HomeHome Metamath Proof Explorer
Theorem List (p. 153 of 315)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-21494)
  Hilbert Space Explorer  Hilbert Space Explorer
(21495-23017)
  Users' Mathboxes  Users' Mathboxes
(23018-31433)
 

Theorem List for Metamath Proof Explorer - 15201-15300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsumadd 15201 The sum of two group sums. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   &    |-  ( ph  ->  ( `' H " ( _V  \  {  .0.  } )
 )  e.  Fin )   =>    |-  ( ph  ->  ( G  gsumg  ( F  o F  .+  H ) )  =  (
 ( G  gsumg 
 F )  .+  ( G  gsumg 
 H ) ) )
 
Theoremgsumzsplit 15202 Split a group sum into two parts. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
 gsumg  ( F  |`  D ) ) ) )
 
Theoremgsumsplit 15203 Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( ( G  gsumg  ( F  |`  C ) )  .+  ( G 
 gsumg  ( F  |`  D ) ) ) )
 
Theoremgsumsplit2 15204* Split a group sum into two parts. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  (
 ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( ph  ->  ( C  i^i  D )  =  (/) )   &    |-  ( ph  ->  A  =  ( C  u.  D ) )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( G  gsumg  ( k  e.  C  |->  X ) )  .+  ( G 
 gsumg  ( k  e.  D  |->  X ) ) ) )
 
Theoremgsumconst 15205* Sum of a constant series. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  Fin  /\  X  e.  B )  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( # `  A )  .x.  X ) )
 
Theoremgsumzmhm 15206 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  H  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  K  e.  ( G MndHom  H ) )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg  F ) ) )
 
Theoremgsummhm 15207 Apply a group homomorphism to a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  H  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  K  e.  ( G MndHom  H ) )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )   =>    |-  ( ph  ->  ( H  gsumg  ( K  o.  F ) )  =  ( K `  ( G  gsumg 
 F ) ) )
 
Theoremgsummhm2 15208* Apply a group homomorphism to a group sum, mapping version with implicit substitution. (Contributed by Mario Carneiro, 5-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  H  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  ( x  e.  B  |->  C )  e.  ( G MndHom  H )
 )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( x  =  X  ->  C  =  D )   &    |-  ( x  =  ( G  gsumg  ( k  e.  A  |->  X ) )  ->  C  =  E )   =>    |-  ( ph  ->  ( H  gsumg  ( k  e.  A  |->  D ) )  =  E )
 
Theoremgsummulglem 15209* Lemma for gsummulg 15210 and gsummulgz 15211. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  ( G  e.  Abel  \/  N  e.  NN0 )
 )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  ( N  .x.  X ) ) )  =  ( N  .x.  ( G  gsumg  (
 k  e.  A  |->  X ) ) ) )
 
Theoremgsummulg 15210* Nonnegative multiple of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 7-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  ( N  .x.  X ) ) )  =  ( N  .x.  ( G  gsumg  (
 k  e.  A  |->  X ) ) ) )
 
Theoremgsummulgz 15211* Integer multiple of a group sum. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .x. 
 =  (.g `  G )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  ( `' ( k  e.  A  |->  X ) " ( _V  \  {  .0.  }
 ) )  e.  Fin )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  N  e.  ZZ )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  A  |->  ( N  .x.  X ) ) )  =  ( N  .x.  ( G  gsumg  (
 k  e.  A  |->  X ) ) ) )
 
Theoremgsumzoppg 15212 The opposite of a group sum is the same as the original. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  O  =  (oppg `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ran 
 F  C_  ( Z ` 
 ran  F ) )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( O  gsumg 
 F )  =  ( G  gsumg 
 F ) )
 
Theoremgsumzinv 15213 Inverse of a group sum. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  I  =  ( inv
 g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ran  F  C_  ( Z `  ran  F ) )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } )
 )  e.  Fin )   =>    |-  ( ph  ->  ( G  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg 
 F ) ) )
 
Theoremgsuminv 15214 Inverse of a group sum. (Contributed by Mario Carneiro, 15-Dec-2014.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( inv g `
  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   =>    |-  ( ph  ->  ( G  gsumg  ( I  o.  F ) )  =  ( I `  ( G  gsumg  F ) ) )
 
Theoremgsumsub 15215 The difference of two group sums. (Contributed by Mario Carneiro, 28-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  H : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  e. 
 Fin )   &    |-  ( ph  ->  ( `' H " ( _V  \  {  .0.  } )
 )  e.  Fin )   =>    |-  ( ph  ->  ( G  gsumg  ( F  o F  .-  H ) )  =  (
 ( G  gsumg 
 F )  .-  ( G  gsumg 
 H ) ) )
 
Theoremgsumsn 15216* Group sum of a singleton. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 24-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  ( k  =  M  ->  A  =  C )   =>    |-  ( ( G  e.  Mnd  /\  M  e.  V  /\  C  e.  B )  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
 
Theoremgsumunsn 15217* Append an element to a finite group sum. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  X  e.  B )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  -.  M  e.  A )   &    |-  ( ph  ->  Y  e.  B )   &    |-  (
 k  =  M  ->  X  =  Y )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  ( A  u.  { M }
 )  |->  X ) )  =  ( ( G 
 gsumg  ( k  e.  A  |->  X ) )  .+  Y ) )
 
Theoremgsumpt 15218 Sum of a family that is nonzero at at most one point. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 25-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  X  e.  A )   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( `' F " ( _V  \  {  .0.  } ) )  C_  { X } )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( F `  X ) )
 
Theoremgsum2d 15219* 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 15220* Lemma for gsum2d2 15221: 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 15221* 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 15222* 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 15223* 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 15224* 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 15225* 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 15226* 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 15227 Internal direct product of a family of subgroups.
 class DProd
 
Syntaxcdpj 15228 Internal direct product of a family of subgroups.
 class dProj
 
Definitiondf-dprd 15229* 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 15230* 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 15231 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 15232* 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 15233* 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 15234* 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 15235* 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 15236 Reverse closure for the internal direct product. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  ( G  dom DProd  S  ->  G  e.  Grp )
 
Theoremdprdf 15237 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 15238 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 15239 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 15240 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 15241* 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 15242* 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 15243* 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 15244* 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 15245* 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 15246* 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 15247 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 15248* 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 15249* 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 15250* 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 15251* 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 15252* 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 15253* 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 15254* 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 15255 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 15256 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 15257* 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 15258 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 15259 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 15260* 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 15261* 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 15262 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 15263 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 15264 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 15265 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 15266 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 15267 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 15268* Lemma for dmdprdsplit 15278. (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 15269 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 15270 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 15271* 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 15272* 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 15273* 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 15274* 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 15275* 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 15276 Lemma for dmdprdsplit 15278. (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 15277 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 15278 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 15279 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 15280 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 15281 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 15282 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 15283 The two subgroups that appear in dpjval 15287 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 15284 The two subgroups that appear in dpjval 15287 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 15285 The two subgroups that appear in dpjval 15287 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 15286* 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 15287 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 15288 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 15289* 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 15290* 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 15291* 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 15292 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 15293 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 15294 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 15295 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 15296* Lemma for ablfacrp2 15298. (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 15297* 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 15298* The factors  K ,  L of ablfacrp 15297 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 15299* Lemma for ablfac1b 15301. 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 15300* The factors of ablfac1b 15301 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 ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31433
  Copyright terms: Public domain < Previous  Next >