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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-eupa 23201* Define the set of all Eulerian paths on an undirected multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- EulPaths  =  ( v  e.  _V ,  e  e.  _V  |->  { <. f ,  p >.  |  ( v UMGrph  e  /\  E. n  e.  NN0  ( f : ( 1 ... n ) -1-1-onto-> dom  e  /\  p : ( 0 ... n ) --> v  /\  A. k  e.  ( 1
 ... n ) ( e `  ( f `
  k ) )  =  { ( p `
  ( k  -  1 ) ) ,  ( p `  k
 ) } ) ) } )
 
Definitiondf-vdgr 23202* Define the vertex degree function for an undirected multigraph. We have to double-count those edges that contain  u "twice" (i.e. self-loops), this being represented as a singleton as the edge's value. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |- VDeg  =  ( v  e.  _V ,  e  e.  _V  |->  ( u  e.  v  |->  ( ( # `  { x  e. 
 dom  e  |  u  e.  ( e `  x ) } )  +  ( # `
  { x  e. 
 dom  e  |  ( e `  x )  =  { u } } ) ) ) )
 
Theoremrelumgra 23203 The class of all undirected multigraphs is a relation. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  Rel UMGrph
 
Theoremisumgra 23204* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  e.  X ) 
 ->  ( V UMGrph  E  <->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremwrdumgra 23205* The property of being an undirected multigraph. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  e. Word  X )  ->  ( V UMGrph  E  <->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 )
 
Theoremumgraf2 23206* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  E : dom  E --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgraf 23207* The edge function of an undirected multigraph is a function into unordered pairs of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A )  ->  E : A --> { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremumgrass 23208 An edge is a subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  C_  V )
 
Theoremumgran0 23209 An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  =/=  (/) )
 
Theoremumgrale 23210 An edge has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( # `
  ( E `  F ) )  <_ 
 2 )
 
Theoremumgrafi 23211 An edge is a finite subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  ( E `  F )  e. 
 Fin )
 
Theoremumgraex 23212* An edge is an unordered pair of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |-  (
 ( V UMGrph  E  /\  E  Fn  A  /\  F  e.  A )  ->  E. x  e.  V  E. y  e.  V  ( E `  F )  =  { x ,  y }
 )
 
Theoremumgrares 23213 A subgraph of a graph (formed by removing some edges from the original graph) is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V UMGrph  E  ->  V UMGrph  ( E  |`  A ) )
 
Theoremumgra0 23214 The empty graph, with vertices but no edges, is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( V  e.  W  ->  V UMGrph  (/) )
 
Theoremumgra1 23215 The graph with one edge. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( ( V  e.  W  /\  A  e.  X )  /\  ( B  e.  V  /\  C  e.  V ) )  ->  V UMGrph  { <. A ,  { B ,  C } >. } )
 
Theoremumgraun 23216 If  <. V ,  E >. and  <. V ,  F >. are graphs, then  <. V ,  E  u.  F >. is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   =>    |-  ( ph  ->  V UMGrph  ( E  u.  F ) )
 
Theoremreleupa 23217 The set  ( V EulPaths  E ) of all Eulerian paths on  <. V ,  E >. is a set of pairs by our definition of an Eulerian path, and so is a relation. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  Rel  ( V EulPaths  E )
 
Theoremiseupa 23218* The property " <. F ,  P >. is an Eulerian path on the graph  <. V ,  E >.". An Eulerian path is defined as bijection  F from the edges to a set  1 ... N a function  P :
( 0 ... N
) --> V into the vertices such that for each 
1  <_  k  <_  N,  F ( k ) is an edge from  P ( k  -  1 ) to  P
( k ). (Since the edges are undirected and there are possibly many edges between any two given vertices, we need to list both the edges and the vertices of the path separately.) (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( dom  E  =  A  ->  ( F ( V EulPaths  E ) P  <->  ( V UMGrph  E  /\  E. n  e.  NN0  ( F : ( 1
 ... n ) -1-1-onto-> A  /\  P : ( 0 ... n ) --> V  /\  A. k  e.  ( 1
 ... n ) ( E `  ( F `
  k ) )  =  { ( P `
  ( k  -  1 ) ) ,  ( P `  k
 ) } ) ) ) )
 
Theoremeupagra 23219 If an eulerian path exists, then 
<. V ,  E >. is a graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  V UMGrph  E )
 
Theoremeupai 23220* Properties of an Eulerian path. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  ( ( ( # `  F )  e.  NN0  /\  F : ( 1
 ... ( # `  F ) ) -1-1-onto-> A  /\  P :
 ( 0 ... ( # `
  F ) ) --> V )  /\  A. k  e.  ( 1 ... ( # `  F ) ) ( E `
  ( F `  k ) )  =  { ( P `  ( k  -  1
 ) ) ,  ( P `  k ) }
 ) )
 
Theoremeupacl 23221 An Eulerian path has length 
# ( F ), which is an integer. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  ( # `  F )  e.  NN0 )
 
Theoremeupaf1o 23222 The  F function in an Eulerian path is a bijection from a one-based sequence to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  F : ( 1 ... ( # `  F ) ) -1-1-onto-> A )
 
Theoremeupafi 23223 Any graph with an Eulerian path is finite. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  E  Fn  A )  ->  A  e.  Fin )
 
Theoremeupapf 23224 The  P function in an Eulerian path is a function from a zero-based finite sequence to the vertices. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( F ( V EulPaths  E ) P  ->  P :
 ( 0 ... ( # `
  F ) ) --> V )
 
Theoremeupaseg 23225 The  N-th edge in an eulerian path is the edge from  P ( N  - 
1 ) to  P ( N ). (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( F ( V EulPaths  E ) P  /\  N  e.  ( 1 ... ( # `  F ) ) )  ->  ( E `  ( F `
  N ) )  =  { ( P `
  ( N  -  1 ) ) ,  ( P `  N ) } )
 
Theoremvdgrfval 23226* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  Fn  A  /\  A  e.  X )  ->  ( V VDeg  E )  =  ( u  e.  V  |->  ( ( # ` 
 { x  e.  A  |  u  e.  ( E `  x ) }
 )  +  ( # ` 
 { x  e.  A  |  ( E `  x )  =  { u } } ) ) ) )
 
Theoremvdgrval 23227* The value of the vertex degree function. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( ( V  e.  W  /\  E  Fn  A  /\  A  e.  X ) 
 /\  U  e.  V )  ->  ( ( V VDeg 
 E ) `  U )  =  ( ( # `
  { x  e.  A  |  U  e.  ( E `  x ) } )  +  ( # `
  { x  e.  A  |  ( E `
  x )  =  { U } }
 ) ) )
 
Theoremvdgrf 23228 The vertex degree function on finite graphs is a function from vertices to nonnegative integers. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  E  Fn  A  /\  A  e.  Fin )  ->  ( V VDeg  E ) : V --> NN0 )
 
Theoremvdgr0 23229 The degree of a vertex in an empty graph is zero, because there are no edges. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  (
 ( V  e.  W  /\  U  e.  V ) 
 ->  ( ( V VDeg  (/) ) `  U )  =  0
 )
 
Theoremvdgrun 23230 The degree of a vertex in the union of two graphs on the same vertex set is the sum of the degrees of the vertex in each graph. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F  Fn  B )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( A  i^i  B )  =  (/) )   &    |-  ( ph  ->  V UMGrph  E )   &    |-  ( ph  ->  V UMGrph  F )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  ( E  u.  F ) ) `
  U )  =  ( ( ( V VDeg 
 E ) `  U )  +  ( ( V VDeg  F ) `  U ) ) )
 
Theoremvdgr1d 23231 The vertex degree of a one-edge graph, case 4: an edge from a vertex to itself contributes two to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   =>    |-  ( ph  ->  ( ( V VDeg  { <. A ,  { U } >. } ) `  U )  =  2 )
 
Theoremvdgr1b 23232 The vertex degree of a one-edge graph, case 2: an edge from the given vertex to some other vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { U ,  B } >. } ) `  U )  =  1 )
 
Theoremvdgr1c 23233 The vertex degree of a one-edge graph, case 3: an edge from some other vertex to the given vertex contributes one to the vertex's degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { B ,  U } >. } ) `  U )  =  1 )
 
Theoremvdgr1a 23234 The vertex degree of a one-edge graph, case 1: an edge between two vertices other than the given vertex contributes nothing to the vertex degree. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  ( ph  ->  V  e.  _V )   &    |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  U )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  C  =/=  U )   =>    |-  ( ph  ->  (
 ( V VDeg  { <. A ,  { B ,  C } >. } ) `  U )  =  0 )
 
Theoremeupa0 23235 There is an Eulerian path on the empty graph. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( V  e.  W  /\  A  e.  V ) 
 ->  (/) ( V EulPaths  (/) ) { <. 0 ,  A >. } )
 
Theoremeupares 23236 The restriction of an Eulerian path to an initial segment of the path forms an Eulerian path on the subgraph consisting of the edges in the initial segment. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.)
 |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  ( 0 ... ( # `
  G ) ) )   &    |-  F  =  ( E  |`  ( G " ( 1 ... N ) ) )   &    |-  H  =  ( G  |`  ( 1
 ... N ) )   &    |-  Q  =  ( P  |`  ( 0 ... N ) )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupap1 23237 Append one path segment to an Eulerian path (enlarging the graph to add the new edge). (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  -.  B  e.  A )   &    |-  ( ph  ->  G ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  =  ( # `  G ) )   &    |-  F  =  ( E  u.  { <. B ,  { ( P `
  N ) ,  C } >. } )   &    |-  H  =  ( G  u.  { <. ( N  +  1 ) ,  B >. } )   &    |-  Q  =  ( P  u.  { <. ( N  +  1 ) ,  C >. } )   =>    |-  ( ph  ->  H ( V EulPaths  F ) Q )
 
Theoremeupath2lem1 23238 Lemma for eupath2 23241. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupath2lem2 23239 Lemma for eupath2 23241. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  B  e.  _V   =>    |-  ( ( B  =/=  C 
 /\  B  =  U )  ->  ( -.  U  e.  if ( A  =  B ,  (/) ,  { A ,  B }
 ) 
 <->  U  e.  if ( A  =  C ,  (/)
 ,  { A ,  C } ) ) )
 
Theoremeupath2lem3 23240* Lemma for eupath2 23241. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  ( N  +  1 ) 
 <_  ( # `  F ) )   &    |-  ( ph  ->  U  e.  V )   &    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  ( E  |`  ( F "
 ( 1 ... N ) ) ) ) `
  x ) }  =  if ( ( P `
  0 )  =  ( P `  N ) ,  (/) ,  {
 ( P `  0
 ) ,  ( P `
  N ) }
 ) )   =>    |-  ( ph  ->  ( -.  2  ||  ( ( V VDeg  ( E  |`  ( F
 " ( 1 ... ( N  +  1 ) ) ) ) ) `  U )  <->  U  e.  if (
 ( P `  0
 )  =  ( P `
  ( N  +  1 ) ) ,  (/) ,  { ( P `
  0 ) ,  ( P `  ( N  +  1 )
 ) } ) ) )
 
Theoremeupath2 23241* The only vertices of odd degree in a graph with an Eulerian path are the endpoints, and then only if the endpoints are distinct. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( ph  ->  E  Fn  A )   &    |-  ( ph  ->  F ( V EulPaths  E ) P )   =>    |-  ( ph  ->  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }  =  if ( ( P `
  0 )  =  ( P `  ( # `
  F ) ) ,  (/) ,  { ( P `  0 ) ,  ( P `  ( # `
  F ) ) } ) )
 
Theoremeupath 23242* A graph with an Eulerian path has either zero or two vertices of odd degree. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  (
 ( V EulPaths  E )  =/=  (/)  ->  ( # `  { x  e.  V  |  -.  2  ||  ( ( V VDeg  E ) `  x ) }
 )  e.  { 0 ,  2 } )
 
Theoremvdeg0i 23243 The base case for the induction for calculating the degree of a vertex. The degree of  U in the empty graph is  0. (Contributed by Mario Carneiro, 12-Mar-2015.)
 |-  V  e.  _V   &    |-  U  e.  V   =>    |-  (
 ( V VDeg  (/) ) `  U )  =  0
 
Theoremumgrabi 23244* Show that an unordered pair is a valid edge in a graph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  X  e.  V   &    |-  Y  e.  V   =>    |-  ( ph  ->  { X ,  Y }  e.  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )
 
Theoremvdegp1ai 23245* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  Y } to the edge set, where  X  =/=  U  =/= 
Y, yields degree  P as well. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  Y  e.  V   &    |-  Y  =/=  U   &    |-  F  =  ( E concat  <" { X ,  Y } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  P
 
Theoremvdegp1bi 23246* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { U ,  X } to the edge set, where 
X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { U ,  X } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremvdegp1ci 23247* The induction step for a vertex degree calculation. If the degree of  U in the edge set  E is  P, then adding  { X ,  U } to the edge set, where  X  =/=  U, yields degree  P  + 
1. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  e.  _V   &    |-  (  T.  ->  E  e. Word  { x  e.  ( ~P V  \  { (/) } )  |  ( # `  x )  <_  2 } )   &    |-  U  e.  V   &    |-  ( ( V VDeg 
 E ) `  U )  =  P   &    |-  Q  =  ( P  +  1 )   &    |-  X  e.  V   &    |-  X  =/=  U   &    |-  F  =  ( E concat  <" { X ,  U } "> )   =>    |-  ( ( V VDeg  F ) `  U )  =  Q
 
Theoremkonigsberg 23248 The Konigsberg Bridge problem. If  <. V ,  E >. is the graph on four vertices  0 ,  1 ,  2 ,  3, with edges  { 0 ,  1 } ,  { 0 ,  2 } ,  { 0 ,  3 } ,  {
1 ,  2 } ,  { 1 ,  2 } ,  {
2 ,  3 } ,  { 2 ,  3 }, then vertices  0 ,  1 ,  3 each have degree three, and  2 has degree five, so there are four vertices of odd degree and thus by eupath 23242 the graph cannot have an Eulerian path. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  V  =  ( 0 ... 3
 )   &    |-  E  =  <" {
 0 ,  1 }  { 0 ,  2 }  { 0 ,  3 }  { 1 ,  2 }  {
 1 ,  2 }  { 2 ,  3 }  { 2 ,  3 } ">   =>    |-  ( V EulPaths  E )  =  (/)
 
16.4.11  Normal numbers
 
Theoremsnmlff 23249* The function  F from snmlval 23251 is a mapping from positive integers to real numbers in the range 
[ 0 ,  1 ]. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  F : NN --> ( 0 [,] 1 )
 
Theoremsnmlfval 23250* The function  F from snmlval 23251 maps  N to the relative density of  B in the first  N digits of the digit string of  A in base  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  ( N  e.  NN  ->  ( F `  N )  =  ( ( # `
  { k  e.  ( 1 ... N )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  N )
 )
 
Theoremsnmlval 23251* The property " A is simply normal in base  R". A number is simply normal if each digit  0  <_  b  <  R occurs in the base-  R digit string of  A with frequency  1  /  R (which is consistent with the expectation in an infinite random string of numbers selected from  0 ... R  -  1). (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  S  =  ( r  e.  ( ZZ>=
 `  2 )  |->  { x  e.  RR  |  A. b  e.  (
 0 ... ( r  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( x  x.  ( r ^ k
 ) )  mod  r
 ) )  =  b } )  /  n ) )  ~~>  ( 1  /  r ) } )   =>    |-  ( A  e.  ( S `  R )  <->  ( R  e.  ( ZZ>= `  2 )  /\  A  e.  RR  /\  A. b  e.  ( 0
 ... ( R  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  b }
 )  /  n )
 )  ~~>  ( 1  /  R ) ) )
 
Theoremsnmlflim 23252* If  A is simply normal, then the function  F of relative density of  B in the digit string converges to  1  /  R, i.e. the set of occurences of  B in the digit string has natural density  1  /  R. (Contributed by Mario Carneiro, 6-Apr-2015.)
 |-  S  =  ( r  e.  ( ZZ>=
 `  2 )  |->  { x  e.  RR  |  A. b  e.  (
 0 ... ( r  -  1 ) ) ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( x  x.  ( r ^ k
 ) )  mod  r
 ) )  =  b } )  /  n ) )  ~~>  ( 1  /  r ) } )   &    |-  F  =  ( n  e.  NN  |->  ( ( # `  { k  e.  ( 1 ... n )  |  ( |_ `  ( ( A  x.  ( R ^ k ) )  mod  R ) )  =  B }
 )  /  n )
 )   =>    |-  ( ( A  e.  ( S `  R ) 
 /\  B  e.  (
 0 ... ( R  -  1 ) ) ) 
 ->  F  ~~>  ( 1  /  R ) )
 
16.4.12  Godel-sets of formulas
 
Syntaxcgoe 23253 The Godel-set of membership.
 class  e.g
 
Syntaxcgna 23254 The Godel-set for the Sheffer stroke.
 class  | g
 
Syntaxcgol 23255 The Godel-set of universal quantification. (Note that this is not a wff.)
 class  A.g N U
 
Syntaxcsat 23256 The satisfaction function.
 class  Sat
 
Syntaxcfmla 23257 The formula set predicate.
 class  Fmla
 
Syntaxcsate 23258 The  e.-satisfaction function.
 class  Sat E
 
Syntaxcprv 23259 The "proves" relation.
 class  |=
 
Definitiondf-goel 23260 Define the Godel-set of membership. Here the arguments  x  =  <. N ,  P >. correspond to vN and vP , so  ( (/)  e.g 
1o ) actually means v0  e. v1 , not  0  e.  1. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  e.g  =  ( x  e.  ( om  X.  om )  |->  <. (/)
 ,  x >. )
 
Definitiondf-gona 23261 Define the Godel-set for the Sheffer stroke NAND. Here the arguments  x  =  <. U ,  V >. are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  | g  =  ( x  e.  ( _V  X.  _V )  |->  <. 1o ,  x >. )
 
Definitiondf-goal 23262 Define the Godel-set of universal quantification. Here  N  e.  om corresponds to vN , and  U represents another formula, and this expression is  [ A. x ph ]  =  A.g N U where 
x is the  N-th variable,  U  =  [ ph ] is the code for  ph. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  A.g N U  =  <. 2o ,  <. N ,  U >. >.
 
Definitiondf-sat 23263* Define the satisfaction predicate. This recursive construction builds up a function over wff codes and simultaneously defines the set of assignments to all variables from  M that makes the coded wff true in the model  M, where  e. is interpreted as the binary relation  E on  M. The interpretation of the statement  S  e.  ( ( ( M  Sat  E ) `  n ) `  U ) is that for the model  <. M ,  E >.,  S : om --> M is an valuation of the variables (v0  =  ( S `  (/) ), v1  =  ( S `  1o ), etc.) and  U is a code for a wff using  =  ,  e.  ,  \/  ,  -.  ,  A. that is true under the assignment  S. The function is defined by finite recursion;  ( ( M  Sat  E ) `  n ) only operates on wffs of depth at most  n  e.  om, and  ( ( M  Sat  E ) `  om )  =  U_ n  e.  om ( ( M  Sat  E ) `  n ) operates on all wffs. The coding scheme for the wffs is defined so that
  • vi  e. vj is coded as  <. (/) ,  <. i ,  j >. >.,
  •  ( ph  -/\  ps ) is coded as  <. 1o ,  <. ph ,  ps >. >., and
  •  A. vi  ph is coded as  <. 2o ,  <. i ,  ph >. >..

(Contributed by Mario Carneiro, 14-Jul-2013.)

 |-  Sat  =  ( m  e.  _V ,  e  e.  _V  |->  ( rec ( ( f  e.  _V  |->  ( f  u.  { <. x ,  y >.  |  E. u  e.  f  ( E. v  e.  f  ( x  =  ( ( 1st `  u )  | g  ( 1st `  v
 ) )  /\  y  =  ( ( m  ^m  om )  \  ( ( 2nd `  u )  i^i  ( 2nd `  v
 ) ) ) )  \/  E. i  e. 
 om  ( x  = 
 A.g i ( 1st `  u )  /\  y  =  { a  e.  ( m  ^m  om )  | 
 A. z  e.  m  ( { <. i ,  z >. }  u.  ( a  |`  ( om  \  {
 i } ) ) )  e.  ( 2nd `  u ) } )
 ) } ) ) ,  { <. x ,  y >.  |  E. i  e.  om  E. j  e. 
 om  ( x  =  ( i  e.g  j
 )  /\  y  =  { a  e.  ( m  ^m  om )  |  ( a `  i
 ) e ( a `
  j ) }
 ) } )  |`  suc  om ) )
 
Definitiondf-sate 23264* A simplified version of the satisfaction predicate, using the standard membership relation and eliminating the extra variable  n. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  Sat E  =  ( m  e. 
 _V ,  u  e. 
 _V  |->  ( ( ( m  Sat  (  _E 
 i^i  ( m  X.  m ) ) ) `
  om ) `  u ) )
 
Definitiondf-fmla 23265 Define the predicate which defines the set of valid Godel formulas. The parameter  n defines the maximum height of the formulas: the set  ( Fmla `  (/) ) is all formulas of the form  x  =  y or  x  e.  y (which in our coding scheme is the set  ( { (/) ,  1o }  X.  ( om  X.  om ) ); see df-sat 23263 for the full coding scheme), and each extra level adds to the complexity of the formulas in  ( Fmla `  n
).  ( Fmla `  om )  =  U_ n  e. 
om ( Fmla `  n
) is the set of all valid formulas. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  Fmla  =  ( n  e.  suc  om 
 |->  dom  ( ( (/)  Sat  (/) ) `  n ) )
 
Syntaxcgon 23266 The Godel-set of negation. (Note that this is not a wff.)
 class  -.g U
 
Syntaxcgoa 23267 The Godel-set of conjunction.
 class  /\g
 
Syntaxcgoi 23268 The Godel-set of implication.
 class  ->g
 
Syntaxcgoo 23269 The Godel-set of disjunction.
 class  \/g
 
Syntaxcgob 23270 The Godel-set of equivalence.
 class  <->g
 
Syntaxcgoq 23271 The Godel-set of equality.
 class  =g
 
Syntaxcgox 23272 The Godel-set of existential quantification. (Note that this is not a wff.)
 class  E.g N U
 
Definitiondf-gonot 23273 Define the Godel-set of negation. Here the argument  U is also a Godel-set corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  -.g U  =  ( U  | g  U )
 
Definitiondf-goan 23274* Define the Godel-set of conjunction. Here the arguments  U and  V are also Godel-sets corresponding to smaller formulae. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  /\g  =  ( u  e.  _V ,  v  e.  _V  |->  -.g ( u  | g  v ) )
 
Definitiondf-goim 23275* Define the Godel-set of implication. Here the arguments  U and  V are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  ->g  =  ( u  e.  _V ,  v  e.  _V  |->  ( u  | g  -.g v ) )
 
Definitiondf-goor 23276* Define the Godel-set of disjunction. Here the arguments  U and  V are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  \/g  =  ( u  e.  _V ,  v  e.  _V  |->  ( -.g u  ->g  v ) )
 
Definitiondf-gobi 23277* Define the Godel-set of equivalence. Here the arguments  U and  V are also Godel-sets corresponding to smaller formulae. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  <->g  =  ( u  e.  _V ,  v  e.  _V  |->  ( ( u 
 ->g  v )  /\g  (
 v  ->g  u ) ) )
 
Definitiondf-goeq 23278* Define the Godel-set of equality. Here the arguments  x  =  <. N ,  P >. correspond to vN and vP , so  ( (/)  =g  1o ) actually means v0  = v1 , not  0  = 
1. Here we use the trick mentioned in ax-ext 2237 to introduce equality as a defined notion in terms of  e.g. The expression  suc  ( u  u.  v )  = max  ( u ,  v )  +  1 here is a convenient way of getting a dummy variable distinct from  u and  v. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  =g  =  ( u  e.  om ,  v  e.  om  |->  [_ suc  ( u  u.  v
 )  /  w ]_ A.g w ( ( w  e.g  u )  <->g  ( w  e.g  v
 ) ) )
 
Definitiondf-goex 23279 Define the Godel-set of existential quantification. Here  N  e.  om corresponds to vN , and  U represents another formula, and this expression is  [ E. x ph ]  =  E.g N U where 
x is the  N-th variable,  U  =  [ ph ] is the code for  ph. Note that this is a class expression, not a wff. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  E.g N U  =  -.g A.g N -.g U
 
Definitiondf-prv 23280* Define the "proves" relation on a set. A wff is true in a model  M if for every valuation  s  e.  ( M  ^m  om ), the interpretation of the wff using the membership relation on  M is true. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  |=  =  { <. m ,  u >.  |  ( m  Sat E  u )  =  ( m  ^m  om ) }
 
16.4.13  Models of ZF
 
Syntaxcgze 23281 The Axiom of Extensionality.
 class  AxExt
 
Syntaxcgzr 23282 The Axiom Scheme of Replacement.
 class  AxRep
 
Syntaxcgzp 23283 The Axiom of Power Sets.
 class  AxPow
 
Syntaxcgzu 23284 The Axiom of Unions.
 class  AxUn
 
Syntaxcgzg 23285 The Axiom of Regularity.
 class  AxReg
 
Syntaxcgzi 23286 The Axiom of Infinity.
 class  AxInf
 
Syntaxcgzf 23287 The set of models of ZF.
 class  ZF
 
Definitiondf-gzext 23288 The Godel-set version of the Axiom of Extensionality. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxExt  =  (
 A.g 2o ( ( 2o 
 e.g  (/) )  <->g  ( 2o  e.g 
 1o ) )  ->g  ( (/)  =g  1o ) )
 
Definitiondf-gzrep 23289 The Godel-set version of the Axiom Scheme of Replacement. Since this is a scheme and not a single axiom, it manifests as a function on wffs, each giving rise to a different axiom. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxRep  =  ( u  e.  ( Fmla ` 
 om )  |->  ( A.g 3o E.g 1o A.g 2o ( A.g 1o u  ->g  ( 2o  =g  1o ) )  ->g  A.g 1o A.g 2o ( ( 2o  e.g  1o )  <->g  E.g 3o ( ( 3o  e.g  (/) )  /\g  A.g
 1o u ) ) ) )
 
Definitiondf-gzpow 23290 The Godel-set version of the Axiom of Power Sets. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxPow  =  E.g 1o A.g 2o ( A.g 1o (
 ( 1o  e.g  2o ) 
 <->g  ( 1o  e.g  (/) ) ) 
 ->g  ( 2o  e.g  1o ) )
 
Definitiondf-gzun 23291 The Godel-set version of the Axiom of Unions. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxUn  = 
 E.g 1o A.g 2o ( E.g 1o ( ( 2o  e.g  1o )  /\g  ( 1o  e.g  (/) ) )  ->g  ( 2o 
 e.g  1o ) )
 
Definitiondf-gzreg 23292 The Godel-set version of the Axiom of Regularity. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxReg  =  (
 E.g 1o ( 1o  e.g  (/) )  ->g  E.g 1o ( ( 1o  e.g  (/) )  /\g  A.g
 2o ( ( 2o 
 e.g  1o )  ->g  -.g ( 2o  e.g  (/) ) ) ) )
 
Definitiondf-gzinf 23293 The Godel-set version of the Axiom of Infinity. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  AxInf  =  E.g 1o ( ( (/)  e.g  1o )  /\g  A.g 2o ( ( 2o  e.g  1o )  ->g 
 E.g (/) ( ( 2o 
 e.g  (/) )  /\g  ( (/) 
 e.g  1o ) ) ) )
 
Definitiondf-gzf 23294* Define the class of all (transitive) models of ZF. (Contributed by Mario Carneiro, 14-Jul-2013.)
 |-  ZF  =  { m  |  ( ( Tr  m  /\  m  |=  AxExt  /\  m  |=  AxPow
 )  /\  ( m  |= 
 AxUn  /\  m  |=  AxReg  /\  m  |= 
 AxInf )  /\  A. u  e.  ( Fmla `  om ) m 
 |=  ( AxRep `  u ) ) }
 
16.4.14  Splitting fields
 
Syntaxcitr 23295 Integral subring of a ring.
 class IntgRing
 
Syntaxccpms 23296 Completion of a metric space.
 class cplMetSp
 
Syntaxchlb 23297 Embeddings for a direct limit.
 class HomLimB
 
Syntaxchlim 23298 Direct limit structure.
 class HomLim
 
Syntaxcpfl 23299 Polynomial extension field.
 class polyFld
 
Syntaxcsf1 23300 Splitting field for a single polynomial (auxiliary).
 class splitFld1
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