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Theorem List for Metamath Proof Explorer - 15501-15600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsum2d 15501* Write a sum over a two-dimensional region as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Rel 
 A )   &    |-  ( ph  ->  D  e.  W )   &    |-  ( ph  ->  dom  A  C_  D )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( G  gsumg  ( j  e.  D  |->  ( G  gsumg  ( k  e.  ( A " { j }
 )  |->  ( j F k ) ) ) ) ) )
 
Theoremgsum2d2lem 15502* Lemma for gsum2d2 15503: show the function is finitely supported. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  C  e.  W )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  C )
 )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  (
 ( ph  /\  ( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   =>    |-  ( ph  ->  ( `' ( j  e.  A ,  k  e.  C  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )
 
Theoremgsum2d2 15503* Write a group sum over a two-dimensional region as a double sum. (Note that  C ( j ) is a function of  j.) (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  C  e.  W )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  C )
 )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  (
 ( ph  /\  ( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A ,  k  e.  C  |->  X ) )  =  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  X ) ) ) ) )
 
Theoremgsumcom2 15504* Two-dimensional commutation of a group sum. Note that while  A and  D are constants w.r.t.  j ,  k,  C ( j ) and 
E ( k ) are not. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  j  e.  A ) 
 ->  C  e.  W )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  C )
 )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  (
 ( ph  /\  ( ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ph  ->  ( ( j  e.  A  /\  k  e.  C )  <->  ( k  e.  D  /\  j  e.  E ) ) )   =>    |-  ( ph  ->  ( G  gsumg  (
 j  e.  A ,  k  e.  C  |->  X ) )  =  ( G 
 gsumg  ( k  e.  D ,  j  e.  E  |->  X ) ) )
 
Theoremgsumxp 15505* Write a group sum over a cartesian product as a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  ( ph  ->  F : ( A  X.  C ) --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )   =>    |-  ( ph  ->  ( G  gsumg  F )  =  ( G  gsumg  ( j  e.  A  |->  ( G 
 gsumg  ( k  e.  C  |->  ( j F k ) ) ) ) ) )
 
Theoremgsumcom 15506* Commute the arguments of a double sum. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  (
 ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A ,  k  e.  C  |->  X ) )  =  ( G  gsumg  ( k  e.  C ,  j  e.  A  |->  X ) ) )
 
Theoremprdsgsum 15507* Finite commutative sums in a product structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.)
 |-  Y  =  ( S
 X_s ( x  e.  I  |->  R ) )   &    |-  B  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  S  e.  X )   &    |-  ( ( ph  /\  x  e.  I )  ->  R  e. CMnd )   &    |-  ( ( ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  B )   &    |-  ( ph  ->  ( `' ( y  e.  J  |->  ( x  e.  I  |->  U ) ) "
 ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
 
Theorempwsgsum 15508* Finite commutative sums in a power structure are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 3-Jul-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  R  e. CMnd )   &    |-  (
 ( ph  /\  ( x  e.  I  /\  y  e.  J ) )  ->  U  e.  B )   &    |-  ( ph  ->  ( `' (
 y  e.  J  |->  ( x  e.  I  |->  U ) ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   =>    |-  ( ph  ->  ( Y  gsumg  ( y  e.  J  |->  ( x  e.  I  |->  U ) ) )  =  ( x  e.  I  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
 
10.3.4  Internal direct products
 
Syntaxcdprd 15509 Internal direct product of a family of subgroups.
 class DProd
 
Syntaxcdpj 15510 Internal direct product of a family of subgroups.
 class dProj
 
Definitiondf-dprd 15511* Define the internal direct product of a family of subgroups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- DProd  =  ( g  e.  Grp ,  s  e.  { h  |  ( h : dom  h --> (SubGrp `  g )  /\  A. x  e.  dom  h ( A. y  e.  ( dom  h  \  { x } ) ( h `  x ) 
 C_  ( (Cntz `  g ) `  ( h `  y ) ) 
 /\  ( ( h `
  x )  i^i  ( (mrCls `  (SubGrp `  g ) ) `  U. ( h " ( dom  h  \  { x } ) ) ) )  =  { ( 0g `  g ) }
 ) ) }  |->  ran  ( f  e.  { h  e.  X_ x  e. 
 dom  s ( s `
  x )  |  ( `' h "
 ( _V  \  {
 ( 0g `  g
 ) } ) )  e.  Fin }  |->  ( g  gsumg  f ) ) )
 
Definitiondf-dpj 15512* Define the projection operator for a direct product. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- dProj  =  ( g  e.  Grp ,  s  e.  ( dom DProd  " { g } )  |->  ( i  e.  dom  s  |->  ( ( s `
  i ) (
 proj 1 `  g ) ( g DProd  ( s  |`  ( dom  s  \  { i } )
 ) ) ) ) )
 
Theoremreldmdprd 15513 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.)
 |- 
 Rel  dom DProd
 
Theoremdmdprd 15514* 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.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( I  e.  V  /\  dom  S  =  I )  ->  ( G dom DProd  S  <->  ( G  e.  Grp  /\  S : I --> (SubGrp `  G )  /\  A. x  e.  I  ( A. y  e.  ( I  \  { x } ) ( S `
  x )  C_  ( Z `  ( S `
  y ) ) 
 /\  ( ( S `
  x )  i^i  ( K `  U. ( S " ( I  \  { x } ) ) ) )  =  {  .0.  } ) ) ) )
 
Theoremdmdprdd 15515* Show that a given family is a direct product decomposition. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S : I --> (SubGrp `  G ) )   &    |-  ( ( ph  /\  ( x  e.  I  /\  y  e.  I  /\  x  =/=  y
 ) )  ->  ( S `  x )  C_  ( Z `  ( S `
  y ) ) )   &    |-  ( ( ph  /\  x  e.  I ) 
 ->  ( ( S `  x )  i^i  ( K `
  U. ( S "
 ( I  \  { x } ) ) ) )  C_  {  .0.  } )   =>    |-  ( ph  ->  G dom DProd  S )
 
Theoremdprdval 15516* 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 15517* 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 15518 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( G dom DProd  S  ->  G  e.  Grp )
 
Theoremdprdf 15519 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 15520 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 15521 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 15522 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 15523* 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 15524* 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 15525* 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 15526* 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 15527* 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 15528* 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 15529 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 15530* 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 15531* 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 15532* 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 15533* 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 15534* 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 15535* 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 15536* 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 15537 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 15538 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 15539* 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 15540 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 15541 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 15542* 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 15543* 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 15544 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 15545 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 15546 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 15547 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 15548 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 15549 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 15550* Lemma for dmdprdsplit 15560. (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 15551 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 15552 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 15553* 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 15554* 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 15555* 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 15556* 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 15557* 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 15558 Lemma for dmdprdsplit 15560. (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 15559 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 15560 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 15561 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 15562 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 15563 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 15564 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 15565 The two subgroups that appear in dpjval 15569 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 15566 The two subgroups that appear in dpjval 15569 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 15567 The two subgroups that appear in dpjval 15569 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 15568* 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 15569 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 15570 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 15571* 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 15572* 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 15573* 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 15574 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 15575 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 15576 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 15577 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 15578* Lemma for ablfacrp2 15580. (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 15579* 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 15580* The factors  K ,  L of ablfacrp 15579 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 15581* Lemma for ablfac1b 15583. 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 15582* The factors of ablfac1b 15583 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 15583* 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 15584* The factors of ablfac1b 15583 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 15585* Lemma for ablfac1eu 15586. (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 15586* The factorization of ablfac1b 15583 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 15587* Lemma for pgpfac1 15593. (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 15588* Lemma for pgpfac1 15593. (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 15589* Lemma for pgpfac1 15593. (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 15590* Lemma for pgpfac1 15593. (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 15591* Lemma for pgpfac1 15593. (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 15592* Lemma for pgpfac1 15593 (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 15593* 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 15594* Lemma for pgpfac 15597. (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 15595* Lemma for pgpfac 15597. (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 15596* Lemma for pgpfac 15597. (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 15597* Full factorization of a finite abelian p-group, by iterating pgpfac1 15593. 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 15598* Lemma for ablfac 15601. (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 15599* Lemma for ablfac 15601. (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 15600* Lemma for ablfac 15601. (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 )  =/=  (/) )
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