Hypertext links. We strongly encourage comments to have many links to related material, with accompanying text that explains the relationship. These can help readers understand the context. Links to other statements, or to HTTP/HTTPS URLs, can be inserted in ASCII source text by prepending a space-separated tilde. When metamath.exe is used to generate HTML it automatically inserts hypertext links for syntax used (e.g., every symbol used), every axiom and definition depended on, the justification for each step in a proof, and to both the next and previous assertion.
  • Bibliography references. Please include a bibliographic reference to any external material used. A name in square brackets in a comment indicates a bibliographic reference. The full reference must be of the form KEYWORD IDENTIFIER? NOISEWORD(S)* [AUTHOR(S)] p. NUMBER - note that this is a very specific form that requires a page number. There should be no comma between the author reference and the "p." (a constant indicator). Whitespace, comma, period, or semicolon should follow NUMBER. An example is Theorem 3.1 of [Monk1] p. 22, The KEYWORD, which is not case-sensitive, must be one of the following: Axiom, Chapter, Compare, Condition, Corollary, Definition, Equation, Example, Exercise, Figure, Item, Lemma, Lemmas, Line, Lines, Notation, Part, Postulate, Problem, Property, Proposition, Remark, Rule, Scheme, Section, or Theorem. The IDENTIFIER is optional, as in for example "Remark in [Monk1] p. 22". The NOISEWORDS(S) are zero or more from the list: from, in, of, on. The AUTHOR(S) must be present in the file identified with the htmlbibliography assignment (e.g. mmset.html) as a named anchor (NAME=). If there is more than one document by the same author(s), add a numeric suffix (as shown here). The NUMBER is a page number, and may be any alphanumeric string such as an integer or Roman numeral. Note that we require page numbers in comments for individual $a or $p statements. We allow names in square brackets without page numbers (a reference to an entire document) in heading comments. If this is a new reference, please also add it to the "Bibliography" section of mmset.html. (The file mmbiblio.html is automatically rebuilt, e.g., using the metamath.exe "write bibliography" command.)
  • Input format. The input is in ASCII with two-space indents. Tab characters are not allowed. Use embedded math comments or HTML entities for non-ASCII characters (e.g., "&eacute;" for "é").
  • Information on syntax, axioms, and definitions. For a hyperlinked list of syntax, axioms, and definitions, see http://us.metamath.org/mpeuni/mmdefinitions.html. If you have questions about a specific symbol or axiom, it is best to go directly to its definition to learn more about it. The generated HTML for each theorem and axiom includes hypertext links to each symbol's definition.

Naming conventions

Every statement has a unique identifying label. We use various label naming conventions to provide easy-to-remember hints about their contents. Labels are not a 1-to-1 mapping, because that would create long names that would be difficult to remember and tedious to type. Instead, label names are relatively short while suggesting their purpose. Names are occasionally changed to make them more consistent or as we find better ways to name them. Here are a few of the label naming conventions:

  • Axioms, definitions, and wff syntax. As noted earlier, axioms are named "ax-NAME", proofs of proven axioms are named "axNAME", and definitions are named "df-NAME". Wff syntax declarations have labels beginning with "w" followed by short fragment suggesting its purpose.
  • Hypotheses. Hypotheses have the name of the final axiom or theorem, followed by ".", followed by a unique id.
  • Common names. If a theorem has a well-known name, that name (or a short version of it) is sometimes used directly. Examples include barbara 2377 and stirling 27747.
  • Syntax label fragments. Most theorems are named using syntax label fragments. Almost every syntactic construct has a definition labelled "df-NAME", and NAME normally is the syntax label fragment. For example, the difference construct  ( A  \  B ) is defined in df-dif 3310, and thus its syntax label fragment is "dif". Similarly, the singleton construct  { A } has syntax label fragment "sn" (because it is defined in df-sn 3807), the subclass (subset) relation  A  C_  B has "ss" (because it is defined in df-ss 3321), and the pair construct  { A ,  B } has "pr" (df-pr 3808). Theorem names are often a concatenation of the syntax label fragments (omitting variables). For example, a theorem about  ( A  \  B )  C_  A involves a difference ("dif") of a subset ("ss"), and thus is named difss 3461. Digits are used to represent themselves. Suffixes (e.g., with numbers) are sometimes used to distinguish multiple theorems that would otherwise produce the same label.
  • Phantom definitions. In some cases there are common label fragments for something that could be in a definition, but for technical reasons is not. The is-element-of (is member of) construct  A  e.  B does not have a df-NAME definition; in this case its syntax label fragment is "el". Thus, because the theorem beginning with  ( A  e.  ( B  \  { C } ) uses is-element-of ("el") of a difference ("dif") of a singleton ("sn"), it is named eldifsn 3914. An "n" is often used for negation ( -.), e.g., nan 564.
  • Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The table below attempts to list all such cases and marks them in bold. For example, label fragment "cn" represents complex numbers  CC (even though its definition is in df-c 8980) and "re" represents real numbers  RR. The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3616. Syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 8985), but "p" is used as the fragment for constant theorems. Equality  ( A  =  B ) often uses "e" as the fragment. As a result, "two plus two equals four" is named 2p2e4 10082.
  • Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
  • Principia Mathematica. Proofs of theorems from Principia Mathematica often use a different naming convention. They are instead often named "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 37.
  • Closures and values. As noted above, if a function df-NAME is defined, there is typically a proof of its value named "NAMEval" and its closure named "NAMEcl". E.g., for cosine (df-cos 12656) we have value cosval 12707 and closure coscl 12711.
  • Special cases. Sometimes syntax and related markings are insufficient to distinguish different theorems. For example, there are over 100 different implication-exclusive theorems. These are then grouped in a more ad-hoc way that attempts to make their distinctions clearer. These often use abbreviations such as "mp" for "modus ponens", "syl" for syllogism, and "id" for "identity". It's especially hard to give good names in the propositional calculus section because there are so few primitives. However, in most cases this is not a serious problem. There are a few very common theorems like ax-mp 8 and syl 16 that you will have no trouble remembering, a few theorem series like syl*anc and simp* that you can use parametrically, and a few other useful glue things for destructuring 'and's and 'or's (see natded 21694 for a list), and that's about all you need for most things. As for the rest, you can just assume that if it involves three or less connectives we probably already have a proof, and searching for it will give you the name.
  • Suffixes. We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1814 via the use of distinct variable conditions combined with nfv 1629. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 2286 derived from df-eu 2284. The "f" stands for "not free in" which is less restrictive than "does not occur in." We sometimes suffix with "s" the label of an inference that manipulates an antecedent, leaving the consequent unchanged. The "s" means that the inference eliminates the need for a syllogism (syl 16) -type inference in a proof. When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the theorem form with "g" (for "more general") as in uniex 4691 vs. uniexg 4692. A theorem name is suffixed with "ALT" if it's an alternative less-preferred proof of a theorem.
  • Reuse. When creating a new theorem or axiom, try to reuse abbreviations used elsewhere. A comment should explain the first use of an abbreviation.

The following table shows some commonly-used abbreviations in labels, in alphabetical order. For each abbreviation we provide a mnenomic to help you remember it, the source theorem/assumption defining it, an expression showing what it looks like, whether or not it is a "syntax fragment" (an abbreviation that indicates a particular kind of syntax), and hyperlinks to label examples that use the abbreviation. The abbreviation is bolded if there is a df-NAME definition but the label fragment is not NAME. This is not a complete list of abbreviations, though we do want this to eventually be a complete list of exceptions.

AbbreviationMnenomicSource ExpressionSyntax?Example(s)
addadd (see "p") df-add 8985  ( A  +  B ) Yes addcl 9056, addcom 9236, addass 9061
ALTalternative/less preferred (suffix) No
anand df-an 361  ( ph  /\  ps ) Yes anor 476, iman 414, imnan 412
assassociative No biass 349, orass 511, mulass 9062
bibiconditional df-bi 178  ( ph  <->  ps ) Yes impbid 184
cncomplex numbers df-c 8980  CC Yes nnsscn 9989, nncn 9992
comcommutative No orcom 377, bicomi 194, eqcomi 2434
ddeduction form No idd 22, impbid 184
di, distrdistributive No andi 838, imdi 353, ordi 835, difindi 3582, ndmovdistr 6222
difdifference df-dif 3310  ( A  \  B ) Yes difss 3461, difindi 3582
divdivision df-div 9662  ( A  /  B ) Yes divcl 9668, divval 9664, divmul 9665
e, eqequals df-cleq 2423  A  =  B Yes 2p2e4 10082, uneqri 3476
elelement of  A  e.  B Yes eldif 3317, eldifsn 3914, elssuni 4030
f"not free in" (suffix) No
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 4692
ididentity No
idmidempotent No anidm 626, tpidm13 3893
im, impimplication (label often omitted) df-im 11889  ( A  ->  B ) Yes iman 414, imnan 412, impbidd 182
inintersection df-in 3314  ( A  i^i  B ) Yes elin 3517, incom 3520
is...is (something a) ...? No isrng 15651
mpmodus ponens ax-mp 8 No mpd 15, mpi 17
mulmultiplication (see "t") df-mul 8986  ( A  x.  B ) Yes mulcl 9058, divmul 9665, mulcom 9060, mulass 9062
n, notnot  -.  ph Yes nan 564, notnot2 106
ne0not equal to zero (see n0)  =/=  0 No negne0d 9393, ine0 9453, gt0ne0 9477
nnnatural numbers df-nn 9985  NN Yes nnsscn 9989, nncn 9992
n0not the empty set (see ne0)  =/=  (/) No n0i 3620, vn0 3622, ssn0 3647
oror df-or 360  ( ph  \/  ps ) Yes orcom 377, anor 476
pplus (see "add"), for all-constant theorems df-add 8985  ( 3  +  2 )  =  5 Yes 3p2e5 10095
pmPrincipia Mathematica No pm2.27 37
prpair df-pr 3808  { A ,  B } Yes elpr 3819, prcom 3869, prid1g 3897, prnz 3910
q  QQ (quotients) df-q 10559  QQ Yes elq 10560
rereal numbers df-r 8984  RR Yes recn 9064, 0re 9075
rngring df-rng 15646  Ring Yes rngidval 15649, isrng 15651, rnggrp 15652
rotrotation No 3anrot 941, 3orrot 942
seliminates need for syllogism (suffix) No
snsingleton df-sn 3807  { A } Yes eldifsn 3914
sssubset df-ss 3321  A  C_  B Yes difss 3461
subsubtract df-sub 9277  ( A  -  B ) Yes subval 9281, subaddi 9371
sylsyllogism syl 16 No 3syl 19
t times (see "mul"), for all-constant theorems df-mul 8986  ( 3  x.  2 )  =  6 Yes 3t2e6 10112
tptriple df-tp 3809  { A ,  B ,  C } Yes eltpi 3839, tpeq1 3879
ununion df-un 3312  ( A  u.  B ) Yes uneqri 3476, uncom 3478
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 1963
xreXtended reals df-xr 9108  RR* Yes ressxr 9113, rexr 9114, 0xr 9115
z  ZZ (integers, from German Zahlen) df-z 10267  ZZ Yes elz 10268, zcn 10271
0, z slashed zero (empty set) (see n0) df-nul 3616  (/) Yes n0i 3620, vn0 3622; snnz 3909, prnz 3910

Distinctness or freeness

Here are some conventions that address distinctness or freeness of a variable:

  •  F/ x ph is read "  x is not free in (wff)  ph"; see df-nf 1554 (whose description has some important technical details). Similarly,  F/_ x A is read  x is not free in (class)  A, see df-nfc 2555.
  • "$d x y $." should be read "Assume x and y are distinct variables."
  • "$d x  ph $." should be read "Assume x does not occur in phi $." Sometimes a theorem is proved using  F/ x ph (df-nf 1554) in place of "$d  x ph $." when a more general result is desired; ax-17 1626 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2286 from df-eu 2284.
  • "$d x A $." should be read "Assume x is not a variable occurring in class A."
  • "$d x A $. $d x ps $. $e |-  ( x  =  A  ->  ( ph  <->  ps ) ) $." is an idiom often used instead of explicit substitution, meaning "Assume psi results from the proper substitution of A for x in phi."
  • "  |-  ( -.  A. x x  =  y  ->  ..." occurs early in some cases, and should be read "If x and y are distinct variables, then..." This antecedent provides us with a technical device (called a "distinctor" in Section 7 of [Megill] p. 444) to avoid the need for the $d statement early in our development of predicate calculus, permitting unrestricted substitutions as conceptually simple as those in propositional calculus. However, the $d eventually becomes a requirement, and after that this device is rarely used.

Here is more information about our processes for checking and contributing to this work:

  • Multiple verifiers. This entire file is verified by multiple independently-implemented verifiers when it is checked in, giving us extremely high confidence that all proofs follow from the assumptions. The checkers also check for various other problems such as overly long lines.
  • Rewrapped line length. The input file routinely has its text wrapped using metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' (so please do the same).
  • Discouraged information. A separate file named "discouraged" lists all discouraged statements and uses of them, and this file is checked. If you change the use of discouraged things, you will need to change this file. This makes it obvious when there is a change to anything discouraged (triggering further review).
  • Proposing specific changes. Please propose specific changes as pull requests (PRs) against the "develop" branch of set.mm, at: https://github.com/metamath/set.mm/tree/develop
  • Community. We encourage anyone interested in Metamath to join our mailing list: https://groups.google.com/forum/#!forum/metamath.

(Contributed by DAW, 27-Dec-2016.)

 |-  ph   =>    |-  ph
 
15.1.2  Natural deduction
 
Theoremnatded 21694 Here are typical natural deduction (ND) rules in the style of Gentzen and Jaśkowski, along with MPE translations of them. This also shows the recommended theorems when you find yourself needing these rules (the recommendations encourage a slightly different proof style that works more naturally with metamath). A decent list of the standard rules of natural deduction can be found beginning with definition /\I in [Pfenning] p. 18. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. Many more citations could be added.

NameNatural Deduction RuleTranslation RecommendationComments
IT  _G |-  ps =>  _G |-  ps idi 2 nothing Reiteration is always redundant in Metamath. Definition "new rule" in [Pfenning] p. 18, definition IT in [Clemente] p. 10.
 /\I  _G |-  ps &  _G |-  ch =>  _G |-  ps  /\  ch jca 519 jca 519, pm3.2i 442 Definition  /\I in [Pfenning] p. 18, definition I /\m,n in [Clemente] p. 10, and definition  /\I in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\EL  _G |-  ps  /\  ch =>  _G |-  ps simpld 446 simpld 446, adantr 452 Definition  /\EL in [Pfenning] p. 18, definition E /\(1) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 /\ER  _G |-  ps  /\  ch =>  _G |-  ch simprd 450 simpr 448, adantl 453 Definition  /\ER in [Pfenning] p. 18, definition E /\(2) in [Clemente] p. 11, and definition  /\E in [Indrzejczak] p. 34 (representing both Gentzen's system NK and Jaśkowski)
 ->I  _G ,  ps |-  ch =>  _G |-  ps  ->  ch ex 424 ex 424 Definition  ->I in [Pfenning] p. 18, definition I=>m,n in [Clemente] p. 11, and definition  ->I in [Indrzejczak] p. 33.
 ->E  _G |-  ps  ->  ch &  _G |-  ps =>  _G |-  ch mpd 15 ax-mp 8, mpd 15, mpdan 650, imp 419 Definition  ->E in [Pfenning] p. 18, definition E=>m,n in [Clemente] p. 11, and definition  ->E in [Indrzejczak] p. 33.
 \/IL  _G |-  ps =>  _G |-  ps  \/  ch olcd 383 olc 374, olci 381, olcd 383 Definition  \/I in [Pfenning] p. 18, definition I \/n(1) in [Clemente] p. 12
 \/IR  _G |-  ch =>  _G |-  ps  \/  ch orcd 382 orc 375, orci 380, orcd 382 Definition  \/IR in [Pfenning] p. 18, definition I \/n(2) in [Clemente] p. 12.
 \/E  _G |-  ps  \/  ch &  _G ,  ps |-  th &  _G ,  ch |-  th =>  _G |-  th mpjaodan 762 mpjaodan 762, jaodan 761, jaod 370 Definition  \/E in [Pfenning] p. 18, definition E \/m,n,p in [Clemente] p. 12.
 -.I  _G ,  ps |-  F. =>  _G |-  -.  ps inegd 1342 pm2.01d 163
 -.I  _G ,  ps |-  th &  _G |-  -.  th =>  _G |-  -.  ps mtand 641 mtand 641 definition I -.m,n,p in [Clemente] p. 13.
 -.I  _G ,  ps |-  ch &  _G ,  ps |-  -.  ch =>  _G |-  -.  ps pm2.65da 560 pm2.65da 560 Contradiction.
 -.I  _G ,  ps |-  -.  ps =>  _G |-  -.  ps pm2.01da 430 pm2.01d 163, pm2.65da 560, pm2.65d 168 For an alternative falsum-free natural deduction ruleset
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  F. pm2.21fal 1344 pm2.21dd 101
 -.E  _G ,  -.  ps |-  F. =>  _G |-  ps pm2.21dd 101 definition  ->E in [Indrzejczak] p. 33.
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  th pm2.21dd 101 pm2.21dd 101, pm2.21d 100, pm2.21 102 For an alternative falsum-free natural deduction ruleset. Definition  -.E in [Pfenning] p. 18.
 T.I  _G |-  T. a1tru 1339 tru 1330, a1tru 1339, trud 1332 Definition  T.I in [Pfenning] p. 18.
 F.E  _G ,  F.  |-  th falimd 1338 falim 1337 Definition  F.E in [Pfenning] p. 18.
 A.I  _G |-  [ a  /  x ] ps =>  _G |-  A. x ps alrimiv 1641 alrimiv 1641, ralrimiva 2776 Definition  A.Ia in [Pfenning] p. 18, definition I A.n in [Clemente] p. 32.
 A.E  _G |-  A. x ps =>  _G |-  [ t  /  x ] ps spsbcd 3161 spcv 3029, rspcv 3035 Definition  A.E in [Pfenning] p. 18, definition E A.n,t in [Clemente] p. 32.
 E.I  _G |-  [ t  /  x ] ps =>  _G |-  E. x ps spesbcd 3230 spcev 3030, rspcev 3039 Definition  E.I in [Pfenning] p. 18, definition I E.n,t in [Clemente] p. 32.
 E.E  _G |-  E. x ps &  _G ,  [ a  /  x ] ps |-  th =>  _G |-  th exlimddv 1648 exlimddv 1648, exlimdd 1912, exlimdv 1646, rexlimdva 2817 Definition  E.Ea,u in [Pfenning] p. 18, definition E E.m,n,p,a in [Clemente] p. 32.
 F.C  _G ,  -.  ps |-  F. =>  _G |-  ps efald 1343 efald 1343 Proof by contradiction (classical logic), definition  F.C in [Pfenning] p. 17.
 F.C  _G ,  -.  ps |-  ps =>  _G |-  ps pm2.18da 431 pm2.18da 431, pm2.18d 105, pm2.18 104 For an alternative falsum-free natural deduction ruleset
 -.  -.C  _G |-  -.  -.  ps =>  _G |-  ps notnotrd 107 notnotrd 107, notnot2 106 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E -.n in [Clemente] p. 14.
EM  _G |-  ps  \/  -.  ps exmidd 406 exmid 405 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
 =I  _G |-  A  =  A eqidd 2431 eqid 2430, eqidd 2431 Introduce equality, definition =I in [Pfenning] p. 127.
 =E  _G |-  A  =  B &  _G [. A  /  x ]. ps =>  _G |-  [. B  /  x ]. ps sbceq1dd 3154 sbceq1d 3153, equality theorems Eliminate equality, definition =E in [Pfenning] p. 127. (Both E1 and E2.)

Note that MPE uses classical logic, not intuitionist logic. As is conventional, the "I" rules are introduction rules, "E" rules are elimination rules, the "C" rules are conversion rules, and  _G represents the set of (current) hypotheses. We use wff variable names beginning with  ps to provide a closer representation of the Metamath equivalents (which typically use the antedent  ph to represent the context  _G).

Most of this information was developed by Mario Carneiro and posted on 3-Feb-2017. For more information, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer.

For annotated examples where some traditional ND rules are directly applied in MPE, see ex-natded5.2 21695, ex-natded5.3 21698, ex-natded5.5 21701, ex-natded5.7 21702, ex-natded5.8 21704, ex-natded5.13 21706, ex-natded9.20 21708, and ex-natded9.26 21710.

(Contributed by DAW, 4-Feb-2017.)

 |-  ph   =>    |-  ph
 
15.1.3  Natural deduction examples

These are examples of how natural deduction rules can be applied in metamath (both as line-for-line translations of ND rules, and as a way to apply deduction forms without being limited to applying ND rules). For more information, see natded 21694 and http://us.metamath.org/mpeuni/mmnatded.html.

 
Theoremex-natded5.2 21695 Theorem 5.2 of [Clemente] p. 15, translated line by line using the interpretation of natural deduction in Metamath. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer. The original proof, which uses Fitch style, was written as follows:
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
15  ( ( ps  /\  ch )  ->  th )  ( ph  ->  ( ( ps  /\  ch )  ->  th ) ) Given $e.
22  ( ch  ->  ps )  ( ph  ->  ( ch  ->  ps ) ) Given $e.
31  ch  ( ph  ->  ch ) Given $e.
43  ps  ( ph  ->  ps )  ->E 2,3 mpd 15, the MPE equivalent of  ->E, 1,2
54  ( ps  /\  ch )  ( ph  ->  ( ps  /\  ch ) )  /\I 4,3 jca 519, the MPE equivalent of  /\I, 3,1
66  th  ( ph  ->  th )  ->E 1,5 mpd 15, the MPE equivalent of  ->E, 4,5

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. Below is the final metamath proof (which reorders some steps). A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.2-2 21696. A proof without context is shown in ex-natded5.2i 21697. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2-2 21696 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 21695 and ex-natded5.2i 21697. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2i 21697 The same as ex-natded5.2 21695 and ex-natded5.2-2 21696 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ps  /\  ch )  ->  th )   &    |-  ( ch  ->  ps )   &    |-  ch   =>    |- 
 th
 
Theoremex-natded5.3 21698 Theorem 5.3 of [Clemente] p. 16, translated line by line using an interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.3-2 21699. A proof without context is shown in ex-natded5.3i 21700. For information about ND and Metamath, see the page on Deduction Form and Natural Deduction in Metamath Proof Explorer . The original proof, which uses Fitch style, was written as follows:

#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 452 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 452 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 448, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15, the MPE equivalent of  ->E, 1.3. adantr 452 was used to transform its dependency (we could also use imp 419 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15, the MPE equivalent of  ->E, 4,6. adantr 452 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 519, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 424, the MPE equivalent of  ->I, 8

The original used Latin letters for predicates; we have replaced them with Greek letters to follow Metamath naming conventions and so that it is easier to follow the Metamath translation. The Metamath line-for-line translation of this natural deduction approach precedes every line with an antecedent including  ph and uses the Metamath equivalents of the natural deduction rules. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3-2 21699 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 21698 and ex-natded5.3i 21700. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3i 21700 The same as ex-natded5.3 21698 and ex-natded5.3-2 21699 but with no context. (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ch  /\  th ) )
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