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Theorem List for Metamath Proof Explorer - 25501-25600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnumclwlk1lem2 25501* There is a bijection between the set of closed walks (having a fixed length greater than 2 and starting at a fixed vertex) with the last but 2 vertex identical with the first (and therefore last) vertex and the set of closed walks (having a fixed length less by 2 and starting at the same vertex) and the neighbors of this vertex. (Contributed by Alexander van der Vekens, 6-Jul-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   =>    |-  (
 ( V USGrph  E  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 ) )  ->  E. f  f :
 ( X G N )
 -1-1-onto-> ( ( X F ( N  -  2
 ) )  X.  ( <. V ,  E >. Neighbors  X ) ) )
 
Theoremnumclwwlk1 25502* Statement 9 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) with v(n-2) = v is kf(n-2)". Since  <. V ,  E >. is k-regular, the vertex v(n-2) = v has k neighbors v(n-1), so there are k walks from v(n-2) = v to v(n) = v (via each of v's neighbors) completing each of the f(n-2) walks from v=v(0) to v(n-2)=v. This theorem holds even for k=0, but only for finite graphs! (Contributed by Alexander van der Vekens, 26-Sep-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   =>    |-  (
 ( ( V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K )  /\  ( X  e.  V  /\  N  e.  ( ZZ>= `  3 )
 ) )  ->  ( # `
  ( X G N ) )  =  ( K  x.  ( # `
  ( X F ( N  -  2
 ) ) ) ) )
 
Theoremnumclwwlkovq 25503* Value of operation Q, mapping a vertex v and a nonnegative integer n to the not closed walks v(0) ... v(n) of length n from a fixed vertex v = v(0). "Not closed" means v(n) =/= v(0). (Contributed by Alexander van der Vekens, 27-Sep-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   =>    |-  ( ( X  e.  V  /\  N  e.  NN0 )  ->  ( X Q N )  =  { w  e.  ( ( V WWalksN  E ) `  N )  |  ( ( w `  0 )  =  X  /\  ( lastS  `  w )  =/=  X ) }
 )
 
Theoremnumclwwlkqhash 25504* In a k-regular graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex is the difference between k to the power of n and the size of the set of walks of length n starting with this vertex and ending at this vertex. (Contributed by Alexander van der Vekens, 30-Sep-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V  e.  Fin )  /\  ( X  e.  V  /\  N  e.  NN )
 )  ->  ( # `  ( X Q N ) )  =  ( ( K ^ N )  -  ( # `  ( X F N ) ) ) )
 
Theoremnumclwwlkovh 25505* Value of operation H, mapping a vertex v and a nonnegative integer n to the "closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v" according to definition 7 in [Huneke] p. 2. (Contributed by Alexander van der Vekens, 26-Aug-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( X  e.  V  /\  N  e.  NN0 )  ->  ( X H N )  =  { w  e.  ( C `  N )  |  ( ( w `  0
 )  =  X  /\  ( w `  ( N  -  2 ) )  =/=  ( w `  0 ) ) }
 )
 
Theoremnumclwwlk2lem1 25506* In a friendship graph, for each walk of length n starting with a fixed vertex and ending not at this vertex, there is a unique vertex so that the walk extended by an edge to this vertex and an edge from this vertex to the first vertex of the walk is a value of operation H. If the walk is represented as a word, it is sufficient to add one vertex to the word to obtain the closed walk contained in the value of operation H, since in a word representing a closed walk the starting vertex is not repeated at the end. This theorem only generally holds for Friendship Graphs, because these guarantee that for the first and last vertex there is a third vertex "in between". (Contributed by Alexander van der Vekens, 3-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X Q N )  ->  E! v  e.  V  ( W ++  <" v "> )  e.  ( X H ( N  +  2 ) ) ) )
 
Theoremnumclwlk2lem2f 25507* R is a function. (Contributed by Alexander van der Vekens, 5-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   &    |-  R  =  ( x  e.  ( X H ( N  +  2
 ) )  |->  ( x substr  <. 0 ,  ( N  +  1 ) >. ) )   =>    |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  R : ( X H ( N  +  2 ) ) --> ( X Q N ) )
 
Theoremnumclwlk2lem2fv 25508* Value of the function R. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   &    |-  R  =  ( x  e.  ( X H ( N  +  2
 ) )  |->  ( x substr  <. 0 ,  ( N  +  1 ) >. ) )   =>    |-  ( ( X  e.  V  /\  N  e.  NN )  ->  ( W  e.  ( X H ( N  +  2 ) ) 
 ->  ( R `  W )  =  ( W substr  <.
 0 ,  ( N  +  1 ) >. ) ) )
 
Theoremnumclwlk2lem2f1o 25509* R is a 1-1 onto function. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   &    |-  R  =  ( x  e.  ( X H ( N  +  2
 ) )  |->  ( x substr  <. 0 ,  ( N  +  1 ) >. ) )   =>    |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  R : ( X H ( N  +  2 ) ) -1-1-onto-> ( X Q N ) )
 
Theoremnumclwwlk2lem3 25510* In a friendship graph, the size of the set of walks of length n starting with a fixed vertex and ending not at this vertex equals the size of the set of all closed walks of length (n+2) starting with this vertex and not having this vertex as last but 2 vertex. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( V FriendGrph  E  /\  X  e.  V  /\  N  e.  NN )  ->  ( # `  ( X Q N ) )  =  ( # `  ( X H ( N  +  2 ) ) ) )
 
Theoremnumclwwlk2 25511* Statement 10 in [Huneke] p. 2: "If n > 1, then the number of closed n-walks v(0) ... v(n-2) v(n-1) v(n) from v = v(0) = v(n) ... with v(n-2) =/= v is k^(n-2) - f(n-2)." According to rusgranumwlkg 25362, we have k^(n-2) different walks of length (n-2): v(0) ... v(n-2). From this number, the number of closed walks of length (n-2), which is f(n-2) per definition, must be subtracted, because for these walks v(n-2) =/= v(0) = v would hold. Because of the friendship condition, there is exactly one vertex v(n-1) which is a neighbor of v(n-2) as well as of v(n)=v=v(0), because v(n-2) and v(n)=v are different, so the number of walks v(0) ... v(n-2) is identical with the number of walks v(0) ... v(n), that means each (not closed) walk v(0) ... v(n-2) can be extended by two edges to a closed walk v(0) ... v(n)=v=v(0) in exactly one way. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
 ) )  ->  ( # `
  ( X H N ) )  =  ( ( K ^
 ( N  -  2
 ) )  -  ( # `
  ( X F ( N  -  2
 ) ) ) ) )
 
Theoremnumclwwlk3lem 25512* Lemma for numclwwlk3 25513. (Contributed by Alexander van der Vekens, 6-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( ( V USGrph  E  /\  V  e.  Fin  /\  X  e.  V ) 
 /\  N  e.  ( ZZ>=
 `  2 ) ) 
 ->  ( # `  ( X F N ) )  =  ( ( # `  ( X H N ) )  +  ( # `
  ( X G N ) ) ) )
 
Theoremnumclwwlk3 25513* Statement 12 in [Huneke] p. 2: "Thus f(n) = (k - 1)f(n - 2) + k^(n-2)." - the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) is the sum of the number of the closed walks v(0) ... v(n-2) v(n-1) v(n) with v(n-2) = v(n) (see numclwwlk1 25502) and with v(n-2) =/= v(n) ( see numclwwlk2 25511): f(n) = kf(n-2) + k^(n-2) - f(n-2) = (k - 1)f(n - 2) + k^(n-2) (Contributed by Alexander van der Vekens, 26-Aug-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   &    |-  G  =  ( v  e.  V ,  n  e.  ( ZZ>= `  2 )  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `
  ( n  -  2 ) )  =  ( w `  0
 ) ) } )   &    |-  Q  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( ( V WWalksN  E ) `
  n )  |  ( ( w `  0 )  =  v  /\  ( lastS  `  w )  =/=  v ) }
 )   &    |-  H  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( ( w `  0 )  =  v  /\  ( w `  ( n  -  2 ) )  =/=  ( w `  0 ) ) }
 )   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  e.  Fin  /\  X  e.  V  /\  N  e.  ( ZZ>= `  3 )
 ) )  ->  ( # `
  ( X F N ) )  =  ( ( ( K  -  1 )  x.  ( # `  ( X F ( N  -  2 ) ) ) )  +  ( K ^ ( N  -  2 ) ) ) )
 
Theoremnumclwwlk4 25514* The total number of closed walks in a finite undirected simple graph is the sum of the numbers of closed walks starting at each of its vertices. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   =>    |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  N  e.  NN0 )  ->  ( # `  ( C `
  N ) )  =  sum_ x  e.  V  ( # `  ( x F N ) ) )
 
Theoremnumclwwlk5lem 25515* Lemma for numclwwlk5 25516. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   =>    |-  ( ( <. V ,  E >. RegUSGrph  K  /\  2  ||  ( K  -  1
 )  /\  X  e.  V )  ->  ( ( # `  ( X F
 2 ) )  mod  2 )  =  1
 )
 
Theoremnumclwwlk5 25516* Statement 13 in [Huneke] p. 2: "Let p be a prime divisor of k-1; then f(p) = 1 (mod p) [for each vertex v]". (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( X  e.  V  /\  P  e.  Prime  /\  P  ||  ( K  -  1 ) ) )  ->  ( ( # `
  ( X F P ) )  mod  P )  =  1 )
 
Theoremnumclwwlk6 25517* For a prime divisor p of k-1, the total number of closed walks of length p in an undirected simple graph with m vertices mod p is equal to the number of vertices mod p. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  C  =  ( n  e.  NN0  |->  ( ( V ClWWalksN  E ) `  n ) )   &    |-  F  =  ( v  e.  V ,  n  e.  NN0  |->  { w  e.  ( C `  n )  |  ( w `  0 )  =  v } )   =>    |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1
 ) ) )  ->  ( ( # `  ( C `  P ) ) 
 mod  P )  =  ( ( # `  V )  mod  P ) )
 
Theoremnumclwwlk7 25518 Statement 14 in [Huneke] p. 2: "The total number of closed walks of length p [in a friendship graph] is (k(k-1)+1)f(p)=1 (mod p)", since the number of vertices in a friendship graph is (k(k-1)+1), see frgregordn0 25474 or frrusgraord 25475, and p divides (k-1), i.e. (k-1) mod p = 0 => k(k-1) mod p = 0 => k(k-1)+1 mod p = 1. Since the empty graph is a friendship graph, see frgra0 25398, as well as k-regular (for any k), see 0vgrargra 25341, but has no closed walk, see clwlk0 25166, this theorem would be false:  ( ( # `  ( C `  P
) )  mod  P
)  =  0  =/=  1, so this case must be excluded. ( (Contributed by Alexander van der Vekens, 1-Sep-2018.)
 |-  ( ( ( <. V ,  E >. RegUSGrph  K  /\  V FriendGrph  E )  /\  ( V  =/=  (/)  /\  V  e.  Fin )  /\  ( P  e.  Prime  /\  P  ||  ( K  -  1
 ) ) )  ->  ( ( # `  (
 ( V ClWWalksN  E ) `  P ) )  mod  P )  =  1 )
 
Theoremnumclwwlk8 25519 The size of the set of closed walks of length p, p prime, is divisible by p. This corresponds to statement 9 in [Huneke] p. 2: "It follows that, if p is a prime number, then the number of closed walks of length p is divisible by p", see also clwlkndivn 25257. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  ( ( V USGrph  E  /\  V  e.  Fin  /\  P  e.  Prime )  ->  ( ( # `  (
 ( V ClWWalksN  E ) `  P ) )  mod  P )  =  0 )
 
Theoremfrgrareggt1 25520 If a finite friendship graph is k-regular with k > 1, then k must be 2. (Contributed by Alexander van der Vekens, 7-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/= 
 (/) )  ->  (
 ( <. V ,  E >. RegUSGrph  K  /\  1  <  K )  ->  K  =  2 ) )
 
Theoremfrgrareg 25521 If a finite friendship graph is k-regular, then k must be 2 (or 0). (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V  e.  Fin  /\  V  =/=  (/) )  ->  ( ( V FriendGrph  E  /\  <. V ,  E >. RegUSGrph  K )  ->  ( K  =  0  \/  K  =  2 ) ) )
 
Theoremfrgraregord013 25522 If a finite friendship graph is k-regular, then it must have order 0, 1 or 3. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  <. V ,  E >. RegUSGrph  K ) 
 ->  ( ( # `  V )  =  0  \/  ( # `  V )  =  1  \/  ( # `
  V )  =  3 ) )
 
Theoremfrgraregord13 25523 If a nonempty finite friendship graph is k-regular, then it must have order 1 or 3. Special case of frgraregord013 25522. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( ( V FriendGrph  E  /\  V  e.  Fin  /\  V  =/=  (/) )  /\  <. V ,  E >. RegUSGrph  K )  ->  ( ( # `  V )  =  1  \/  ( # `  V )  =  3 )
 )
 
Theoremfrgraogt3nreg 25524* If a finite friendship graph has an order greater than 3, it cannot be k-regular for any k. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  ->  A. k  e.  NN0  -.  <. V ,  E >. RegUSGrph  k )
 
Theoremfriendshipgt3 25525* The friendship theorem for big graphs: In every finite friendship graph with order greater than 3 there is a vertex which is adjacent to all other vertices. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  e.  Fin  /\  3  <  ( # `  V ) )  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E )
 
Theoremfriendship 25526* The friendship theorem: In every finite (nonempty) friendship graph there is a vertex which is adjacent to all other vertices. This is Metamath 100 proof #83. (Contributed by Alexander van der Vekens, 9-Oct-2018.)
 |-  ( ( V FriendGrph  E  /\  V  =/=  (/)  /\  V  e.  Fin )  ->  E. v  e.  V  A. w  e.  ( V  \  {
 v } ) {
 v ,  w }  e.  ran  E )
 
PART 17  GUIDES AND MISCELLANEA
 
17.1  Guides (conventions, explanations, and examples)
 
17.1.1  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 25527 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 13656) 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 13841). 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 as discussed below, except for 4 definitions (df-bi 185, df-cleq 2394, df-clel 2397, df-clab 2388) 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 9579, proven by the preceding theorem ax1cn 9555. 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 5, 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 5 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 9579 (not ax1cn 9555) and ax-1ne0 9590 (not ax1ne0 9566), as these are proven axioms for complex arithmetic. Thus, both ax1cn 9555 and ax1ne0 9566 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.
  • Minimizing axioms and the axiom of choice. We prefer proofs that depend on fewer and/or weaker axioms, even if the proofs are longer. In particular, we prefer proofs that do not use the axiom of choice (df-ac 8528) where such proofs can be found. The axiom of choice 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 7674) does not use the axiom of choice. In some cases, the weaker axiom of countable choice (ax-cc 8846) or axiom of dependent choice (ax-dc 8857) can be used instead.
  • Alternative (ALT) proofs. If a different proof is significantly shorter or clearer but uses more or stronger axioms, we prefer to make that proof an "alternative" proof (marked with an ALT label suffix), even if this alternative proof was formalized first. We then make the proof that requires fewer axioms the main proof. This has the effect of reducing (over time) the number and strength of axioms used by any particular proof. There can be multiple alternatives if it makes sense to do so. Alternative (*ALT) theorems should have "(Proof modification is discouraged.) (New usage is discouraged.)" in their comment and should follow the main statement, so that people reading the text in order will see the main statement first. The alternative and main statement comments should use hyperlinks to refer to each other (so that a reader of one will become easily aware of the other).
  • Alternative (ALTV) versions. If a theorem or definition is an alternative/variant of an already existing theorem resp. definition, its label should have the same name with suffix ALTV. Such alternatives should be temporary only, until it is decided which alternative should be used in the future. Alternative (*ALTV) theorems or definitions are usually contained in mathboxes. Their comments need not to contain "(Proof modification is discouraged.) (New usage is discouraged.)". Alternative statements should follow the main statement, so that people reading the text in order will see the main statement first.
  • Old (OLD) versions or proofs. If a proof, definition, axiom, or theorem is going to be removed, we often stage that change by first renaming its label with an OLD suffix (to make it clear that it is going to be removed). Old (*OLD) statements should have "(Proof modification is discouraged.) (New usage is discouraged.)" and "Obsolete version of ~ xxx as of dd-mmm-yyyy." (not enclosed in parentheses) in the comment. An old statement should follow the main statement, so that people reading the text in order will see the main statement first.
  • Variables. Propositional variables (variables for well-formed formulas or wffs) are represented with lowercase Greek letters and are normally used in this order:  ph = phi,  ps = psi,  ch = chi,  th = theta,  ta = tau,  et = eta,  ze = zeta, and  si = sigma. Individual setvar variables are represented with lowercase Latin letters and are normally used in this order:  x,  y,  z,  w,  v,  u, and  t. Variables that represent classes are often represented by uppercase Latin letters:  A,  B,  C,  D,  E, and so on. There are other symbols that also represent class variables and suggest specific purposes, e.g.,  .0. for poset zero (see p0val 15993) and connective symbols such as  .+ for some group addition operation. (See prdsplusgval 15085 for an example of the use of  .+). Class variables are selected in alphabetical order starting from  A if there is no reason to do otherwise, but many assertions select different class variables or a different order to make their intended meaning clearer.
  • 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 5, there is a corresponding version with a biconditional or a reversed biconditional, like mpbi 208 or mpbir 209. 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 194 are available, in most cases there is already a theorem that combines it with your theorem of choice, like mpbir2an 921, sylbir 213, or 3imtr4i 266.
  • 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 1764 and the related df-sbc 3277 and df-csb 3373.
  • Is-a set. " A  e.  _V" should be read "Class  A is a set (i.e. exists)." This is a convention based on Definition 2.9 of [Quine] p. 19. See df-v 3060 and isset 3062. However, instead of using  I  e.  _V in the antecedent of a theorem for some variable  I, we now prefer to use  I  e.  V (or another variable if  V is not available) to make it more general. That way we can often avoid needing extra uses of elex 3067 and syl 17 in the common case where  I is already a member of something.
  • 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 4830. This can be used to define a subset, e.g., df-tan 14014 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 14092). 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 5576. 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 6280). For example, the  + in  ( 2  +  2 ); see 2p2e4 10693. 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 368, df-an 369, and df-bi 185 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 1843), equality  A  =  B (see df-cleq 2394), subset  A  C_  B (see df-ss 3427), and less-than  A  <  B (see df-lt 9534). For the general definition of a binary relation in the form  A R B, see df-br 4395. For example,  0  <  1 (see 0lt1 10114) 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 9841.
  • 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 14007). The function is then defined labelled df-NAME; definitions are typically given using the maps-to notation (e.g., df-cos 14013). Typically, there are other proofs such as its closure labelled NAMEcl (e.g., coscl 14069), its function application form labelled NAMEval (e.g., cosval 14065), and at least one simple value (e.g., cos0 14092).
  • 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 12396 and fac4 12403).
  • 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 9528), while " 0g" is the group identity element (df-0g 15054), " 0." is the poset zero (df-p0 15991), " 0p" is the zero polynomial (df-0p 22367), " 0vec" is the zero vector in a normed complex vector space (df-0v 25891), and " .0." is a class variable for use as a connective symbol (this is used, for example, in p0val 15993). 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.
  • ASCII representation of symbols. We must have an ASCII representation for each symbol. We generally choose short sequences, ideally digraphs, and generally choose sequences that vaguely resemble the mathematical symbol. Here are some of the conventions we use when selecting an ASCII representation.

    We generally do not include parentheses inside a symbol because that confuses text editors (such as emacs). Greek letters for wff variables always use the first two letters of their English names, making them easy to type and easy to remember. Symbols that almost look like letters, such as  A., are often represented by that letter followed by a period. For example, "A." is used to represent  A., "e." is used to represent  e., and "E." is used to represent  E.. Single letters are now always variable names, so constants that are often shown as single letters are now typically preceded with "_" in their ASCII representation, for example, "_i" is the ASCII representation for the imaginary unit  _i. A script font constant is often the letter preceded by "~" meaning "curly", such as "~P" to represent the power class  ~P.

    Originally, all setvar and class variables used only single letters a-z and A-Z, respectively. A big change in recent years was to allow the use of certain symbols as variable names to make formulas more readable, such as a variable representing an additive group operation. The convention is to take the original constant token (in this case "+" which means complex number addition) and put a period in front of it to result in the ASCII representation of the variable ".+", shown as  .+, that can be used instead of say the letter "P" that had to be used before.

    Choosing tokens for more advanced concepts that have no standard symbols but are represented by words in books, is hard. A few are reasonably obvious, like "Grp" for group and "Top" for topology, but often they seem to end up being either too long or too cryptic. It would be nice if the math community came up with standardized short abbreviations for English math terminology, like they have more or less done with symbols, but that probably won't happen any time soon.

    Another informal convention that we've somewhat followed, that is also not uncommon in the literature, is to start tokens with a capital letter for collection-like objects and lower case for function-like objects. For example, we have the collections On (ordinal numbers), Fin, Prime, Grp, and we have the functions sin, tan, log, sup. Predicates like Ord and Lim also tend to start with upper case, but in a sense they are really collection-like, e.g. Lim indirectly represents the collection of limit ordinals, but it can't be an actual class since not all limit ordinals are sets. This initial capital vs. lower case letter convention is sometimes ambiguous. In the past there's been a debate about whether domain and range are collection-like or function-like, thus whether we should use Dom, Ran or dom, ran. Both are used in the literature. In the end dom, ran won out for aesthetic reasons (Norm Megill simply just felt they looked nicer).
  • Natural numbers. There are different definitions of "natural" numbers in the literature. We use  NN (df-nn 10576) for the set of positive integers starting from 1, and  NN0 (df-n0 10836) 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 11019, e.g., ;;; 4 0 0 1 (see 4001prm 14834 for a proof that 4001 is prime).
  • Theorem forms. We will use the following descriptive terms to categorize theorems:
    • A theorem is in "closed form" if it has no $e hypotheses (e.g. unss 3616). The term "tautology" is also used, especially in propositional calculus. This form was formerly called "theorem form" or "closed theorem form".
    • A theorem is in "deduction form" (or is a "deduction") if it has one or more $e hypotheses, and the hypotheses and the conclusion are implications that share the same antecedent. More precisely, the conclusion is an implication with a wff variable as the antecedent (usually  ph), and every hypothesis ($e statement) is either
      1. an implication with the same antecedent as the conclusion, or
      2. a definition. A definition can be for a class variable (this is a class variable followed by  =, e.g. the definition of  D in lhop 22707) or a wff variable (this is a wff variable followed by  <->); class variable definitions are more common.
      In practice, a proof of a theorem in deduction form will also contain many steps that are implications where the antecedent is either that wff variable (usually  ph) or is a conjunction  ( ph  i^i  ... ) including that wff variable ( ph). E.g. a1d 25, unssd 3618.
    • A theorem is in "inference form" (or is an "inference") if it has one or more $e hypotheses, but is not in deduction form, i.e. there is no common antecedent (e.g. unssi 3617).
    Any theorem whose conclusion is an implication has an associated inference, whose hypotheses are the hypotheses of that theorem together with the antecedent of its conclusion, and whose conclusion is the consequent of that conclusion. When both theorems are in set.mm, then the associated inference is often labeled by adding the suffix "i" to the label of the original theorem (for instance, con3i 135 is the inference associated with con3 134). The inference associated with a theorem is easily derivable from that theorem by a simple use of ax-mp 5. The other direction is the subject of the Deduction Theorem discussed below. We may also use the term "associated inference" when the above process is iterated. For instance, syl 17 is an inference associated with imim1 76 because it is the inference associated with imim1i 57 which is itself the inference associated with imim1 76. "Deduction form" is the preferred form for theorems because this form allows us to easily use the theorem in places where (in traditional textbook formalizations) the standard Deduction Theorem (see below) would be used. We call this approach "deduction style". In contrast, we usually avoid theorems in "inference form" when that would end up requiring us to use the deduction theorem.
    Deductions have a label suffix of "d", especially if there are other forms of the same theorem (e.g. pm2.43d 47). The labels for inferences usually have the suffix "i" (e.g. pm2.43i 46). The labels of theorems in "closed form" would have no special suffix (e.g. pm2.43 50). When an inference is converted to a theorem by eliminating an "is a set" hypothesis, we sometimes suffix the closed form with "g" (for "more general") as in uniex 6577 vs. uniexg 6578.
  • 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 (its associated inference). 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 3935, which works in certain cases in set theory. We also sometimes use dedhb 3218. However, the primary mechanism we use today for emulating the deduction theorem is to write proofs in deduction form (aka "deduction style") as 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 25528.
  • 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 7074,  rec ( F ,  I ) in df-rdg 7112, seq𝜔 ( F ,  I ) in df-seqom 7149, and  seq M (  .+  ,  F ) in df-seq 12150. These have characteristic function  F and initial value  I. ( gsumg in df-gsum 15055 isn't really designed for arbitrary recursion, but you could do it with the right magma.) The logically primary one is df-recs 7074, but for the "average user" the most useful one is probably df-seq 12150- 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 14841.
  • Junk/undefined results. Some expressions are only expected to be meaningful in certain contexts. For example, consider Russell's definition description binder iota, where  ( iota x ph ) is meant to be "the  x such that  ph" (where  ph typically depends on x). What should that expression produce when there is no such  x? In set.mm we primarily use one of two approaches. One approach is to make the expression evaluate to the empty set whenever the expression is being used outside of its expected context. While not perfect, it makes it a bit more clear when something is undefined, and it has the advantage that it makes more things equal outside their domain which can remove hypotheses when you feel like exploiting these so-called junk theorems. Note that Quine does this with iota (his definition of iota evaluates to the empty set when there is no unique value of  x). Quine has no problem with that and we don't see why we should, so we define iota exactly the same way that Quine does. The main place where you see this being systematically exploited is in "reverse closure" theorems like  A  e.  ( F `  B )  ->  B  e.  dom  F, which is useful when  F is a family of sets. (by this we mean it's a set set even in a type theoretic interpretation.) The second approach uses "(New usage is discouraged.)" to prevent unintentional uses of certain properties. For example, you could define some construct df-NAME whose usage is discouraged, and prove only the specific properties you wish to use (and add those proofs to the list of permitted uses of "discouraged" information). From then on, you can only use those specific properties without a warning. Other approaches often have hidden problems. For example, you could try to "not define undefined terms" by creating definitions like ${ $d  y x $. $d  y ph $. df-iota $a  |-  ( E! x ph  ->  ( iota x ph )  =  U. { x  |  ph } ) $. $}. This will be rejected by the definition checker, but the bigger theoretical reason to reject this axiom is that it breaks equality - the metatheorem  ( x  =  y  -> P(x)  = P(y)  ) fails to hold if definitions don't unfold without some assumptions. (That is, iotabidv 5553 is no longer provable and must be added as an axiom.) It is important for every syntax constructor to satisfy equality theorems *unconditionally*, e.g., expressions like  ( 1  /  0 )  =  ( 1  /  0 ) should not be rejected. This is forced on us by the context free term language, and anything else requires a lot more infrastructure (e.g. a type checker) to support without making everything else more painful to use. Another approach would be to try to make nonsensical statements syntactically invalid, but that can create its own complexities; in some cases that would make parsing itself undecidable! In practice this does not seem to be a serious issue. No one does these things deliberately in "real" situations, and some knowledgeable people (such as Mario Carneiro) have never seen this happen accidentally. Norman Megill doesn't agree that these "junk" consequences are necessarily bad anyway, and they can significantly shorten proofs in some cases. This database would be much larger if, for example, we had to condition fvex 5858 on the argument being in the domain of the function. It is impossible to derive a contradiction from sound definitions (i.e. that pass the definition check), assuming ZFC is consistent, and he doesn't see the point of all the extra busy work and huge increase in set.mm size that would result from restricting *all* definitions. So instead of implementing a complex system to counter a problem that does not appear to occur in practice, we use a significantly simpler set of approaches.
  • 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 7664 is the first lemma for sbth 7674. 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.
  • Hypertext links to section headers. Some section headers have text under them that describes or explains the section. However, they are not part of the description of axioms or theorems, and there is no way to link to them directly. To provide for this, section headers with accompanying text (indicated with "*" prefixed to mmtheorems.html#mmdtoc entries) have an anchor in mmtheorems.html whose name is the first $a or $p statement that follows the header. For example there is a glossary under the section heading called GRAPH THEORY. The first $a or $p statement that follows is cuhg 24694, which you can see two lines down. To reference it we link to the anchor using a space-separated tilde followed by the space-separated link mmtheorems.html#cuhg, which will become the hyperlink mmtheorems.html#cuhg. Note that no theorem in set.mm is allowed to begin with "mm" (enforced by "verify markup" in the metamath program). Whenever the software sees a tilde reference beginning with "http:", "https:", or "mm", the reference is assumed to be a link to something other than a statement label, and the tilde reference is used as is. This can also be useful for relative links to other pages such as mmcomplex.html.
  • 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.
  • Reserved symbols: 'LETTER. Some symbols are reserved for potential future use. Symbols with the pattern 'LETTER are reserved for possibly representing characters (this is somewhat similar to Lisp). We would expect '\n to represent newline, 'sp for space, and perhaps '\x24 for the dollar character.

Label naming conventions

Every statement has a unique identifying label, which serves the same purpose as an equation number in a book. 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 (these ids are usually consecutive integers starting with 1, e.g. for rgen 2763"rgen.1 $e |- ( x e. A -> ph ) $." or letters corresponding to the (main) class variable used in the hypothesis, e.g. for mdet0 19398: "mdet0.d $e |- D = ( N maDet R ) $.").
  • 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 2341 and stirling 37220.
  • Principia Mathematica. Proofs of theorems from Principia Mathematica often use a special naming convention: "pm" followed by its identifier. For example, Theorem *2.27 of [WhiteheadRussell] p. 104 is named pm2.27 37.
  • 19.x series of theorems. Similar to the conventions for the theorems from Principia Mathematica, theorems from section 19 of [Margaris] often use a special naming convention: "19." resp. "r19." (for corresponding restricted quantifier versions) followed by its identifier. For example, theorem 38 from section 19 of [Margaris] p. 90 is named 19.38 1683, and the restricted quantifier version of theorem 21 from section 19 of [Margaris] p. 90 is named r19.21 2802.
  • Characters to be used for labels Although the specification of Metamath allows for dots/periods "." in any label, it is usually used only in labels for hypotheses (see above). Exceptions are the names of theorems from Principia Mathematica and the 19.x series of theorems from [Margaris] (see above) and 0.999... 13840. Furthermore, the underscore "_" should not be used.
  • Syntax label fragments. Most theorems are named using a concatenation of syntax label fragments (omitting variables) that represent the important part of the theorem's main conclusion. Almost every syntactic construct has a definition labelled "df-NAME", and normally NAME is the syntax label fragment. For example, the difference construct  ( A  \  B ) is defined in df-dif 3416, and thus its syntax label fragment is "dif". Similarly, the subclass (subset) relation  A  C_  B has syntax label fragment "ss", because it is defined in df-ss 3427. Most theorem names follow from these fragments, for example, theorem proving  ( A  \  B )  C_  A involves a difference ("dif") of a subset ("ss"), and thus is named difss 3569. There are many other syntax label fragments, e.g., singleton construct  { A } has syntax label fragment "sn" (because it is defined in df-sn 3972), and the pair construct  { A ,  B } has fragment "pr" ( from df-pr 3974). 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 4096. An "n" is often used for negation ( -.), e.g., nan 578.
  • Exceptions. Sometimes there is a definition df-NAME but the label fragment is not the NAME part. The definition should note this exception as part of its definition. In addition, 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 9527) and "re" represents real numbers  RR. The empty set  (/) often uses fragment 0, even though it is defined in df-nul 3738. Syntax construct  ( A  +  B ) usually uses the fragment "add" (which is consistent with df-add 9532), 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 10693.
  • Other markings. In labels we sometimes use "com" for "commutative", "ass" for "associative", "rot" for "rotation", and "di" for "distributive".
  • Focus on the important part of the conclusion. Typically the conclusion is the part the user is most interested in. So, a rough guideline is that a label typically provides a hint about only the conclusion; a label rarely says anything about the hypotheses or antecedents. If there are multiple theorems with the same conclusion but different hypotheses/antecedents, then the labels will need to differ; those label differences should emphasize what is different. There is no need to always fully describe the conclusion; just identify the important part. For example, cos0 14092 is the theorem that provides the value for the cosine of 0; we would need to look at the theorem itself to see what that value is. The label "cos0" is concise and we use it instead of "cos0eq1". There is no need to add the "eq1", because there will never be a case where we have to disambiguate between different values produced by the cosine of zero, and we generally prefer shorter labels if they are unambiguous.
  • 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 14013) we have value cosval 14065 and closure coscl 14069.
  • 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 5 and syl 17 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 25528 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. Suffixes are used to indicate the form of a theorem (see above). Additionally, we sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1933 via the use of distinct variable conditions combined with nfv 1728. If two (or three) such hypotheses are eliminated, the suffix "vv" resp. "vvv" is used, e.g. exlimivv 1744. 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 2248 derived from df-eu 2242. The "f" stands for "not free in" which is less restrictive than "does not occur in." The suffix "b" often means "biconditional" ( <->, "iff" , "if and only if"), e.g. sspwb 4639. 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 17) -type inference in a proof. A theorem name is suffixed with "ALT" if it provides an alternative less-preferred proof of a theorem (e.g., the theorem is clearer but uses more axioms than the preferred theorem). The "ALT" may be further suffixed with a number if there is more than one alternate theorem. Furthermore, a theorem name is suffixed with "OLD" if there is a new version of it and the OLD version is obsolete (and will be removed soon). Finally, it should be mentioned that suffixes can be combined, for example in cbvaldva 2058 (cbval 2048 in deduction form "d" with a not free variable replaced by a distinct variable condition "v" with a conjunction as antecedent "a"). In the following, a list of common suffixes is provided:
    • a : theorem having a conjunction as antecedent
    • b : theorem expressing a logical equivalence
    • d : theorem in deduction form
    • f : theorem with a hypothesis such as  F/ x ph
    • g : theorem in closed form having an "is a set" antecedent
    • i : theorem in inference form
    • l : theorem concerning something at the left
    • r : theorem concerning something at the right
    • r : theorem with something reversed (e.g a biconditional)
    • s : inference that manipulates an antecedent
    • v : theorem with one (main) distinct variable
    • vv : theorem with two (main) distinct variables
    • w : weak(er) form of a theorem
    • ALT : alternative proof for a theorem
    • ALTV : alternative for another theorem/definition
    • OLD : old/obsolete version of a theorem/definition/proof
  • 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)
aand (suffix) No biimpa 482, rexlimiva 2891
ablAbelian group df-abl 17123  Abel Yes ablgrp 17125, zringabl 18810
absabsorption No ressabs 14905
absabsolute value (of a complex number) df-abs 13216  ( abs `  A ) Yes absval 13218, absneg 13257, abs1 13277
adadding No adantr 463, ad2antlr 725
addadd (see "p") df-add 9532  ( A  +  B ) Yes addcl 9603, addcom 9799, addass 9608
al"for all"  A. x ph No alim 1653, alex 1668
ALTalternative/less preferred (suffix) No aevALT 2089
anand df-an 369  ( ph  /\  ps ) Yes anor 487, iman 422, imnan 420
antantecedent No adantr 463
assassociative No biass 357, orass 522, mulass 9609
asymasymmetric, antisymmetric No intasym 5202, asymref 5203, posasymb 15904
axaxiom No ax6dgen 1848, ax1cn 9555
bas, base base (set of an extensible structure) df-base 14844  ( Base `  S ) Yes baseval 14886, ressbas 14896, cnfldbas 18742
b, bibiconditional ("iff", "if and only if") df-bi 185  ( ph  <->  ps ) Yes impbid 191, sspwb 4639
brbinary relation df-br 4395  A R B Yes brab1 4439, brun 4442
cbvchange bound variable No cbvalivw 1813, cbvrex 3030
clclosure No ifclda 3916, ovrcl 6310, zaddcl 10944
cncomplex numbers df-c 9527  CC Yes nnsscn 10580, nncn 10583
cnfldfield of complex numbers df-cnfld 18739 fld Yes cnfldbas 18742, cnfldinv 18767
cntzcentralizer df-cntz 16677  (Cntz `  M ) Yes cntzfval 16680, dprdfcntz 17367
cnvconverse df-cnv 4830  `' A Yes opelcnvg 5002, f1ocnv 5810
cocomposition df-co 4831  ( A  o.  B ) Yes cnvco 5008, fmptco 6042
comcommutative No orcom 385, bicomi 202, eqcomi 2415
concontradiction, contraposition No condan 795, con2d 115
csbclass substitution df-csb 3373  [_ A  /  x ]_ B Yes csbid 3380, csbie2g 3403
cygcyclic group df-cyg 17203 CycGrp Yes iscyg 17204, zringcyg 18824
ddeduction form (suffix) No idd 24, impbid 191
df(alternate) definition (prefix) No dfrel2 5273, dffn2 5714
di, distrdistributive No andi 868, imdi 361, ordi 865, difindi 3703, ndmovdistr 6444
difdifference df-dif 3416  ( A  \  B ) Yes difss 3569, difindi 3703
divdivision df-div 10247  ( A  /  B ) Yes divcl 10253, divval 10249, divmul 10250
dmdomain df-dm 4832  dom  A Yes dmmpt 5317, iswrddm0 12615
e, eq, equequals df-cleq 2394  A  =  B Yes 2p2e4 10693, uneqri 3584, equtr 1820
elelement of  A  e.  B Yes eldif 3423, eldifsn 4096, elssuni 4219
eu"there exists exactly one" df-eu 2242  E! x ph Yes euex 2264, euabsn 4043
exexists (i.e. is a set) No brrelex 4861, 0ex 4525
ex"there exists (at least one)" df-ex 1634  E. x ph Yes exim 1675, alex 1668
expexport No expt 156, expcom 433
f"not free in" (suffix) No equs45f 2115, sbf 2145
ffunction df-f 5572  F : A --> B Yes fssxp 5725, opelf 5729
falfalse df-fal 1411 F. Yes bifal 1418, falantru 1431
fifinite intersection df-fi 7904  ( fi `  B ) Yes fival 7905, inelfi 7911
fi, finfinite df-fin 7557  Fin Yes isfi 7576, snfi 7633, onfin 7745
fldfield (Note: there is an alternative definition  Fld of a field, see df-fld 25816) df-field 17717 Field Yes isfld 17723, fldidom 18272
fnfunction with domain df-fn 5571  A  Fn  B Yes ffn 5713, fndm 5660
frgpfree group df-frgp 17050  (freeGrp `  I ) Yes frgpval 17098, frgpadd 17103
fsuppfinitely supported function df-fsupp 7863  R finSupp  Z Yes isfsupp 7866, fdmfisuppfi 7871, fsuppco 7894
funfunction df-fun 5570  Fun  F Yes funrel 5585, ffun 5715
fvfunction value df-fv 5576  ( F `  A ) Yes fvres 5862, swrdfv 12703
fzfinite set of sequential integers df-fz 11725  ( M ... N ) Yes fzval 11726, eluzfz 11735
fz0finite set of sequential nonnegative integers  ( 0 ... N ) Yes nn0fz0 11827, fz0tp 11830
fzohalf-open integer range df-fzo 11853  ( M..^ N ) Yes elfzo 11859, elfzofz 11872
gmore general (suffix); eliminates "is a set" hypothsis No uniexg 6578
gragraph No uhgrav 24700, isumgra 24719, usgrares 24773
grpgroup df-grp 16379  Grp Yes isgrp 16383, tgpgrp 20867
gsumgroup sum df-gsum 15055  ( G  gsumg  F ) Yes gsumval 16220, gsumwrev 16723
hashsize (of a set) df-hash 12451  ( # `  A ) Yes hashgval 12453, hashfz1 12464, hashcl 12473
hbhypothesis builder (prefix) No hbxfrbi 1664, hbald 1872, hbequid 31912
hm(monoid, group, ring) homomorphism No ismhm 16290, isghm 16589, isrhm 17688
iinference (suffix) No eleq1i 2479, tcsni 8205
iimplication (suffix) No brwdomi 8027, infeq5i 8085
ididentity No biid 236
idmidempotent No anidm 642, tpidm13 4073
im, impimplication (label often omitted) df-im 13081  ( A  ->  B ) Yes iman 422, imnan 420, impbidd 189
imaimage df-ima 4835  ( A " B ) Yes resima 5125, imaundi 5235
impimport No biimpa 482, impcom 428
inintersection df-in 3420  ( A  i^i  B ) Yes elin 3625, incom 3631
is...is (something a) ...? No isring 17520
jjoining, disjoining No jc 147, jaoi 377
lleft No olcd 391, simpl 455
mapmapping operation or set exponentiation df-map 7458  ( A  ^m  B ) Yes mapvalg 7466, elmapex 7476
matmatrix df-mat 19200  ( N Mat  R ) Yes matval 19203, matring 19235
mdetdeterminant (of a square matrix) df-mdet 19377  ( N maDet  R ) Yes mdetleib 19379, mdetrlin 19394
mgmmagma df-mgm 16194  Magma Yes mgmidmo 16208, mgmlrid 16215, ismgm 16195
mgpmultiplicative group df-mgp 17460  (mulGrp `  R ) Yes mgpress 17470, ringmgp 17522
mndmonoid df-mnd 16243  Mnd Yes mndass 16252, mndodcong 16888
mo"there exists at most one" df-mo 2243  E* x ph Yes eumo 2269, moim 2291
mpmodus ponens ax-mp 5 No mpd 15, mpi 18
mptmodus ponendo tollens No mptnan 1621, mptxor 1622
mptmaps-to notation for a function df-mpt 4454  ( x  e.  A  |->  B ) Yes fconstmpt 4866, resmpt 5142
mpt2maps-to notation for an operation df-mpt2 6282  ( x  e.  A ,  y  e.  B  |->  C ) Yes mpt2mpt 6374, resmpt2 6380
mulmultiplication (see "t") df-mul 9533  ( A  x.  B ) Yes mulcl 9605, divmul 10250, mulcom 9607, mulass 9609
n, notnot  -.  ph Yes nan 578, notnot2 112
nenot equaldf-ne  A  =/=  B Yes exmidne 2608, neeqtrd 2698
nelnot element ofdf-nel  A  e/  B Yes neli 2738, nnel 2748
ne0not equal to zero (see n0)  =/=  0 No negne0d 9964, ine0 10032, gt0ne0 10057
nf "not free in" (prefix) No nfnd 1930
ngpnormed group df-ngp 21394 NrmGrp Yes isngp 21406, ngptps 21412
nmnorm (on a group or ring) df-nm 21393  ( norm `  W ) Yes nmval 21400, subgnm 21437
nnpositive integers df-nn 10576  NN Yes nnsscn 10580, nncn 10583
nn0nonnegative integers df-n0 10836  NN0 Yes nnnn0 10842, nn0cn 10845
n0not the empty set (see ne0)  =/=  (/) No n0i 3742, vn0 3745, ssn0 3771
OLDold, obsolete (to be removed soon) No 19.43OLD 1715
opordered pair df-op 3978  <. A ,  B >. Yes dfopif 4155, opth 4664
oror df-or 368  ( ph  \/  ps ) Yes orcom 385, anor 487
otordered triple df-ot 3980  <. A ,  B ,  C >. Yes euotd 4690, fnotovb 6318
ovoperation value df-ov 6280  ( A F B ) Yes fnotovb 6318, fnovrn 6430
pplus (see "add"), for all-constant theorems df-add 9532  ( 3  +  2 )  =  5 Yes 3p2e5 10708
pfxprefix df-pfx 37850  ( W prefix  L ) Yes pfxlen 37859, ccatpfx 37877
pmPrincipia Mathematica No pm2.27 37
pmpartial mapping (operation) df-pm 7459  ( A  ^pm  B ) Yes elpmi 7474, pmsspw 7490
prpair df-pr 3974  { A ,  B } Yes elpr 3989, prcom 4049, prid1g 4077, prnz 4090
prm, primeprime (number) df-prm 14425  Prime Yes 1nprm 14429, dvdsprime 14437
pssproper subset df-pss 3429  A  C.  B Yes pssss 3537, sspsstri 3544
q rational numbers ("quotients") df-q 11227  QQ Yes elq 11228
rright No orcd 390, simprl 756
rabrestricted class abstraction df-rab 2762  { x  e.  A  |  ph } Yes rabswap 2986, df-oprab 6281
ralrestricted universal quantification df-ral 2758  A. x  e.  A ph Yes ralnex 2849, ralrnmpt2 6397
rclreverse closure No ndmfvrcl 5873, nnarcl 7301
rereal numbers df-r 9531  RR Yes recn 9611, 0re 9625
relrelation df-rel 4829  Rel  A Yes brrelex 4861, relmpt2opab 6865
resrestriction df-res 4834  ( A  |`  B ) Yes opelres 5098, f1ores 5812
reurestricted existential uniqueness df-reu 2760  E! x  e.  A ph Yes nfreud 2979, reurex 3023
rexrestricted existential quantification df-rex 2759  E. x  e.  A ph Yes rexnal 2851, rexrnmpt2 6398
rmorestricted "at most one" df-rmo 2761  E* x  e.  A ph Yes nfrmod 2980, nrexrmo 3026
rnrange df-rn 4833  ran  A Yes elrng 5014, rncnvcnv 5046
rng(unital) ring df-ring 17518  Ring Yes ringidval 17473, isring 17520, ringgrp 17521
rotrotation No 3anrot 979, 3orrot 980
seliminates need for syllogism (suffix) No ancoms 451
sb(proper) substitution (of a set) df-sb 1764  [ y  /  x ] ph Yes spsbe 1767, sbimi 1769
sbc(proper) substitution of a class df-sbc 3277  [. A  /  x ]. ph Yes sbc2or 3285, sbcth 3291
scascalar df-sca 14923  (Scalar `  H ) Yes resssca 14989, mgpsca 17466
simpsimple, simplification No simpl 455, simp3r3 1107
snsingleton df-sn 3972  { A } Yes eldifsn 4096
spspecialization No spsbe 1767, spei 2039
sssubset df-ss 3427  A  C_  B Yes difss 3569
structstructure df-struct 14841 Struct Yes brstruct 14847, structfn 14852
subsubtract df-sub 9842  ( A  -  B ) Yes subval 9846, subaddi 9942
suppsupport (of a function) df-supp 6902  ( F supp  Z ) Yes ressuppfi 7888, mptsuppd 6925
swapswap (two parts within a theorem) No rabswap 2986, 2reuswap 3251
sylsyllogism syl 17 No 3syl 20
symsymmetric No df-symdif 3669, cnvsym 5201
symgsymmetric group df-symg 16725  ( SymGrp `  A ) Yes symghash 16732, pgrpsubgsymg 16755
t times (see "mul"), for all-constant theorems df-mul 9533  ( 3  x.  2 )  =  6 Yes 3t2e6 10727
ththeorem No nfth 1646, sbcth 3291, weth 8906
tptriple df-tp 3976  { A ,  B ,  C } Yes eltpi 4015, tpeq1 4059
trtransitive No bitrd 253, biantr 932
trutrue df-tru 1408 T. Yes bitru 1417, truanfal 1430
ununion df-un 3418  ( A  u.  B ) Yes uneqri 3584, uncom 3586
unitunit (in a ring) df-unit 17609  (Unit `  R ) Yes isunit 17624, nzrunit 18233
vdistinct variable conditions used when a not-free hypothesis (suffix) No spimv 2036
vv2 distinct variables (in a not-free hypothesis) (suffix) No 19.23vv 1785
wweak (version of a theorem) (suffix) No ax11w 1850, spnfw 1809
wrdword df-word 12589 Word  S Yes iswrdb 12602, wrdfn 12610, ffz0iswrd 12619
xpcross product (Cartesian product) df-xp 4828  ( A  X.  B ) Yes elxp 4839, opelxpi 4854, xpundi 4875
xreXtended reals df-xr 9661  RR* Yes ressxr 9666, rexr 9668, 0xr 9669
z integers (from German "Zahlen") df-z 10905  ZZ Yes elz 10906, zcn 10909
zn ring of integers  mod  n df-zn 18842  (ℤ/n `  N ) Yes znval 18870, zncrng 18879, znhash 18893
zringring of integers df-zring 18807 ring Yes zringbas 18812, zringcrng 18808
0, z slashed zero (empty set) (see n0) df-nul 3738  (/) Yes n0i 3742, vn0 3745; snnz 4089, prnz 4090

The challenge of varying mathematical conventions

We try to follow mathematical conventions, but in many cases different texts use different conventions. In those cases we pick some reasonably common convention and stick to it. We have already mentioned that the term "natural number" has varying definitions (some start from 0, others start from 1), but that is not the only such case.

A useful example is the set of metavariables used to represent arbitrary well-formed formulas (wffs). We use an open phi, φ, to represent the first arbitrary wff in an assertion with one or more wffs; this is a common convention and this symbol is easily distinguished from the empty set symbol. That said, it is impossible to please everyone or simply "follow the literature" because there are many different conventions for a variable that represents any arbitrary wff. To demonstrate the point, here are some conventions for variables that represent an arbitrary wff and some texts that use each convention:

  • open phi φ (and so on): Tarski's papers, Rasiowa & Sikorski's The Mathematics of Metamathematics (1963), Monk's Introduction to Set Theory (1969), Enderton's Elements of Set Theory (1977), Bell & Machover's A Course in Mathematical Logic (1977), Jech's Set Theory (1978), Takeuti & Zaring's Introduction to Axiomatic Set Theory (1982).
  • closed phi ϕ (and so on): Levy's Basic Set Theory (1979), Kunen's Set Theory (1980), Paulson's Isabelle: A Generic Theorem Prover (1994), Huth and Ryan's Logic in Computer Science (2004/2006).
  • Greek α, β, γ: Duffy's Principles of Automated Theorem Proving (1991).
  • Roman A, B, C: Kleene's Introduction to Metamathematics (1974), Smullyan's First-Order Logic (1968/1995).
  • script A, B, C: Hamilton's Logic for Mathematicians (1988).
  • italic A, B, C: Mendelson's Introduction to Mathematical Logic (1997).
  • italic P, Q, R: Suppes's Axiomatic Set Theory (1972), Gries and Schneider's A Logical Approach to Discrete Math (1993/1994), Rosser's Logic for Mathematicians (2008).
  • italic p, q, r: Quine's Set Theory and Its Logic (1969), Kuratowski & Mostowski's Set Theory (1976).
  • italic X, Y, Z: Dijkstra and Scholten's Predicate Calculus and Program Semantics (1990).
  • Fraktur letters: Fraenkel et. al's Foundations of Set Theory (1973).

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 1638 (whose description has some important technical details). Similarly,  F/_ x A is read  x is not free in (class)  A, see df-nfc 2552.
  • "$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 $." Sometimes a theorem is proved using  F/ x ph (df-nf 1638) in place of "$d  x ph $." when a more general result is desired; ax-5 1725 can be used to derive the $d version. For an example of how to get from the $d version back to the $e version, see the proof of euf 2248 from df-eu 2242.
  • "$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.

There is a general technique to replace a $d x A or $d x ph condition in a theorem with the corresponding  F/_ x A or  F/ x ph; here it is.  |- T[x, A] where , and you wish to prove  |-  F/_ x A =>  |- T[x, A]. You apply the theorem substituting  y for  x and  A for  A, where  y is a new dummy variable, so that $d y A is satisfied. You obtain  |- T[y, A], and apply chvar to obtain  |- T[x, A] (or just use mpbir 209 if T[x, A] binds  x). The side goal is  |-  ( x  =  y  ->  ( T[y, A]  <-> T[x, A]  ) ), where you can use equality theorems, except that when you get to a bound variable you use a non-dv bound variable renamer theorem like cbval 2048. The section mmtheorems32.html#mm3146s also describes the metatheorem that underlies this.

Standard Metamath verifiers do not distinguish between axioms and definitions (both are $a statements). In practice, we require that definitions (1) be conservative (a definition should not allow an expression that previously qualified as a wff but was not provable to become provable) and be eliminable (there should exist an algorithmic method for converting any expression using the definition into a logically equivalent expression that previously qualified as a wff). To ensure this, we have additional rules on almost all definitions ($a statements with a label that does not begin with ax-). These additional rules are not applied in a few cases where they are too strict (df-bi 185, df-clab 2388, df-cleq 2394, and df-clel 2397); see those definitions for more information. These additional rules for definitions are checked by at least mmj2's definition check (see mmj2 master file mmj2jar/macros/definitionCheck.js). This definition check relies on the database being very much like set.mm, down to the names of certain constants and types, so it cannot apply to all Metamath databases... but it is useful in set.mm. In this definition check, a $a-statement with a given label and typecode  |- passes the test if and only if it respects the following rules (these rules require that we have an unambiguous tree parse, which is checked separately):

  1. The expression must be a biconditional or an equality (i.e. its root-symbol must be  <-> or  =). If the proposed definition passes this first rule, we then define its definiendum as its left hand side (LHS) and its definiens as its right hand side (RHS). We define the *defined symbol* as the root-symbol of the LHS. We define a *dummy variable* as a variable occurring in the RHS but not in the LHS. Note that the "root-symbol" is the root of the considered tree; it need not correspond to a single token in the database (e.g., see w3o 973 or wsb 1763).
  2. The defined expression must not appear in any statement between its syntax axiom () and its definition, and the defined expression must be not be used in its definiens. See df-3an 976 for an example where the same symbol is used in different ways (this is allowed).
  3. No two variables occurring in the LHS may share a disjoint variable (DV) condition.
  4. All dummy variables are required to be disjoint from any other (dummy or not) variable occurring in this labelled expression.
  5. Either (a) there must be no non-setvar dummy variables, or (b) there must be a justification theorem. The justification theorem must be of form  |-  ( definiens root-symbol definiens'  ) where definiens' is definiens but the dummy variables are all replaced with other unused dummy variables of the same type. Note that root-symbol is  <-> or  =, and that setvar variables are simply variables with the  setvar typecode.
  6. One of the following must be true: (a) there must be no setvar dummy variables, (b) there must be a justification theorem as described in rule 5, or (c) if there are setvar dummy variables, every one must not be free. That is, it must be true that  ( ph  ->  A. x ph ) for each setvar dummy variable  x where  ph is the definiens. We use two different tests for not-freeness; one must succeed for each setvar dummy variable  x. The first test requires that the setvar dummy variable  x be syntactically bound (this is sometimes called the "fast" test, and this implies that we must track binding operators). The second test requires a successful search for the directly-stated proof of  ( ph  ->  A. x ph ) Part c of this rule is how most setvar dummy variables are handled.

Rule 3 may seem unnecessary, but it is needed. Without this rule, you can define something like cbar $a wff Foo x y $. ${ $d x y $. df-foo $a |- ( Foo x y <-> x = y ) $. $} and now "Foo x x" is not eliminable; there is no way to prove that it means anything in particular, because the definitional theorem that is supposed to be responsible for connecting it to the original language wants nothing to do with this expression, even though it is well formed.

A justification theorem for a definition (if used this way) must be proven before the definition that depends on it. One example of a justification theorem is vjust 3059. The definition df-v 3060  |-  _V  =  { x  |  x  =  x } is justified by the justification theorem vjust 3059  |-  { x  |  x  =  x }  =  { y  |  y  =  y }. Another example of a justification theorem is trujust 1407; the definition df-tru 1408  |-  ( T.  <->  ( A. x x  =  x  ->  A. x x  =  x ) ) is justified by trujust 1407  |-  ( ( A. x x  =  x  ->  A. x x  =  x )  <->  ( A. y y  =  y  ->  A. y y  =  y ) ).

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.
  • Maximum text line length is 79 characters. You can fix comment line length by running the commands scripts/rewrap or metamath 'read set.mm' 'save proof */c/f' 'write source set.mm/rewrap' quit . As a general rule, a math string in a comment should be surrounded by backquotes on the same line, and if it is too long it should be broken into multiple adjacent mathstrings on multiple lines. Those commands don't modify the math content of statements. In statements we try to break before the outermost important connective (not including the typecode and perhaps not the antecedent). For examples, see sqrtmulii 13366 and absmax 13309.
  • 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).
  • LRParser check. Metamath verifiers ensure that $p statements follow from previous $a and $p statements. However, by itself the Metamath language permits certain kinds of syntactic ambiguity that we choose to avoid in this database. Thus, we require that this database unambiguously parse using the "LRParser" check (implemented by at least mmj2). (For details, see mmj2 master file src/mmj/verify/LRParser.java). This check counters, for example, a devious ambiguous construct developed by saueran at oregonstate dot edu posted on Mon, 11 Feb 2019 17:32:32 -0800 (PST) based on creating definitions with mismatched parentheses.
  • 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
 
17.1.2  Natural deduction
 
Theoremnatded 25528 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 530 jca 530, pm3.2i 453 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 457 simpld 457, adantr 463 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 461 simpr 459, adantl 464 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 432 ex 432 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 5, mpd 15, mpdan 666, imp 427 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 391 olc 382, olci 389, olcd 391 Definition  \/I in [Pfenning] p. 18, definition I \/n(1) in [Clemente] p. 12
 \/IR  _G |-  ch =>  _G |-  ps  \/  ch orcd 390 orc 383, orci 388, orcd 390 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 787 mpjaodan 787, jaodan 786, jaod 378 Definition  \/E in [Pfenning] p. 18, definition E \/m,n,p in [Clemente] p. 12.
 -.I  _G ,  ps |- F. =>  _G |-  -.  ps inegd 1426 pm2.01d 169
 -.I  _G ,  ps |-  th &  _G |-  -.  th =>  _G |-  -.  ps mtand 657 mtand 657 definition I -.m,n,p in [Clemente] p. 13.
 -.I  _G ,  ps |-  ch &  _G ,  ps |-  -.  ch =>  _G |-  -.  ps pm2.65da 574 pm2.65da 574 Contradiction.
 -.I  _G ,  ps |-  -.  ps =>  _G |-  -.  ps pm2.01da 440 pm2.01d 169, pm2.65da 574, pm2.65d 175 For an alternative falsum-free natural deduction ruleset
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |- F. pm2.21fal 1428 pm2.21dd 174
 -.E  _G ,  -.  ps |- F. =>  _G |-  ps pm2.21dd 174 definition  ->E in [Indrzejczak] p. 33.
 -.E  _G |-  ps &  _G |-  -.  ps =>  _G |-  th pm2.21dd 174 pm2.21dd 174, pm2.21d 106, pm2.21 108 For an alternative falsum-free natural deduction ruleset. Definition  -.E in [Pfenning] p. 18.
T.I  _G |- T. a1tru 1421 tru 1409, a1tru 1421, trud 1414 Definition T.I in [Pfenning] p. 18.
F.E  _G , F.  |-  th falimd 1420 falim 1419 Definition F.E in [Pfenning] p. 18.
 A.I  _G |-  [ a  /  x ] ps =>  _G |-  A. x ps alrimiv 1740 alrimiv 1740, ralrimiva 2817 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 3290 spcv 3149, rspcv 3155 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 3359 spcev 3150, rspcev 3159 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 1747 exlimddv 1747, exlimdd 2008, exlimdv 1745, rexlimdva 2895 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 1427 efald 1427 Proof by contradiction (classical logic), definition F.C in [Pfenning] p. 17.
F.C  _G ,  -.  ps |-  ps =>  _G |-  ps pm2.18da 441 pm2.18da 441, pm2.18d 111, pm2.18 110 For an alternative falsum-free natural deduction ruleset
 -.  -.C  _G |-  -.  -.  ps =>  _G |-  ps notnotrd 113 notnotrd 113, notnot2 112 Double negation rule (classical logic), definition NNC in [Pfenning] p. 17, definition E -.n in [Clemente] p. 14.
EM  _G |-  ps  \/  -.  ps exmidd 414 exmid 413 Excluded middle (classical logic), definition XM in [Pfenning] p. 17, proof 5.11 in [Clemente] p. 14.
 =I  _G |-  A  =  A eqidd 2403 eqid 2402, eqidd 2403 Introduce equality, definition =I in [Pfenning] p. 127.
 =E  _G |-  A  =  B &  _G [. A  /  x ]. ps =>  _G |-  [. B  /  x ]. ps sbceq1dd 3282 sbceq1d 3281, 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 25529, ex-natded5.3 25532, ex-natded5.5 25535, ex-natded5.7 25536, ex-natded5.8 25538, ex-natded5.13 25540, ex-natded9.20 25542, and ex-natded9.26 25544.

(Contributed by DAW, 4-Feb-2017.) (New usage is discouraged.)

 |-  ph   =>    |-  ph
 
17.1.3  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 25528 and http://us.metamath.org/mpeuni/mmnatded.html. Since these examples should not be used within proofs of other theorems, especially in Mathboxes, they are marked with "(New usage is discouraged.)".

 
Theoremex-natded5.2 25529 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 530, 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 25530. A proof without context is shown in ex-natded5.2i 25531. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2-2 25530 A more efficient proof of Theorem 5.2 of [Clemente] p. 15. Compare with ex-natded5.2 25529 and ex-natded5.2i 25531. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  th ) )   &    |-  ( ph  ->  ( ch  ->  ps ) )   &    |-  ( ph  ->  ch )   =>    |-  ( ph  ->  th )
 
Theoremex-natded5.2i 25531 The same as ex-natded5.2 25529 and ex-natded5.2-2 25530 but with no context. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ( ps  /\  ch )  ->  th )   &    |-  ( ch  ->  ps )   &    |-  ch   =>    |- 
 th
 
Theoremex-natded5.3 25532 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 25533. A proof without context is shown in ex-natded5.3i 25534. 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 463 to move it into the ND hypothesis
25;6  ( ch  ->  th )  ( ph  ->  ( ch  ->  th ) ) Given $e; adantr 463 to move it into the ND hypothesis
31 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 459, to access the new assumption
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15, the MPE equivalent of  ->E, 1.3. adantr 463 was used to transform its dependency (we could also use imp 427 to get this directly from 1)
57 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15, the MPE equivalent of  ->E, 4,6. adantr 463 was used to transform its dependency
68 ...  ( ch  /\  th )  ( ( ph  /\  ps )  ->  ( ch  /\  th ) )  /\I 4,5 jca 530, the MPE equivalent of  /\I, 4,7
79  ( ps  ->  ( ch  /\  th ) )  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )  ->I 3,6 ex 432, 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.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3-2 25533 A more efficient proof of Theorem 5.3 of [Clemente] p. 16. Compare with ex-natded5.3 25532 and ex-natded5.3i 25534. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ch  ->  th ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  /\  th ) ) )
 
Theoremex-natded5.3i 25534 The same as ex-natded5.3 25532 and ex-natded5.3-2 25533 but with no context. Identical to jccir 537, which should be used instead. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ps  ->  ch )   &    |-  ( ch  ->  th )   =>    |-  ( ps  ->  ( ch  /\  th ) )
 
Theoremex-natded5.5 25535 Theorem 5.5 of [Clemente] p. 18, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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 (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
12;3  ( ps  ->  ch )  ( ph  ->  ( ps  ->  ch ) ) Given $e; adantr 463 to move it into the ND hypothesis
25  -.  ch  ( ph  ->  -.  ch ) Given $e; we'll use adantr 463 to move it into the ND hypothesis
31 ...|  ps  ( ph  ->  ps ) ND hypothesis assumption simpr 459
44 ...  ch  ( ( ph  /\  ps )  ->  ch )  ->E 1,3 mpd 15 1,3
56 ...  -.  ch  ( ( ph  /\  ps )  ->  -.  ch ) IT 2 adantr 463 5
67  -.  ps  ( ph  ->  -.  ps )  /\I 3,4,5 pm2.65da 574 4,6

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 463; simpr 459 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is mtod 177; a proof without context is shown in mto 176.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.7 25536 Theorem 5.7 of [Clemente] p. 19, translated line by line using the interpretation of natural deduction in Metamath. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.7-2 25537. 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
16  ( ps  \/  ( ch  /\  th ) )  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) ) Given $e. No need for adantr 463 because we do not move this into an ND hypothesis
21 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption (new scope) simpr 459
32 ...  ( ps  \/  ch )  ( ( ph  /\  ps )  ->  ( ps  \/  ch ) )  \/IL 2 orcd 390, the MPE equivalent of  \/IL, 1
43 ...|  ( ch  /\  th )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ch  /\  th ) ) ND hypothesis assumption (new scope) simpr 459
54 ...  ch  ( ( ph  /\  ( ch  /\  th ) )  ->  ch )  /\EL 4 simpld 457, the MPE equivalent of  /\EL, 3
66 ...  ( ps  \/  ch )  ( ( ph  /\  ( ch  /\  th ) )  ->  ( ps  \/  ch ) )  \/IR 5 olcd 391, the MPE equivalent of  \/IR, 4
77  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) )  \/E 1,3,6 mpjaodan 787, the MPE equivalent of  \/E, 2,5,6

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.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.7-2 25537 A more efficient proof of Theorem 5.7 of [Clemente] p. 19. Compare with ex-natded5.7 25536. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ( ch  /\  th ) ) )   =>    |-  ( ph  ->  ( ps  \/  ch )
 )
 
Theoremex-natded5.8 25538 Theorem 5.8 of [Clemente] p. 20, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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 (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
110;11  ( ( ps  /\  ch )  ->  -.  th )  ( ph  ->  ( ( ps  /\  ch )  ->  -.  th ) ) Given $e; adantr 463 to move it into the ND hypothesis
23;4  ( ta  ->  th )  ( ph  ->  ( ta  ->  th ) ) Given $e; adantr 463 to move it into the ND hypothesis
37;8  ch  ( ph  ->  ch ) Given $e; adantr 463 to move it into the ND hypothesis
41;2  ta  ( ph  ->  ta ) Given $e. adantr 463 to move it into the ND hypothesis
56 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND Hypothesis/Assumption simpr 459. New ND hypothesis scope, each reference outside the scope must change antecedent  ph to  ( ph  /\  ps ).
69 ...  ( ps  /\  ch )  ( ( ph  /\  ps )  ->  ( ps  /\  ch ) )  /\I 5,3 jca 530 ( /\I), 6,8 (adantr 463 to bring in scope)
75 ...  -.  th  ( ( ph  /\  ps )  ->  -.  th )  ->E 1,6 mpd 15 ( ->E), 2,4
812 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15 ( ->E), 9,11; note the contradiction with ND line 7 (MPE line 5)
913  -.  ps  ( ph  ->  -.  ps )  -.I 5,7,8 pm2.65da 574 ( -.I), 5,12; proof by contradiction. MPE step 6 (ND#5) does not need a reference here, because the assumption is embedded in the antecedents

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 463; simpr 459 is useful when you want to depend directly on the new assumption). 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-natded5.8-2 25539.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.8-2 25539 A more efficient proof of Theorem 5.8 of [Clemente] p. 20. For a longer line-by-line translation, see ex-natded5.8 25538. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  (
 ( ps  /\  ch )  ->  -.  th )
 )   &    |-  ( ph  ->  ( ta  ->  th ) )   &    |-  ( ph  ->  ch )   &    |-  ( ph  ->  ta )   =>    |-  ( ph  ->  -.  ps )
 
Theoremex-natded5.13 25540 Theorem 5.13 of [Clemente] p. 20, 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. A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded5.13-2 25541. The original proof, which uses Fitch style, was written as follows (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
115  ( ps  \/  ch )  ( ph  ->  ( ps  \/  ch ) ) Given $e.
2;32  ( ps  ->  th )  ( ph  ->  ( ps  ->  th ) ) Given $e. adantr 463 to move it into the ND hypothesis
39  ( -.  ta  ->  -.  ch )  ( ph  ->  ( -.  ta  ->  -.  ch ) ) Given $e. ad2antrr 724 to move it into the ND sub-hypothesis
41 ...|  ps  ( ( ph  /\  ps )  ->  ps ) ND hypothesis assumption simpr 459
54 ...  th  ( ( ph  /\  ps )  ->  th )  ->E 2,4 mpd 15 1,3
65 ...  ( th  \/  ta )  ( ( ph  /\  ps )  ->  ( th  \/  ta ) )  \/I 5 orcd 390 4
76 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 459
88 ... ...|  -.  ta  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ta ) (sub) ND hypothesis assumption simpr 459
911 ... ...  -.  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  -.  ch )  ->E 3,8 mpd 15 8,10
107 ... ...  ch  ( ( ( ph  /\  ch )  /\  -.  ta )  ->  ch ) IT 7 adantr 463 6
1112 ...  -.  -.  ta  ( ( ph  /\  ch )  ->  -.  -.  ta )  -.I 8,9,10 pm2.65da 574 7,11
1213 ...  ta  ( ( ph  /\  ch )  ->  ta )  -.E 11 notnotrd 113 12
1314 ...  ( th  \/  ta )  ( ( ph  /\  ch )  ->  ( th  \/  ta ) )  \/I 12 olcd 391 13
1416  ( th  \/  ta )  ( ph  ->  ( th  \/  ta ) )  \/E 1,6,13 mpjaodan 787 5,14,15

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 463; simpr 459 is useful when you want to depend directly on the new assumption). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)

 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded5.13-2 25541 A more efficient proof of Theorem 5.13 of [Clemente] p. 20. Compare with ex-natded5.13 25540. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( -.  ta  ->  -.  ch ) )   =>    |-  ( ph  ->  ( th  \/  ta ) )
 
Theoremex-natded9.20 25542 Theorem 9.20 of [Clemente] p. 43, translated line by line using the usual translation of natural deduction (ND) in the Metamath Proof Explorer (MPE) notation. 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 (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
11  ( ps  /\  ( ch  \/  th ) )  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) ) Given $e
22  ps  ( ph  ->  ps )  /\EL 1 simpld 457 1
311  ( ch  \/  th )  ( ph  ->  ( ch  \/  th ) )  /\ER 1 simprd 461 1
44 ...|  ch  ( ( ph  /\  ch )  ->  ch ) ND hypothesis assumption simpr 459
55 ...  ( ps  /\  ch )  ( ( ph  /\  ch )  ->  ( ps  /\  ch ) )  /\I 2,4 jca 530 3,4
66 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  ch )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IR 5 orcd 390 5
78 ...|  th  ( ( ph  /\  th )  ->  th ) ND hypothesis assumption simpr 459
89 ...  ( ps  /\  th )  ( ( ph  /\  th )  ->  ( ps  /\  th ) )  /\I 2,7 jca 530 7,8
910 ...  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ( ph  /\  th )  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/IL 8 olcd 391 9
1012  ( ( ps  /\  ch )  \/  ( ps  /\  th ) )  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )  \/E 3,6,9 mpjaodan 787 6,10,11

The original used Latin letters; 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. To add an assumption, the antecedent is modified to include it (typically by using adantr 463; simpr 459 is useful when you want to depend directly on the new assumption). Below is the final metamath proof (which reorders some steps).

A much more efficient proof is ex-natded9.20-2 25543. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)

 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.20-2 25543 A more efficient proof of Theorem 9.20 of [Clemente] p. 45. Compare with ex-natded9.20 25542. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by David A. Wheeler, 19-Feb-2017.)
 |-  ( ph  ->  ( ps  /\  ( ch  \/  th ) ) )   =>    |-  ( ph  ->  ( ( ps  /\  ch )  \/  ( ps  /\  th ) ) )
 
Theoremex-natded9.26 25544* Theorem 9.26 of [Clemente] p. 45, translated line by line using an interpretation of natural deduction in Metamath. This proof has some additional complications due to the fact that Metamath's existential elimination rule does not change bound variables, so we need to verify that  x is bound in the conclusion. 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 (the leading "..." shows an embedded ND hypothesis, beginning with the initial assumption of the ND hypothesis):
#MPE#ND Expression MPE TranslationND Rationale MPE Rationale
13  E. x A. y ps ( x ,  y )  ( ph  ->  E. x A. y ps ) Given $e.
26 ...|  A. y ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  A. y ps ) ND hypothesis assumption simpr 459. Later statements will have this scope.
37;5,4 ...  ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  ps )  A.E 2,y spsbcd 3290 ( A.E), 5,6. To use it we need a1i 11 and vex 3061. This could be immediately done with 19.21bi 1893, but we want to show the general approach for substitution.
412;8,9,10,11 ...  E. x ps ( x ,  y )  ( ( ph  /\  A. y ps )  ->  E. x ps )  E.I 3,a spesbcd 3359 ( E.I), 11. To use it we need sylibr 212, which in turn requires sylib 196 and two uses of sbcid 3293. This could be more immediately done using 19.8a 1881, but we want to show the general approach for substitution.
513;1,2  E. x ps ( x ,  y )  ( ph  ->  E. x ps )  E.E 1,2,4,a exlimdd 2008 ( E.E), 1,2,3,12. We'll need supporting assertions that the variable is free (not bound), as provided in nfv 1728 and nfe1 1864 (MPE# 1,2)
614  A. y E. x ps ( x ,  y )  ( ph  ->  A. y E. x ps )  A.I 5 alrimiv 1740 ( A.I), 13

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).

Note that in the original proof,  ps ( x ,  y ) has explicit parameters. In Metamath, these parameters are always implicit, and the parameters upon which a wff variable can depend are recorded in the "allowed substitution hints" below.

A much more efficient proof, using more of Metamath and MPE's capabilities, is shown in ex-natded9.26-2 25545.

(Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.) (Revised by David A. Wheeler, 18-Feb-2017.)

 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
Theoremex-natded9.26-2 25545* A more efficient proof of Theorem 9.26 of [Clemente] p. 45. Compare with ex-natded9.26 25544. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Mario Carneiro, 9-Feb-2017.)
 |-  ( ph  ->  E. x A. y ps )   =>    |-  ( ph  ->  A. y E. x ps )
 
17.1.4  Definitional examples
 
Theoremex-or 25546 Example for df-or 368. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  3  \/  4  =  4 )
 
Theoremex-an 25547 Example for df-an 369. Example by David A. Wheeler. (Contributed by Mario Carneiro, 9-May-2015.)
 |-  ( 2  =  2 
 /\  3  =  3 )
 
Theoremex-dif 25548 Example for df-dif 3416. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  \  {
 1 ,  8 } )  =  { 3 }
 
Theoremex-un 25549 Example for df-un 3418. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  u.  {
 1 ,  8 } )  =  { 1 ,  3 ,  8 }
 
Theoremex-in 25550 Example for df-in 3420. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( { 1 ,  3 }  i^i  {
 1 ,  8 } )  =  { 1 }
 
Theoremex-uni 25551 Example for df-uni 4191. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |- 
 U. { { 1 ,  3 } ,  { 1 ,  8 } }  =  {
 1 ,  3 ,  8 }
 
Theoremex-ss 25552 Example for df-ss 3427. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C_  { 1 ,  2 ,  3 }
 
Theoremex-pss 25553 Example for df-pss 3429. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |- 
 { 1 ,  2 }  C.  { 1 ,  2 ,  3 }
 
Theoremex-pw 25554 Example for df-pw 3956. Example by David A. Wheeler. (Contributed by Mario Carneiro, 2-Jul-2016.)
 |-  ( A  =  {
 3 ,  5 ,  7 }  ->  ~P A  =  ( ( { (/) }  u.  { { 3 } ,  { 5 } ,  { 7 } }
 )  u.  ( { { 3 ,  5 } ,  { 3 ,  7 } ,  { 5 ,  7 } }  u.  { { 3 ,  5 ,  7 } }
 ) ) )
 
Theoremex-pr 25555 Example for df-pr 3974. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( A  e.  {
 1 ,  -u 1 }  ->  ( A ^
 2 )  =  1 )
 
Theoremex-br 25556 Example for df-br 4395. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  ( R  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  3 R
 9 )
 
Theoremex-opab 25557* Example for df-opab 4453. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( R  =  { <. x ,  y >.  |  ( x  e.  CC  /\  y  e.  CC  /\  ( x  +  1
 )  =  y ) }  ->  3 R
 4 )
 
Theoremex-eprel 25558 Example for df-eprel 4733. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  5  _E  { 1 ,  5 }
 
Theoremex-id 25559 Example for df-id 4737. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 5  _I  5  /\  -.  4  _I  5
 )
 
Theoremex-po 25560 Example for df-po 4743. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  (  <  Po  RR  /\ 
 -.  <_  Po  RR )
 
Theoremex-xp 25561 Example for df-xp 4828. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( { 1 ,  5 }  X.  {
 2 ,  7 } )  =  ( { <. 1 ,  2 >. ,  <. 1 ,  7
 >. }  u.  { <. 5 ,  2 >. ,  <. 5 ,  7 >. } )
 
Theoremex-cnv 25562 Example for df-cnv 4830. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  `' { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  =  { <. 6 ,  2 >. ,  <. 9 ,  3 >. }
 
Theoremex-co 25563 Example for df-co 4831. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( exp  o.  cos ) `  0 )  =  _e
 
Theoremex-dm 25564 Example for df-dm 4832. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  dom  F  =  { 2 ,  3 } )
 
Theoremex-rn 25565 Example for df-rn 4833. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ran  F  =  { 6 ,  9 } )
 
Theoremex-res 25566 Example for df-res 4834. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F  |`  B )  =  { <. 2 ,  6
 >. } )
 
Theoremex-ima 25567 Example for df-ima 4835. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( ( F  =  { <. 2 ,  6
 >. ,  <. 3 ,  9
 >. }  /\  B  =  { 1 ,  2 } )  ->  ( F " B )  =  { 6 } )
 
Theoremex-fv 25568 Example for df-fv 5576. Example by David A. Wheeler. (Contributed by Mario Carneiro, 7-May-2015.)
 |-  ( F  =  { <. 2 ,  6 >. ,  <. 3 ,  9
 >. }  ->  ( F `  3 )  =  9 )
 
Theoremex-1st 25569 Example for df-1st 6783. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 1st `  <. 3 ,  4 >. )  =  3
 
Theoremex-2nd 25570 Example for df-2nd 6784. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( 2nd `  <. 3 ,  4 >. )  =  4
 
Theorem1kp2ke3k 25571 Example for df-dec 11019, 1000 + 2000 = 3000.

This proof disproves (by counter-example) the assertion of Hao Wang, who stated, "There is a theorem in the primitive notation of set theory that corresponds to the arithmetic theorem 1000 + 2000 = 3000. The formula would be forbiddingly long... even if (one) knows the definitions and is asked to simplify the long formula according to them, chances are he will make errors and arrive at some incorrect result." (Hao Wang, "Theory and practice in mathematics" , In Thomas Tymoczko, editor, New Directions in the Philosophy of Mathematics, pp 129-152, Birkauser Boston, Inc., Boston, 1986. (QA8.6.N48). The quote itself is on page 140.)

This is noted in Metamath: A Computer Language for Pure Mathematics by Norman Megill (2007) section 1.1.3. Megill then states, "A number of writers have conveyed the impression that the kind of absolute rigor provided by Metamath is an impossible dream, suggesting that a complete, formal verification of a typical theorem would take millions of steps in untold volumes of books... These writers assume, however, that in order to achieve the kind of complete formal verification they desire one must break down a proof into individual primitive steps that make direct reference to the axioms. This is not necessary. There is no reason not to make use of previously proved theorems rather than proving them over and over... A hierarchy of theorems and definitions permits an exponential growth in the formula sizes and primitive proof steps to be described with only a linear growth in the number of symbols used. Of course, this is how ordinary informal mathematics is normally done anyway, but with Metamath it can be done with absolute rigor and precision."

The proof here starts with  ( 2  +  1 )  =  3, commutes it, and repeatedly multiplies both sides by ten. This is certainly longer than traditional mathematical proofs, e.g., there are a number of steps explicitly shown here to show that we're allowed to do operations such as multiplication. However, while longer, the proof is clearly a manageable size - even though every step is rigorously derived all the way back to the primitive notions of set theory and logic. And while there's a risk of making errors, the many independent verifiers make it much less likely that an incorrect result will be accepted.

This proof heavily relies on the decimal constructor df-dec 11019 developed by Mario Carneiro in 2015. The underlying Metamath language has an intentionally very small set of primitives; it doesn't even have a built-in construct for numbers. Instead, the digits are defined using these primitives, and the decimal constructor is used to make it easy to express larger numbers as combinations of digits.

(Contributed by David A. Wheeler, 29-Jun-2016.) (Shortened by Mario Carneiro using the arithmetic algorithm in mmj2, 30-Jun-2016.)

 |-  (;;; 1 0 0 0  + ;;; 2 0 0 0 )  = ;;; 3 0 0 0
 
Theoremex-fl 25572 Example for df-fl 11964. Example by David A. Wheeler. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( ( |_ `  (
 3  /  2 )
 )  =  1  /\  ( |_ `  -u (
 3  /  2 )
 )  =  -u 2
 )
 
Theoremex-dvds 25573 3 divides into 6. A demonstration of df-dvds 14194. (Contributed by David A. Wheeler, 19-May-2015.)
 |-  3  ||  6
 
17.1.5  Other examples
 
Theoremex-ind-dvds 25574 Example of a proof by induction (divisibility result). (Contributed by Stanislas Polu, 9-Mar-2020.) (Revised by BJ, 24-Mar-2020.)
 |-  ( N  e.  NN0  -> 
 3  ||  ( (
 4 ^ N )  +  2 ) )
 
17.2  Humor
 
17.2.1  April Fool's theorem
 
Theoremavril1 25575 Poisson d'Avril's Theorem. This theorem is noted for its Selbstdokumentieren property, which means, literally, "self-documenting" and recalls the principle of quidquid german dictum sit, altum viditur, often used in set theory. Starting with the seemingly simple yet profound fact that any object  x equals itself (proved by Tarski in 1965; see Lemma 6 of [Tarski] p. 68), we demonstrate that the power set of the real numbers, as a relation on the value of the imaginary unit, does not conjoin with an empty relation on the product of the additive and multiplicative identity elements, leading to this startling conclusion that has left even seasoned professional mathematicians scratching their heads. (Contributed by Prof. Loof Lirpa, 1-Apr-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

A reply to skeptics can be found at http://us.metamath.org/mpeuni/mmnotes.txt, under the 1-Apr-2006 entry.

 |- 
 -.  ( A ~P RR ( _i `  1
 )  /\  F (/) ( 0  x.  1 ) )
 
Theorem2bornot2b 25576 The law of excluded middle. Act III, Theorem 1 of Shakespeare, Hamlet, Prince of Denmark (1602). Its author leaves its proof as an exercise for the reader - "To be, or not to be: that is the question" - starting a trend that has become standard in modern-day textbooks, serving to make the frustrated reader feel inferior, or in some cases to mask the fact that the author does not know its solution. (Contributed by Prof. Loof Lirpa, 1-Apr-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 2  x.  B  \/  -.  2  x.  B )
 
Theoremhelloworld 25577 The classic "Hello world" benchmark has been translated into 314 computer programming languages - see http://www.roesler-ac.de/wolfram/hello.htm. However, for many years it eluded a proof that it is more than just a conjecture, even though a wily mathematician once claimed, "I have discovered a truly marvelous proof of this, which this margin is too narrow to contain." Using an IBM 709 mainframe, a team of mathematicians led by Prof. Loof Lirpa, at the New College of Tahiti, were finally able put it rest with a remarkably short proof only 4 lines long. (Contributed by Prof. Loof Lirpa, 1-Apr-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 -.  ( h  e.  ( L L 0 )  /\  W (/) ( R. 1 d ) )
 
Theorem1p1e2apr1 25578 One plus one equals two. Using proof-shortening techniques pioneered by Mr. Mel L. O'Cat, along with the latest supercomputer technology, Prof. Loof Lirpa and colleagues were able to shorten Whitehead and Russell's 360-page proof that 1+1=2 in Principia Mathematica to this remarkable proof only two steps long, thus establishing a new world's record for this famous theorem. (Contributed by Prof. Loof Lirpa, 1-Apr-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  +  1 )  =  2
 
Theoremeqid1 25579 Law of identity (reflexivity of class equality). Theorem 6.4 of [Quine] p. 41.

This law is thought to have originated with Aristotle (Metaphysics, Book VII, Part 17). It is one of the three axioms of Ayn Rand's philosophy (Atlas Shrugged, Part Three, Chapter VII). While some have proposed extending Rand's axiomatization to include Compassion and Kindness, others fear that such an extension may flirt with logical inconsistency. (Contributed by Stefan Allan, 1-Apr-2009.) (Proof modification is discouraged.) (New usage is discouraged.)

 |-  A  =  A
 
Theorem1div0apr 25580 Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( 1  /  0
 )  =  (/)
 
17.3  (Future - to be reviewed and classified)
 
17.3.1  Planar incidence geometry
 
Syntaxcplig 25581 Extend class notation with the class of all planar incidence geometries.
 class  Plig
 
Definitiondf-plig 25582* Planar incidence geometry. I use Hilbert's "axioms" adapted to planar geometry.  e. is the incidence relation. I could take a generic incidence relation but I'm lazy and I'm not sure the gain is worth the extra work. Much of what follows is directly borrowed from Aitken. http://public.csusm.edu/aitken_html/m410/betweenness.08.pdf (Contributed by FL, 2-Aug-2009.)
 |- 
 Plig  =  { x  |  ( A. a  e. 
 U. x A. b  e.  U. x ( a  =/=  b  ->  E! l  e.  x  (
 a  e.  l  /\  b  e.  l )
 )  /\  A. l  e.  x  E. a  e. 
 U. x E. b  e.  U. x ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  U. x E. b  e.  U. x E. c  e.  U. x A. l  e.  x  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l
 ) ) }
 
Theoremisplig 25583* The predicate "is a planar incidence geometry". (Contributed by FL, 2-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  A  ->  ( L  e.  Plig  <->  ( A. a  e.  P  A. b  e.  P  ( a  =/=  b  ->  E! l  e.  L  ( a  e.  l  /\  b  e.  l ) )  /\  A. l  e.  L  E. a  e.  P  E. b  e.  P  ( a  =/=  b  /\  a  e.  l  /\  b  e.  l )  /\  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) ) ) )
 
Theoremtncp 25584* There exist three non colinear points. (Contributed by FL, 3-Aug-2009.)
 |-  P  =  U. L   =>    |-  ( L  e.  Plig  ->  E. a  e.  P  E. b  e.  P  E. c  e.  P  A. l  e.  L  -.  ( a  e.  l  /\  b  e.  l  /\  c  e.  l ) )
 
Theoremlpni 25585* For any line, there exists a point not on the line. (Contributed by Jeff Hankins, 15-Aug-2009.)
 |-  P  =  U. G   =>    |-  (
 ( G  e.  Plig  /\  L  e.  G ) 
 ->  E. a  e.  P  a  e/  L )
 
17.3.2  Algebra preliminaries
 
Syntaxcrpm 25586 Ring primes.
 class RPrime
 
Definitiondf-rprm 25587* Define the set of prime elements in a ring. A prime element is a nonzero non-unit that satisfies an equivalent of Euclid's lemma euclemma 14456. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |- RPrime  =  ( w  e.  _V  |->  [_ ( Base `  w )  /  b ]_ { p  e.  ( b  \  (
 (Unit `  w )  u.  { ( 0g `  w ) } )
 )  |  A. x  e.  b  A. y  e.  b  [. ( ||r `  w )  /  d ]. ( p d ( x ( .r `  w ) y )  ->  ( p d x  \/  p d y ) ) } )
 
PART 18  ADDITIONAL MATERIAL ON GROUPS, RINGS, AND FIELDS (DEPRECATED)

This part contains an earlier development of groups, rings, and fields that was defined before extensible structures were introduced.

Theorem grpo2grp 25636 shows the relationship between the older group definition and the extensible structure definition.

The intent is for this deprecated section to be deleted once its theorems have extensible structure versions (or are not useful). You can make a list of "terminal" theorems (i.e. theorems not referenced by anything else) and for each theorem see if there exists an extensible structure version (or decide it's not useful), and if so, delete it. Then repeat this recursively. One way to search for terminal theorems, for example in deprecated group theory, is to log the output ("open log x.txt") of "show usage cgr~circgrp" in metamath.exe and search for "(None)".

 
18.1  Additional material on group theory
 
18.1.1  Definitions and basic properties for groups
 
Syntaxcgr 25588 Extend class notation with the class of all group operations.
 class  GrpOp
 
Syntaxcgi 25589 Extend class notation with a function mapping a group operation to the group's identity element.
 class GId
 
Syntaxcgn 25590 Extend class notation with a function mapping a group operation to the inverse function for the group.
 class  inv
 
Syntaxcgs 25591 Extend class notation with a function mapping a group operation to the division (or subtraction) operation for the group.
 class  /g
 
Syntaxcgx 25592 Extend class notation with a function mapping a group operation to the power operation for the group.
 class  ^g
 
Definitiondf-grpo 25593* Define the class of all group operations. The base set for a group can be determined from its group operation. Based on the definition in Exercise 28 of [Herstein] p. 54. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |- 
 GrpOp  =  { g  |  E. t ( g : ( t  X.  t ) --> t  /\  A. x  e.  t  A. y  e.  t  A. z  e.  t  (
 ( x g y ) g z )  =  ( x g ( y g z ) )  /\  E. u  e.  t  A. x  e.  t  (
 ( u g x )  =  x  /\  E. y  e.  t  ( y g x )  =  u ) ) }
 
Definitiondf-gid 25594* Define a function that maps a group operation to the group's identity element. (Contributed by FL, 5-Feb-2010.) (Revised by Mario Carneiro, 15-Dec-2013.) (New usage is discouraged.)
 |- GId 
 =  ( g  e. 
 _V  |->  ( iota_ u  e. 
 ran  g A. x  e.  ran  g ( ( u g x )  =  x  /\  ( x g u )  =  x ) ) )
 
Definitiondf-ginv 25595* Define a function that maps a group operation to the group's inverse function. (Contributed by NM, 26-Oct-2006.) (New usage is discouraged.)
 |- 
 inv  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g  |->  ( iota_ z  e.  ran  g (
 z g x )  =  (GId `  g
 ) ) ) )
 
Definitiondf-gdiv 25596* Define a function that maps a group operation to the group's division (or subtraction) operation. (Contributed by NM, 15-Feb-2008.) (New usage is discouraged.)
 |- 
 /g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ran  g  |->  ( x g ( ( inv `  g ) `  y
 ) ) ) )
 
Definitiondf-gx 25597* Define a function that maps a group operation to the group's power operation. (Contributed by Paul Chapman, 17-Apr-2009.) (New usage is discouraged.)
 |- 
 ^g  =  ( g  e.  GrpOp  |->  ( x  e. 
 ran  g ,  y  e.  ZZ  |->  if ( y  =  0 ,  (GId `  g ) ,  if ( 0  <  y ,  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  y
 ) ,  ( ( inv `  g ) `  (  seq 1
 ( g ,  ( NN  X.  { x }
 ) ) `  -u y
 ) ) ) ) ) )
 
Theoremisgrpo 25598* The predicate "is a group operation." Note that  X is the base set of the group. (Contributed by NM, 10-Oct-2006.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y ) G z )  =  ( x G ( y G z ) ) 
 /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. y  e.  X  ( y G x )  =  u ) ) ) )
 
Theoremisgrpo2 25599* The predicate "is a group operation." (Contributed by NM, 23-Oct-2012.) (New usage is discouraged.)
 |-  X  =  ran  G   =>    |-  ( G  e.  A  ->  ( G  e.  GrpOp  <->  ( G :
 ( X  X.  X )
 --> X  /\  A. x  e.  X  A. y  e.  X  A. z  e.  X  ( ( x G y )  e.  X  /\  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )  /\  E. u  e.  X  A. x  e.  X  ( ( u G x )  =  x  /\  E. n  e.  X  ( n G x )  =  u ) ) ) )
 
Theoremisgrpoi 25600* Properties that determine a group operation. Read  N as  N ( x ). (Contributed by NM, 4-Nov-2006.) (New usage is discouraged.)
 |-  X  e.  _V   &    |-  G : ( X  X.  X ) --> X   &    |-  (
 ( x  e.  X  /\  y  e.  X  /\  z  e.  X )  ->  ( ( x G y ) G z )  =  ( x G ( y G z ) ) )   &    |-  U  e.  X   &    |-  ( x  e.  X  ->  ( U G x )  =  x )   &    |-  ( x  e.  X  ->  N  e.  X )   &    |-  ( x  e.  X  ->  ( N G x )  =  U )   =>    |-  G  e.  GrpOp
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