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Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdprdval 15201* The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  (
 ( G dom DProd  S  /\  dom 
 S  =  I ) 
 ->  ( G DProd  S )  =  ran  (  f  e.  W  |->  ( G 
 gsumg  f ) ) )
 
Theoremeldprd 15202* The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   =>    |-  ( dom  S  =  I  ->  ( A  e.  ( G DProd  S )  <->  ( G dom DProd  S 
 /\  E. f  e.  W  A  =  ( G  gsumg  f
 ) ) ) )
 
Theoremdprdgrp 15203 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( G dom DProd  S  ->  G  e.  Grp )
 
Theoremdprdf 15204 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( G dom DProd  S  ->  S : dom  S --> (SubGrp `  G ) )
 
Theoremdprdf2 15205 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   =>    |-  ( ph  ->  S : I
 --> (SubGrp `  G )
 )
 
Theoremdprdcntz 15206 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  Y  e.  I )   &    |-  ( ph  ->  X  =/=  Y )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( S `  X )  C_  ( Z `  ( S `
  Y ) ) )
 
Theoremdprddisj 15207 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  .0.  =  ( 0g `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ph  ->  ( ( S `  X )  i^i  ( K `  U. ( S " ( I  \  { X }
 ) ) ) )  =  {  .0.  }
 )
 
Theoremdprdw 15208* The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   =>    |-  ( ph  ->  ( F  e.  W  <->  ( F  Fn  I  /\  A. x  e.  I  ( F `  x )  e.  ( S `  x )  /\  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )
 ) )
 
Theoremdprdwd 15209* The property of being a finitely supported function in the family  S. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ( ph  /\  x  e.  I )  ->  A  e.  ( S `  x ) )   &    |-  ( ph  ->  ( `' ( x  e.  I  |->  A ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( x  e.  I  |->  A )  e.  W )
 
Theoremdprdff 15210* A finitely supported function in  S is a function into the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  B  =  (
 Base `  G )   =>    |-  ( ph  ->  F : I --> B )
 
Theoremdprdfcl 15211* A finitely supported function in  S has its  X-th element in  S ( X ). (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F  e.  W )   =>    |-  ( ( ph  /\  X  e.  I )  ->  ( F `  X )  e.  ( S `  X ) )
 
Theoremdprdffi 15212* The function  S is a family of subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F  e.  W )   =>    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )
 
Theoremdprdfcntz 15213* A function on the elements of an internal direct product has pairwise-commuting values. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )
 
Theoremdprdssv 15214 The internal direct product of a family of subgroups is a subset of the base. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   =>    |-  ( G DProd  S ) 
 C_  B
 
Theoremdprdfid 15215* The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  A  e.  ( S `  X ) )   &    |-  F  =  ( n  e.  I  |->  if ( n  =  X ,  A ,  .0.  )
 )   =>    |-  ( ph  ->  ( F  e.  W  /\  ( G  gsumg 
 F )  =  A ) )
 
Theoremeldprdi 15216* The domain of definition of the internal direct product, which states that  S is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  e.  ( G DProd  S ) )
 
Theoremdprdfinv 15217* Take the inverse of a group sum over a family of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ph  ->  (
 ( N  o.  F )  e.  W  /\  ( G  gsumg  ( N  o.  F ) )  =  ( N `  ( G  gsumg  F ) ) ) )
 
Theoremdprdfadd 15218* Take the sum of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  H  e.  W )   &    |- 
 .+  =  ( +g  `  G )   =>    |-  ( ph  ->  (
 ( F  o F  .+  H )  e.  W  /\  ( G  gsumg  ( F  o F  .+  H ) )  =  ( ( G  gsumg  F ) 
 .+  ( G  gsumg  H ) ) ) )
 
Theoremdprdfsub 15219* Take the difference of group sums over two families of elements of disjoint subgroups. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  H  e.  W )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ph  ->  (
 ( F  o F  .-  H )  e.  W  /\  ( G  gsumg  ( F  o F  .-  H ) )  =  ( ( G  gsumg  F ) 
 .-  ( G  gsumg  H ) ) ) )
 
Theoremdprdfeq0 15220* The zero function is the only function that sums two zero in a direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   =>    |-  ( ph  ->  ( ( G  gsumg 
 F )  =  .0.  <->  F  =  ( x  e.  I  |->  .0.  ) )
 )
 
Theoremdprdf11 15221* Two group sums over a direct product that give the same value are equal as functions. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  H  e.  W )   =>    |-  ( ph  ->  (
 ( G  gsumg 
 F )  =  ( G  gsumg 
 H )  <->  F  =  H ) )
 
Theoremdprdsubg 15222 The internal direct product of a family of subgroups is a subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( G dom DProd  S  ->  ( G DProd  S )  e.  (SubGrp `  G )
 )
 
Theoremdprdub 15223 Each factor is a subset of the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( S `  X )  C_  ( G DProd  S ) )
 
Theoremdprdlub 15224* The direct product is smaller than any subgroup which contains the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ( ph  /\  k  e.  I )  ->  ( S `  k )  C_  T )   =>    |-  ( ph  ->  ( G DProd  S )  C_  T )
 
Theoremdprdspan 15225 The direct product is the span of the union of the factors. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( G dom DProd  S  ->  ( G DProd  S )  =  ( K `  U. ran  S ) )
 
Theoremdprdres 15226 Restriction of a direct product (dropping factors). (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  A  C_  I
 )   =>    |-  ( ph  ->  ( G dom DProd  ( S  |`  A ) 
 /\  ( G DProd  ( S  |`  A ) ) 
 C_  ( G DProd  S ) ) )
 
Theoremdprdss 15227* Create a direct product by finding subgroups inside each factor of another direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  T )   &    |-  ( ph  ->  dom 
 T  =  I )   &    |-  ( ph  ->  S : I
 --> (SubGrp `  G )
 )   &    |-  ( ( ph  /\  k  e.  I )  ->  ( S `  k )  C_  ( T `  k ) )   =>    |-  ( ph  ->  ( G dom DProd  S  /\  ( G DProd  S )  C_  ( G DProd  T ) ) )
 
Theoremdprdz 15228* A family consisting entirely of trivial groups is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  I  e.  V )  ->  ( G dom DProd  ( x  e.  I  |->  {  .0.  } )  /\  ( G DProd  ( x  e.  I  |->  {  .0.  } ) )  =  {  .0.  } ) )
 
Theoremdprd0 15229 The empty family is an internal direct product, the product of which is the trivial subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  ( G dom DProd  (/)  /\  ( G DProd 
 (/) )  =  {  .0.  } ) )
 
Theoremdprdf1o 15230 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F : J
 -1-1-onto-> I )   =>    |-  ( ph  ->  ( G dom DProd  ( S  o.  F )  /\  ( G DProd 
 ( S  o.  F ) )  =  ( G DProd  S ) ) )
 
Theoremdprdf1 15231 Rearrange the index set of a direct product family. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  F : J -1-1-> I )   =>    |-  ( ph  ->  ( G dom DProd  ( S  o.  F )  /\  ( G DProd  ( S  o.  F ) )  C_  ( G DProd  S ) ) )
 
Theoremsubgdmdprd 15232 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  H  =  ( Gs  A )   =>    |-  ( A  e.  (SubGrp `  G )  ->  ( H dom DProd  S  <->  ( G dom DProd  S 
 /\  ran  S  C_  ~P A ) ) )
 
Theoremsubgdprd 15233 A direct product in a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  H  =  ( Gs  A )   &    |-  ( ph  ->  A  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  ran 
 S  C_  ~P A )   =>    |-  ( ph  ->  ( H DProd  S )  =  ( G DProd  S ) )
 
Theoremdprdsn 15234 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ( A  e.  V  /\  S  e.  (SubGrp `  G ) )  ->  ( G dom DProd  { <. A ,  S >. }  /\  ( G DProd  { <. A ,  S >. } )  =  S ) )
 
Theoremdmdprdsplitlem 15235* Lemma for dmdprdsplit 15245. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } )
 )  e.  Fin }   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom  S  =  I )   &    |-  ( ph  ->  A 
 C_  I )   &    |-  ( ph  ->  F  e.  W )   &    |-  ( ph  ->  ( G  gsumg 
 F )  e.  ( G DProd  ( S  |`  A ) ) )   =>    |-  ( ( ph  /\  X  e.  ( I  \  A ) )  ->  ( F `
  X )  =  .0.  )
 
Theoremdprdcntz2 15236 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  C  C_  I
 )   &    |-  ( ph  ->  D  C_  I )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )
 
Theoremdprddisj2 15237 The function  S is a family of subgroups. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  C  C_  I
 )   &    |-  ( ph  ->  D  C_  I )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd 
 ( S  |`  D ) ) )  =  {  .0.  } )
 
Theoremdprd2dlem2 15238* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ( ph  /\  X  e.  A )  ->  ( S `  X )  C_  ( G DProd  ( j  e.  ( A " {
 ( 1st `  X ) } )  |->  ( ( 1st `  X ) S j ) ) ) )
 
Theoremdprd2dlem1 15239* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   &    |-  ( ph  ->  C  C_  I )   =>    |-  ( ph  ->  ( K `  U. ( S
 " ( A  |`  C ) ) )  =  ( G DProd  ( i  e.  C  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) ) )
 
Theoremdprd2da 15240* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremdprd2db 15241* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  Rel  A )   &    |-  ( ph  ->  S : A --> (SubGrp `  G )
 )   &    |-  ( ph  ->  dom  A  C_  I )   &    |-  ( ( ph  /\  i  e.  I ) 
 ->  G dom DProd  ( j  e.  ( A " {
 i } )  |->  ( i S j ) ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( G DProd  ( i  e.  I  |->  ( G DProd  (
 j  e.  ( A
 " { i }
 )  |->  ( i S j ) ) ) ) ) )
 
Theoremdprd2d2 15242* The direct product of a collection of direct products. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ( ph  /\  (
 i  e.  I  /\  j  e.  J )
 )  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ( ph  /\  i  e.  I )  ->  G dom DProd  ( j  e.  J  |->  S ) )   &    |-  ( ph  ->  G dom DProd  ( i  e.  I  |->  ( G DProd 
 ( j  e.  J  |->  S ) ) ) )   =>    |-  ( ph  ->  ( G dom DProd  ( i  e.  I ,  j  e.  J  |->  S )  /\  ( G DProd  ( i  e.  I ,  j  e.  J  |->  S ) )  =  ( G DProd  (
 i  e.  I  |->  ( G DProd  ( j  e.  J  |->  S ) ) ) ) ) )
 
Theoremdmdprdsplit2lem 15243 Lemma for dmdprdsplit 15245. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G dom DProd  ( S  |`  C ) )   &    |-  ( ph  ->  G dom DProd  ( S  |`  D ) )   &    |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )   &    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  }
 )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( ph  /\  X  e.  C )  ->  (
 ( Y  e.  I  ->  ( X  =/=  Y  ->  ( S `  X )  C_  ( Z `  ( S `  Y ) ) ) )  /\  ( ( S `  X )  i^i  ( K `
  U. ( S "
 ( I  \  { X } ) ) ) )  C_  {  .0.  } ) )
 
Theoremdmdprdsplit2 15244 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G dom DProd  ( S  |`  C ) )   &    |-  ( ph  ->  G dom DProd  ( S  |`  D ) )   &    |-  ( ph  ->  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) )   &    |-  ( ph  ->  ( ( G DProd  ( S  |`  C ) )  i^i  ( G DProd  ( S  |`  D ) ) )  =  {  .0.  }
 )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremdmdprdsplit 15245 The direct product splits into the direct product of any partition of the index set. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( G dom DProd  S  <->  ( ( G dom DProd  ( S  |`  C ) 
 /\  G dom DProd  ( S  |`  D ) )  /\  ( G DProd  ( S  |`  C ) )  C_  ( Z `  ( G DProd  ( S  |`  D ) ) ) 
 /\  ( ( G DProd 
 ( S  |`  C ) )  i^i  ( G DProd 
 ( S  |`  D ) ) )  =  {  .0.  } ) ) )
 
Theoremdprdsplit 15246 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( ph  ->  S : I --> (SubGrp `  G )
 )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  I  =  ( C  u.  D ) )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  ( ph  ->  G dom DProd  S )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( ( G DProd  ( S  |`  C ) )  .(+)  ( G DProd  ( S  |`  D )
 ) ) )
 
Theoremdmdprdpr 15247 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   =>    |-  ( ph  ->  ( G dom DProd  `' ( { S }  +c  { T } )  <->  ( S  C_  ( Z `  T )  /\  ( S  i^i  T )  =  {  .0.  } )
 ) )
 
Theoremdprdpr 15248 A singleton family is an internal direct product, the product of which is the given subgroup. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  S  C_  ( Z `  T ) )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  }
 )   =>    |-  ( ph  ->  ( G DProd  `' ( { S }  +c  { T } )
 )  =  ( S 
 .(+)  T ) )
 
Theoremdpjlem 15249 Lemma for theorems about direct product projection. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( G DProd  ( S  |`  { X } ) )  =  ( S `  X ) )
 
Theoremdpjcntz 15250 The two subgroups that appear in dpjval 15254 commute. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( S `  X )  C_  ( Z `  ( G DProd 
 ( S  |`  ( I 
 \  { X }
 ) ) ) ) )
 
Theoremdpjdisj 15251 The two subgroups that appear in dpjval 15254 are disjoint. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  ( ( S `  X )  i^i  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) )  =  {  .0.  } )
 
Theoremdpjlsm 15252 The two subgroups that appear in dpjval 15254 add to the full direct product. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  ( ph  ->  X  e.  I )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ph  ->  ( G DProd  S )  =  ( ( S `  X )  .(+)  ( G DProd  ( S  |`  ( I  \  { X } ) ) ) ) )
 
Theoremdpjfval 15253* Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  Q  =  (
 proj 1 `  G )   =>    |-  ( ph  ->  P  =  ( i  e.  I  |->  ( ( S `  i ) Q ( G DProd  ( S  |`  ( I 
 \  { i }
 ) ) ) ) ) )
 
Theoremdpjval 15254 Value of the direct product projection (defined in terms of binary projection). (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  Q  =  (
 proj 1 `  G )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  =  ( ( S `  X ) Q ( G DProd  ( S  |`  ( I 
 \  { X }
 ) ) ) ) )
 
Theoremdpjf 15255 The  X-th index projection is a function from the direct product to the  X-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X ) : ( G DProd  S )
 --> ( S `  X ) )
 
Theoremdpjidcl 15256* The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   =>    |-  ( ph  ->  (
 ( x  e.  I  |->  ( ( P `  x ) `  A ) )  e.  W  /\  A  =  ( G 
 gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A ) ) ) ) )
 
Theoremdpjeq 15257* Decompose a group sum into projections. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  W  =  { h  e.  X_ i  e.  I  ( S `  i )  |  ( `' h " ( _V  \  {  .0.  } ) )  e. 
 Fin }   &    |-  ( ph  ->  ( x  e.  I  |->  C )  e.  W )   =>    |-  ( ph  ->  ( A  =  ( G  gsumg  ( x  e.  I  |->  C ) )  <->  A. x  e.  I  ( ( P `  x ) `  A )  =  C )
 )
 
Theoremdpjid 15258* The key property of projections: the sum of all the projections of  A is  A. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  A  e.  ( G DProd  S ) )   =>    |-  ( ph  ->  A  =  ( G  gsumg  ( x  e.  I  |->  ( ( P `  x ) `  A ) ) ) )
 
Theoremdpjlid 15259 The  X-th index projection acts as the identity on elements of the  X-th factor. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  A  e.  ( S `  X ) )   =>    |-  ( ph  ->  ( ( P `  X ) `  A )  =  A )
 
Theoremdpjrid 15260 The  Y-th index projection annihilates elements of other factors. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   &    |-  ( ph  ->  A  e.  ( S `  X ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  Y  e.  I )   &    |-  ( ph  ->  Y  =/=  X )   =>    |-  ( ph  ->  (
 ( P `  Y ) `  A )  =  .0.  )
 
Theoremdpjghm 15261 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  e.  ( ( Gs  ( G DProd  S ) ) 
 GrpHom  G ) )
 
Theoremdpjghm2 15262 The direct product is the binary subgroup product ("sum") of the direct products of the partition. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  dom 
 S  =  I )   &    |-  P  =  ( GdProj S )   &    |-  ( ph  ->  X  e.  I )   =>    |-  ( ph  ->  ( P `  X )  e.  ( ( Gs  ( G DProd  S ) ) 
 GrpHom  ( Gs  ( S `  X ) ) ) )
 
10.3.5  The Fundamental Theorem of Abelian Groups
 
Theoremablfacrplem 15263* Lemma for ablfacrp2 15265. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   =>    |-  ( ph  ->  ( ( # `  K )  gcd  N )  =  1 )
 
Theoremablfacrp 15264* A finite abelian group whose order factors into relatively prime integers, itself "factors" into two subgroups  K ,  L that have trivial intersection and whose product is the whole group. Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ph  ->  (
 ( K  i^i  L )  =  {  .0.  } 
 /\  ( K  .(+)  L )  =  B ) )
 
Theoremablfacrp2 15265* The factors  K ,  L of ablfacrp 15264 have the expected orders (which allows for repeated application to decompose  G into subgroups of prime-power order). Lemma 6.1C.2 of [Shapiro], p. 199. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  K  =  { x  e.  B  |  ( O `
  x )  ||  M }   &    |-  L  =  { x  e.  B  |  ( O `  x ) 
 ||  N }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  M  e.  NN )   &    |-  ( ph  ->  N  e.  NN )   &    |-  ( ph  ->  ( M  gcd  N )  =  1 )   &    |-  ( ph  ->  ( # `  B )  =  ( M  x.  N ) )   =>    |-  ( ph  ->  ( ( # `  K )  =  M  /\  ( # `  L )  =  N ) )
 
Theoremablfac1lem 15266* Lemma for ablfac1b 15268. Satisfy the assumptions of ablfacrp. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  M  =  ( P ^ ( P 
 pCnt  ( # `  B ) ) )   &    |-  N  =  ( ( # `  B )  /  M )   =>    |-  ( ( ph  /\  P  e.  A ) 
 ->  ( ( M  e.  NN  /\  N  e.  NN )  /\  ( M  gcd  N )  =  1  /\  ( # `  B )  =  ( M  x.  N ) ) )
 
Theoremablfac1a 15267* The factors of ablfac1b 15268 are of prime power order. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   =>    |-  ( ( ph  /\  P  e.  A )  ->  ( # `
  ( S `  P ) )  =  ( P ^ ( P  pCnt  ( # `  B ) ) ) )
 
Theoremablfac1b 15268* Any abelian group is the direct product of factors of prime power order (with the exact order further matching the prime factorization of the group order). (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremablfac1c 15269* The factors of ablfac1b 15268 cover the entire group. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   =>    |-  ( ph  ->  ( G DProd  S )  =  B )
 
Theoremablfac1eulem 15270* Lemma for ablfac1eu 15271. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   &    |-  ( ph  ->  ( G dom DProd  T 
 /\  ( G DProd  T )  =  B )
 )   &    |-  ( ph  ->  dom  T  =  A )   &    |-  ( ( ph  /\  q  e.  A ) 
 ->  C  e.  NN0 )   &    |-  (
 ( ph  /\  q  e.  A )  ->  ( # `
  ( T `  q ) )  =  ( q ^ C ) )   &    |-  ( ph  ->  P  e.  Prime )   &    |-  ( ph  ->  A  e.  Fin )   =>    |-  ( ph  ->  -.  P  ||  ( # `  ( G DProd  ( T  |`  ( A 
 \  { P }
 ) ) ) ) )
 
Theoremablfac1eu 15271* The factorization of ablfac1b 15268 is unique, in that any other factorization into prime power factors (even if the exponents are different) must be equal to 
S. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x )  ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  A 
 C_  Prime )   &    |-  D  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  ( ph  ->  D 
 C_  A )   &    |-  ( ph  ->  ( G dom DProd  T 
 /\  ( G DProd  T )  =  B )
 )   &    |-  ( ph  ->  dom  T  =  A )   &    |-  ( ( ph  /\  q  e.  A ) 
 ->  C  e.  NN0 )   &    |-  (
 ( ph  /\  q  e.  A )  ->  ( # `
  ( T `  q ) )  =  ( q ^ C ) )   =>    |-  ( ph  ->  T  =  S )
 
Theorempgpfac1lem1 15272* Lemma for pgpfac1 15278. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   =>    |-  ( ( ph  /\  C  e.  ( U 
 \  ( S  .(+)  W ) ) )  ->  ( ( S  .(+)  W )  .(+)  ( K ` 
 { C } )
 )  =  U )
 
Theorempgpfac1lem2 15273* Lemma for pgpfac1 15278. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ph  ->  ( P  .x.  C )  e.  ( S  .(+)  W ) )
 
Theorempgpfac1lem3a 15274* Lemma for pgpfac1 15278. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( ( P 
 .x.  C ) ( +g  `  G ) ( M 
 .x.  A ) )  e.  W )   =>    |-  ( ph  ->  ( P  ||  E  /\  P  ||  M ) )
 
Theorempgpfac1lem3 15275* Lemma for pgpfac1 15278. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  ( ( P 
 .x.  C ) ( +g  `  G ) ( M 
 .x.  A ) )  e.  W )   &    |-  D  =  ( C ( +g  `  G ) ( ( M 
 /  P )  .x.  A ) )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  U ) )
 
Theorempgpfac1lem4 15276* Lemma for pgpfac1 15278. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  W  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( S  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  ( S  .(+)  W )  C_  U )   &    |-  ( ph  ->  A. w  e.  (SubGrp `  G ) ( ( w 
 C.  U  /\  A  e.  w )  ->  -.  ( S  .(+)  W )  C.  w ) )   &    |-  ( ph  ->  C  e.  ( U  \  ( S  .(+)  W ) ) )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  U ) )
 
Theorempgpfac1lem5 15277* Lemma for pgpfac1 15278 (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  A  e.  U )   &    |-  ( ph  ->  A. s  e.  (SubGrp `  G ) ( ( s 
 C.  U  /\  A  e.  s )  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  s ) ) )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G ) ( ( S  i^i  t )  =  {  .0.  }  /\  ( S  .(+)  t )  =  U ) )
 
Theorempgpfac1 15278* Factorization of a finite abelian p-group. There is a direct product decomposition of any abelian group of prime-power order where one of the factors is cyclic and generated by an element of maximal order. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  S  =  ( K ` 
 { A } )   &    |-  B  =  ( Base `  G )   &    |-  O  =  ( od `  G )   &    |-  E  =  (gEx `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .(+) 
 =  ( LSSum `  G )   &    |-  ( ph  ->  P pGrp  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  ( O `  A )  =  E )   &    |-  ( ph  ->  A  e.  B )   =>    |-  ( ph  ->  E. t  e.  (SubGrp `  G )
 ( ( S  i^i  t )  =  {  .0.  }  /\  ( S 
 .(+)  t )  =  B ) )
 
Theorempgpfaclem1 15279* Lemma for pgpfac 15282. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   &    |-  H  =  ( Gs  U )   &    |-  K  =  (mrCls `  (SubGrp `  H )
 )   &    |-  O  =  ( od
 `  H )   &    |-  E  =  (gEx `  H )   &    |-  .0.  =  ( 0g `  H )   &    |-  .(+)  =  ( LSSum `  H )   &    |-  ( ph  ->  E  =/=  1 )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  ( O `  X )  =  E )   &    |-  ( ph  ->  W  e.  (SubGrp `  H )
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  .(+)  W )  =  U )   &    |-  ( ph  ->  S  e. Word  C )   &    |-  ( ph  ->  G dom DProd  S )   &    |-  ( ph  ->  ( G DProd  S )  =  W )   &    |-  T  =  ( S concat  <" ( K `
  { X }
 ) "> )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd  s )  =  U ) )
 
Theorempgpfaclem2 15280* Lemma for pgpfac 15282. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   &    |-  H  =  ( Gs  U )   &    |-  K  =  (mrCls `  (SubGrp `  H )
 )   &    |-  O  =  ( od
 `  H )   &    |-  E  =  (gEx `  H )   &    |-  .0.  =  ( 0g `  H )   &    |-  .(+)  =  ( LSSum `  H )   &    |-  ( ph  ->  E  =/=  1 )   &    |-  ( ph  ->  X  e.  U )   &    |-  ( ph  ->  ( O `  X )  =  E )   &    |-  ( ph  ->  W  e.  (SubGrp `  H )
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  i^i  W )  =  {  .0.  }
 )   &    |-  ( ph  ->  (
 ( K `  { X } )  .(+)  W )  =  U )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  U ) )
 
Theorempgpfaclem3 15281* Lemma for pgpfac 15282. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  A. t  e.  (SubGrp `  G ) ( t 
 C.  U  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  t ) ) )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s 
 /\  ( G DProd  s
 )  =  U ) )
 
Theorempgpfac 15282* Full factorization of a finite abelian p-group, by iterating pgpfac1 15278. There is a direct product decomposition of any abelian group of prime-power order into cyclic subgroups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  P pGrp 
 G )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  B ) )
 
Theoremablfaclem1 15283* Lemma for ablfac 15286. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   =>    |-  ( U  e.  (SubGrp `  G )  ->  ( W `  U )  =  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  U ) } )
 
Theoremablfaclem2 15284* Lemma for ablfac 15286. (Contributed by Mario Carneiro, 27-Apr-2016.) (Proof shortened by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   &    |-  ( ph  ->  F : A -->Word  C )   &    |-  ( ph  ->  A. y  e.  A  ( F `  y )  e.  ( W `  ( S `  y ) ) )   &    |-  L  =  U_ y  e.  A  ( { y }  X.  dom  (  F `  y
 ) )   &    |-  ( ph  ->  H : ( 0..^ ( # `  L ) ) -1-1-onto-> L )   =>    |-  ( ph  ->  ( W `  B )  =/=  (/) )
 
Theoremablfaclem3 15285* Lemma for ablfac 15286. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  O  =  ( od `  G )   &    |-  A  =  { w  e.  Prime  |  w  ||  ( # `  B ) }   &    |-  S  =  ( p  e.  A  |->  { x  e.  B  |  ( O `  x ) 
 ||  ( p ^
 ( p  pCnt  ( # `
  B ) ) ) } )   &    |-  W  =  ( g  e.  (SubGrp `  G )  |->  { s  e. Word  C  |  ( G dom DProd  s  /\  ( G DProd 
 s )  =  g ) } )   =>    |-  ( ph  ->  ( W `  B )  =/=  (/) )
 
Theoremablfac 15286* The Fundamental Theorem of (finite) Abelian Groups. Any finite abelian group is a direct product of cyclic p-groups. (Contributed by Mario Carneiro, 27-Apr-2016.) (Revised by Mario Carneiro, 3-May-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   =>    |-  ( ph  ->  E. s  e. Word  C ( G dom DProd  s  /\  ( G DProd 
 s )  =  B ) )
 
Theoremablfac2 15287* Choose generators for each cyclic group in ablfac 15286. (Contributed by Mario Carneiro, 28-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  C  =  {
 r  e.  (SubGrp `  G )  |  ( Gs  r
 )  e.  (CycGrp  i^i  ran pGrp  ) }   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  B  e.  Fin )   &    |-  .x.  =  (.g `  G )   &    |-  S  =  ( k  e.  dom  w  |->  ran  (  n  e.  ZZ  |->  ( n  .x.  ( w `
  k ) ) ) )   =>    |-  ( ph  ->  E. w  e. Word  B ( S : dom  w --> C  /\  G dom DProd  S  /\  ( G DProd  S )  =  B ) )
 
10.4  Rings
 
10.4.1  Multiplicative Group
 
Syntaxcmgp 15288 Multiplicative group.
 class mulGrp
 
Definitiondf-mgp 15289 Define a structure that puts the multiplication operation of a ring in the addition slot. Note that this will not actually be a group for the average ring, or even for a field, but it will be a monoid, and unitgrp 15412 shows that we get a group if we restrict to the elements that have inverses. This allows us to formalize such notions as "the multiplication operation of a ring is a monoid" (rngmgp 15310) or "the multiplicative identity" in terms of the identity of a monoid (df-1r 8655). (Contributed by Mario Carneiro, 21-Dec-2014.)
 |- mulGrp  =  ( w  e.  _V  |->  ( w sSet  <. ( +g  ` 
 ndx ) ,  ( .r `  w ) >. ) )
 
Theoremfnmgp 15290 The multiplicative group operator is a function. (Contributed by Mario Carneiro, 11-Mar-2015.)
 |- mulGrp  Fn  _V
 
Theoremmgpval 15291 Value of the multiplication group operation. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |-  M  =  ( R sSet  <. ( +g  `  ndx ) ,  .x.  >. )
 
Theoremmgpplusg 15292 Value of the group operation of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.)
 |-  M  =  (mulGrp `  R )   &    |- 
 .x.  =  ( .r `  R )   =>    |- 
 .x.  =  ( +g  `  M )
 
Theoremmgplem 15293 Lemma for mgpbas 15294. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  E  = Slot  N   &    |-  N  e.  NN   &    |-  N  =/=  2   =>    |-  ( E `  R )  =  ( E `  M )
 
Theoremmgpbas 15294 Base set of the multiplication group. (Contributed by Mario Carneiro, 21-Dec-2014.) (Revised by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( Base `  R )   =>    |-  B  =  ( Base `  M )
 
Theoremmgpsca 15295 The multiplication monoid has the same (if any) scalars as the original ring. Mostly to simplify pwsmgp 15364. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 5-May-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  S  =  (Scalar `  R )   =>    |-  S  =  (Scalar `  M )
 
Theoremmgptset 15296 Topology component of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   =>    |-  (TopSet `  R )  =  (TopSet `  M )
 
Theoremmgptopn 15297 Topology of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  J  =  ( TopOpen `  R )   =>    |-  J  =  ( TopOpen `  M )
 
Theoremmgpds 15298 Distance function of the multiplication group. (Contributed by Mario Carneiro, 5-Oct-2015.)
 |-  M  =  (mulGrp `  R )   &    |-  B  =  ( dist `  R )   =>    |-  B  =  ( dist `  M )
 
Theoremmgpress 15299 Subgroup commutes with the multiplication group operator. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  S  =  ( Rs  A )   &    |-  M  =  (mulGrp `  R )   =>    |-  ( ( R  e.  V  /\  A  e.  W )  ->  ( Ms  A )  =  (mulGrp `  S ) )
 
10.4.2  Definition and basic properties
 
Syntaxcrg 15300 Extend class notation with class of all (unital) rings.
 class  Ring
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