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Theorem List for Metamath Proof Explorer - 23901-24000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremeupares 23901 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 23902 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 23903 Lemma for eupath2 23906. (Contributed by Mario Carneiro, 8-Apr-2015.)
 |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
 ,  { A ,  B } )  <->  ( A  =/=  B 
 /\  ( U  =  A  \/  U  =  B ) ) ) )
 
Theoremeupath2lem2 23904 Lemma for eupath2 23906. (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 23905* Lemma for eupath2 23906. (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 23906* 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 23907* 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 23908 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 23909* 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 23910* 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 23911* 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 23912* 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 23913 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 23907 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 )  =  (/)
 
18.4.11  Normal numbers
 
Theoremsnmlff 23914* The function  F from snmlval 23916 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 23915* The function  F from snmlval 23916 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 23916* 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 23917* 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 ) )
 
18.4.12  Godel-sets of formulas
 
Syntaxcgoe 23918 The Godel-set of membership.
 class  e.g
 
Syntaxcgna 23919 The Godel-set for the Sheffer stroke.
 class  | g
 
Syntaxcgol 23920 The Godel-set of universal quantification. (Note that this is not a wff.)
 class  A.g N U
 
Syntaxcsat 23921 The satisfaction function.
 class  Sat
 
Syntaxcfmla 23922 The formula set predicate.
 class  Fmla
 
Syntaxcsate 23923 The  e.-satisfaction function.
 class  Sat E
 
Syntaxcprv 23924 The "proves" relation.
 class  |=
 
Definitiondf-goel 23925 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 23926 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 23927 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 23928* 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 a 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 23929* 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 23930 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 23928 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 23931 The Godel-set of negation. (Note that this is not a wff.)
 class  -.g U
 
Syntaxcgoa 23932 The Godel-set of conjunction.
 class  /\g
 
Syntaxcgoi 23933 The Godel-set of implication.
 class  ->g
 
Syntaxcgoo 23934 The Godel-set of disjunction.
 class  \/g
 
Syntaxcgob 23935 The Godel-set of equivalence.
 class  <->g
 
Syntaxcgoq 23936 The Godel-set of equality.
 class  =g
 
Syntaxcgox 23937 The Godel-set of existential quantification. (Note that this is not a wff.)
 class  E.g N U
 
Definitiondf-gonot 23938 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 23939* 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 23940* 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 23941* 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 23942* 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 23943* 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 2266 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 23944 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 23945* 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 ) }
 
18.4.13  Models of ZF
 
Syntaxcgze 23946 The Axiom of Extensionality.
 class  AxExt
 
Syntaxcgzr 23947 The Axiom Scheme of Replacement.
 class  AxRep
 
Syntaxcgzp 23948 The Axiom of Power Sets.
 class  AxPow
 
Syntaxcgzu 23949 The Axiom of Unions.
 class  AxUn
 
Syntaxcgzg 23950 The Axiom of Regularity.
 class  AxReg
 
Syntaxcgzi 23951 The Axiom of Infinity.
 class  AxInf
 
Syntaxcgzf 23952 The set of models of ZF.
 class  ZF
 
Definitiondf-gzext 23953 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 23954 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 23955 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 23956 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 23957 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 23958 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 23959* 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 ) ) }
 
18.4.14  Splitting fields
 
Syntaxcitr 23960 Integral subring of a ring.
 class IntgRing
 
Syntaxccpms 23961 Completion of a metric space.
 class cplMetSp
 
Syntaxchlb 23962 Embeddings for a direct limit.
 class HomLimB
 
Syntaxchlim 23963 Direct limit structure.
 class HomLim
 
Syntaxcpfl 23964 Polynomial extension field.
 class polyFld
 
Syntaxcsf1 23965 Splitting field for a single polynomial (auxiliary).
 class splitFld1
 
Syntaxcsf 23966 Splitting field for a finite set of polynomials.
 class splitFld
 
Syntaxcpsl 23967 Splitting field for a sequence of polynomials.
 class polySplitLim
 
Definitiondf-irng 23968* Define the subring of elements of  r integral over  s in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- IntgRing  =  ( r  e.  _V ,  s  e.  _V  |->  U_ f  e.  (Monic1p `
  ( rs  s ) ) ( `' f " { ( 0g `  r ) } )
 )
 
Definitiondf-cplmet 23969* A function which completes the given metric space. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- cplMetSp  =  ( w  e.  _V  |->  [_ ( ( w  ^s  NN )s  ( Cau `  ( dist `  w ) ) ) 
 /  r ]_ [_ ( Base `  r )  /  v ]_ [_ { <. f ,  g >.  |  ( { f ,  g }  C_  v  /\  A. x  e.  RR+  E. j  e.  ZZ  ( f  |`  ( ZZ>= `  j )
 ) : ( ZZ>= `  j ) --> ( ( g `  j ) ( ball `  ( dist `  w ) ) x ) ) }  /  e ]_ ( ( r 
 /.s 
 e ) sSet  { <. (
 dist `  ndx ) ,  { <. <. x ,  y >. ,  z >.  |  E. p  e.  v  E. q  e.  v  (
 ( x  =  [ p ] e  /\  y  =  [ q ] e
 )  /\  ( p  o F ( dist `  r
 ) q )  ~~>  z ) } >. } ) )
 
Definitiondf-homlimb 23970* The input to this function is a sequence (on  NN) of homomorphisms  F ( n ) : R ( n ) --> R ( n  +  1 ). The resulting structure is the direct limit of the direct system so defined. This function returns the pair  <. S ,  G >. where 
S is the terminal object and  G is a sequence of functions such that  G ( n ) : R ( n ) --> S and  G ( n )  =  F ( n )  o.  G
( n  +  1 ). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- HomLimB  =  ( f  e.  _V  |->  [_ U_ n  e.  NN  ( { n }  X.  dom  ( f `  n ) )  /  v ]_ [_ |^| { s  |  ( s  Er  v  /\  ( x  e.  v  |-> 
 <. ( ( 1st `  x )  +  1 ) ,  ( ( f `  ( 1st `  x )
 ) `  ( 2nd `  x ) ) >. ) 
 C_  s ) }  /  e ]_ <. ( v
 /. e ) ,  ( n  e.  NN  |->  ( x  e.  dom  ( f `  n )  |->  [ <. n ,  x >. ] e ) )
 >. )
 
Definitiondf-homlim 23971* The input to this function is a sequence (on  NN) of structures  R ( n ) and homomorphisms  F ( n ) : R ( n ) --> R ( n  +  1 ). The resulting structure is the direct limit of the direct system so defined, and maintains any structures that were present in the original objects. TODO: generalize to directed sets? (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- HomLim  =  ( r  e.  _V ,  f  e.  _V  |->  [_ ( HomLimB  `  f )  /  e ]_ [_ ( 1st `  e
 )  /  v ]_ [_ ( 2nd `  e
 )  /  g ]_ ( { <. ( Base `  ndx ) ,  v >. , 
 <. ( +g  `  ndx ) ,  U_ n  e. 
 NN  ran  ( x  e.  dom  ( g `  n ) ,  y  e.  dom  ( g `  n )  |->  <. <. ( ( g `  n ) `
  x ) ,  ( ( g `  n ) `  y
 ) >. ,  ( ( g `  n ) `
  ( x (
 +g  `  ( r `  n ) ) y ) ) >. ) >. , 
 <. ( .r `  ndx ) ,  U_ n  e. 
 NN  ran  ( x  e.  dom  ( g `  n ) ,  y  e.  dom  ( g `  n )  |->  <. <. ( ( g `  n ) `
  x ) ,  ( ( g `  n ) `  y
 ) >. ,  ( ( g `  n ) `
  ( x ( .r `  ( r `
  n ) ) y ) ) >. )
 >. }  u.  { <. (
 TopOpen `  ndx ) ,  { s  e.  ~P v  |  A. n  e. 
 NN  ( `' (
 g `  n ) " s )  e.  ( TopOpen `  ( r `  n ) ) } >. , 
 <. ( dist `  ndx ) , 
 U_ n  e.  NN  ran  ( x  e.  dom  ( ( g `  n ) `  n ) ,  y  e.  dom  ( ( g `  n ) `  n )  |->  <. <. ( ( g `
  n ) `  x ) ,  (
 ( g `  n ) `  y ) >. ,  ( x ( dist `  ( r `  n ) ) y )
 >. ) >. ,  <. ( le ` 
 ndx ) ,  U_ n  e.  NN  ( `' ( g `  n )  o.  ( ( le `  ( r `  n ) )  o.  (
 g `  n )
 ) ) >. } )
 )
 
Definitiondf-plfl 23972* Define the field extension that augments a field with the root of the given irreducible polynomial, and extends the norm if one exists and the extension is unique. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- polyFld  =  ( r  e.  _V ,  p  e.  _V  |->  [_ (Poly1 `  r )  /  s ]_ [_ ( (RSpan `  s ) `  { p } )  /  i ]_ [_ ( z  e.  ( Base `  r )  |->  [ ( z ( .s `  s ) ( 1r `  s
 ) ) ] (
 s ~QG 
 i ) )  /  f ]_ <. [_ ( s  /.s  (
 s ~QG 
 i ) )  /  t ]_ ( ( t toNrmGrp  ( iota_ n  e.  (AbsVal `  t ) ( n  o.  f )  =  ( norm `  r )
 ) ) sSet  <. ( le ` 
 ndx ) ,  [_ ( z  e.  ( Base `  t )  |->  (
 iota_ q  e.  z
 ( r deg1  q )  < 
 ( r deg1  p ) ) )  /  g ]_ ( `' g  o.  (
 ( le `  s
 )  o.  g ) ) >. ) ,  f >. )
 
Definitiondf-sfl1 23973* Temporary construction for the splitting field of a polynomial. The inputs are a field  r and a polynomial  p that we want to split, along with a tuple  j in the same format as the output. The output is a tuple  <. S ,  F >. where 
S is the splitting field and  F is an injective homomorphism from the original field  r.

The function works by repeatedly finding the smallest monic irreducible factor, and extending the field by that factor using the polyFld construction. We keep track of a total order in each of the splitting fields so that we can pick an element definably without needing global choice. (Contributed by Mario Carneiro, 2-Dec-2014.)

 |- splitFld1  =  (
 r  e.  _V ,  j  e.  _V  |->  ( p  e.  (Poly1 `  r )  |->  ( rec ( ( s  e.  _V ,  f  e.  _V  |->  [_ ( mPoly  `  s
 )  /  m ]_ [_ { g  e.  ( (Monic1p `
  s )  i^i  (Irred `  m )
 )  |  ( g ( ||r
 `  m ) ( p  o.  f ) 
 /\  1  <  (
 s deg1  g ) ) }  /  b ]_ if (
 ( ( p  o.  f )  =  ( 0g `  m )  \/  b  =  (/) ) , 
 <. s ,  f >. , 
 [_ ( glb `  b
 )  /  h ]_ [_ (
 s polyFld  h )  /  t ]_ <. ( 1st `  t
 ) ,  ( f  o.  ( 2nd `  t
 ) ) >. ) ) ,  j ) `  ( card `  ( 1 ... ( r deg1  p ) ) ) ) ) )
 
Definitiondf-sfl 23974* Define the splitting field of a finite collection of polynomials, given a total ordered base field. The output is a tuple  <. S ,  F >. where  S is the totally ordered splitting field and  F is an injective homomorphism from the original field  r. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- splitFld  =  ( r  e.  _V ,  p  e.  _V  |->  ( iota
 x E. f ( f  Isom  <  ,  ( lt `  r ) ( ( 1 ... ( # `
  p ) ) ,  p )  /\  x  =  (  seq  0 ( ( e  e.  _V ,  g  e.  _V  |->  ( ( r splitFld1  e ) `  g ) ) ,  ( f  u.  { <. 0 ,  <. r ,  (  _I  |`  ( Base `  r
 ) ) >. >. } )
 ) `  ( # `  p ) ) ) ) )
 
Definitiondf-psl 23975* Define the direct limit of an increasing sequence of fields produced by pasting together the splitting fields for each sequence of polynomials. That is, given a ring  r, a strict order on  r, and a sequence  p : NN --> ( ~P r  i^i  Fin ) of finite sets of polynomials to split, we construct the direct limit system of field extensions by splitting one set at a time and passing the resulting construction to HomLim. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- polySplitLim  =  ( r  e.  _V ,  p  e.  ( ( ~P ( Base `  r )  i^i  Fin )  ^m  NN )  |->  [_ ( 1st  o.  seq  0 ( ( g  e.  _V ,  q  e.  _V  |->  [_ ( 1st `  g
 )  /  e ]_ [_ ( 1st `  e
 )  /  s ]_ [_ ( s splitFld  ran  ( x  e.  q  |->  ( x  o.  ( 2nd `  g
 ) ) ) ) 
 /  f ]_ <. f ,  ( ( 2nd `  g
 )  o.  ( 2nd `  f ) ) >. ) ,  ( p  u.  {
 <. 0 ,  <. <. r ,  (/) >. ,  (  _I  |`  ( Base `  r )
 ) >. >. } ) ) )  /  f ]_ ( ( 1st  o.  ( f  shift  1 ) ) HomLim  ( 2nd  o.  f ) ) )
 
18.4.15  p-adic number fields
 
Syntaxczr 23976 Integral elements of a ring.
 class ZRing
 
Syntaxcgf 23977 Galois finite field.
 class GF
 
Syntaxcgfo 23978 Galois limit field.
 class GF
 
Syntaxceqp 23979 Equivalence relation for df-qp 23990.
 class ~Qp
 
Syntaxcrqp 23980 Equivalence relation representatives for df-qp 23990.
 class /Qp
 
Syntaxcqp 23981 The set of  p-adic rational numbers.
 class Qp
 
Syntaxczp 23982 The set of  p-adic integers. (Not to be confused with czn 16456.)
 class Zp
 
Syntaxcqpa 23983 Algebraic completion of the  p-adic rational numbers.
 class _Qp
 
Syntaxccp 23984 Metric completion of _Qp.
 class Cp
 
Definitiondf-zrng 23985 Define the subring of integral elements in a ring. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- ZRing  =  ( r  e.  _V  |->  ( r IntgRing  ran  ( ZRHom `  r
 ) ) )
 
Definitiondf-gf 23986* Define the Galois finite field of order  p ^ n. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- GF  =  ( p  e.  Prime ,  n  e.  NN  |->  [_ (ℤ/n `  p )  /  r ]_ ( 1st `  (
 r splitFld  { [_ (Poly1 `  r
 )  /  s ]_ [_ (var1 `  r )  /  x ]_ ( ( ( p ^ n ) (.g `  (mulGrp `  s
 ) ) x ) ( -g `  s
 ) x ) }
 ) ) )
 
Definitiondf-gfoo 23987* Define the Galois field of order  p ^  +oo, as a direct limit of the Galois finite fields. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- GF  =  ( p  e.  Prime  |->  [_ (ℤ/n `  p )  /  r ]_ (
 r polySplitLim  ( n  e.  NN  |->  {
 [_ (Poly1 `  r )  /  s ]_ [_ (var1 `  r
 )  /  x ]_ (
 ( ( p ^ n ) (.g `  (mulGrp `  s ) ) x ) ( -g `  s
 ) x ) }
 ) ) )
 
Definitiondf-eqp 23988* Define an equivalence relation on 
ZZ-indexed sequences of integers such that two sequences are equivalent iff the difference is equivalent to zero, and a sequence is equivalent to zero iff the sum  sum_ k  <_  n f ( k ) ( p ^
k ) is a multiple of  p ^ (
n  +  1 ) for every  n. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- ~Qp  =  ( p  e.  Prime  |->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( ZZ  ^m  ZZ )  /\  A. n  e.  ZZ  sum_ k  e.  ( ZZ>=
 `  -u n ) ( ( ( f `  -u k )  -  (
 g `  -u k ) )  /  ( p ^ ( k  +  ( n  +  1
 ) ) ) )  e.  ZZ ) }
 )
 
Definitiondf-rqp 23989* There is a unique element of  ( ZZ  ^m  (
0 ... ( p  - 
1 ) ) ) ~Qp -equivalent to any element of 
( ZZ  ^m  ZZ ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- /Qp  =  ( p  e.  Prime  |->  (~Qp  i^i  [_
 { f  e.  ( ZZ  ^m  ZZ )  | 
 E. x  e.  ran  ZZ>= ( `' f " ( ZZ  \  { 0 } )
 )  C_  x }  /  y ]_ ( y  X.  ( y  i^i  ( ZZ  ^m  (
 0 ... ( p  -  1 ) ) ) ) ) ) )
 
Definitiondf-qp 23990* Define the  p-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Qp  =  ( p  e.  Prime  |->  [_
 { h  e.  ( ZZ  ^m  ( 0 ... ( p  -  1
 ) ) )  | 
 E. x  e.  ran  ZZ>= ( `' h " ( ZZ  \  { 0 } )
 )  C_  x }  /  b ]_ ( ( { <. ( Base `  ndx ) ,  b >. , 
 <. ( +g  `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  (
 f  o F  +  g ) ) )
 >. ,  <. ( .r `  ndx ) ,  ( f  e.  b ,  g  e.  b  |->  ( (/Qp `  p ) `  ( n  e.  ZZ  |->  sum_ k  e.  ZZ  ( ( f `
  k )  x.  ( g `  ( n  -  k ) ) ) ) ) )
 >. }  u.  { <. ( le `  ndx ) ,  { <. f ,  g >.  |  ( { f ,  g }  C_  b  /\  sum_ k  e.  ZZ  ( ( f `  -u k )  x.  (
 ( p  +  1 ) ^ -u k
 ) )  <  sum_ k  e.  ZZ  ( ( g `
  -u k )  x.  ( ( p  +  1 ) ^ -u k
 ) ) ) } >. } ) toNrmGrp  ( f  e.  b  |->  if (
 f  =  ( ZZ 
 X.  { 0 } ) ,  0 ,  ( p ^ -u sup ( ( `' f " ( ZZ  \  { 0 } )
 ) ,  RR ,  `'  <  ) ) ) ) ) )
 
Definitiondf-zp 23991 Define the  p-adic integers, as a subset of the  p-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Zp  =  (ZRing  o. Qp )
 
Definitiondf-qpa 23992* Define the completion of the  p-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the  n-th set the collection of polynomials with degree less than  n and with coefficients  <  ( p ^
n )). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial  x ^ (
p ^ n )  -  x, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- _Qp  =  ( p  e.  Prime  |->  [_ (Qp `  p )  /  r ]_ ( r polySplitLim  ( n  e.  NN  |->  { f  e.  (Poly1 `  r )  |  (
 ( r deg1  f )  <_  n  /\  A. d  e. 
 ran  (coe1 `  f ) ( `' d " ( ZZ  \  { 0 } )
 )  C_  ( 0 ... n ) ) }
 ) ) )
 
Definitiondf-cp 23993 Define the metric completion of the algebraic completion of the  p -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- Cp  =  ( cplMetSp  o. _Qp )
 
18.5  Mathbox for Paul Chapman
 
18.5.1  Group homomorphism and isomorphism
 
Theoremghomgrpilem1 23994 Lemma for ghomgrpi 23996. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( F `
  A ) H ( F `  B ) )  =  ( F `  ( A G B ) ) )
 
Theoremghomgrpilem2 23995 Lemma for ghomgrpi 23996. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  N  =  ( inv `  G )   &    |-  W  =  ran  H   &    |-  T  =  (GId `  H )   &    |-  M  =  ( inv `  H )   &    |-  Z  =  ran  F   &    |-  S  =  ( H  |`  ( Z  X.  Z ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomgrpi 23996 The image of a group homomorphism from  G to  H is a subgroup of  H (inference version). (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  G  e.  GrpOp   &    |-  H  e.  GrpOp   &    |-  F  e.  ( G GrpOpHom  H )   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  S  e.  ( SubGrpOp `  H )
 
Theoremghomsn 23997 The endomorphism of the trivial group. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  A  e.  _V   &    |-  G  =  { <.
 <. A ,  A >. ,  A >. }   =>    |-  (  _I  |`  { A } )  e.  ( G GrpOpHom  G )
 
Theoremghomgrplem 23998 Lemma for ghomgrp 23999. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  ( ph  ->  ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 ) )   &    |-  S  =  { <.
 <. z ,  z >. ,  z >. }   &    |-  J  =  (  _I  |`  { z } )   =>    |-  ( ph  ->  ( H  |`  ( ran  F  X.  ran  F ) )  e.  ( SubGrpOp `  H ) )
 
Theoremghomgrp 23999 The image of a group homomorphism from  G to  H is a subgroup of  H. (Contributed by Paul Chapman, 25-Feb-2008.)
 |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  ->  S  e.  ( SubGrpOp `  H ) )
 
Theoremghomfo 24000 A group homomorphism maps onto its image. (Contributed by Paul Chapman, 3-Mar-2008.)
 |-  X  =  ran  G   &    |-  Y  =  ran  F   &    |-  S  =  ( H  |`  ( Y  X.  Y ) )   &    |-  Z  =  ran  S   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  F : X -onto-> Z )
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