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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgrpsubpropd 14401 Weak property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 27-Mar-2015.)
 |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  ( +g  `  G )  =  ( +g  `  H ) )   =>    |-  ( ph  ->  ( -g `  G )  =  ( -g `  H ) )
 
Theoremgrpsubpropd2 14402* Strong property deduction for the group subtraction operation. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  B  =  ( Base `  H )
 )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B ) )  ->  ( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )   =>    |-  ( ph  ->  ( -g `  G )  =  ( -g `  H ) )
 
Theoremmulgfval 14403* Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( inv
 g `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  =  ( n  e.  ZZ ,  x  e.  B  |->  if ( n  =  0 ,  .0.  ,  if ( 0  <  n ,  (  seq  1 ( 
 .+  ,  ( NN  X. 
 { x } )
 ) `  n ) ,  ( I `  (  seq  1 (  .+  ,  ( NN  X.  { x } ) ) `  -u n ) ) ) ) )
 
Theoremmulgval 14404 Group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  I  =  ( inv
 g `  G )   &    |-  .x.  =  (.g `  G )   &    |-  S  =  seq  1 (  .+  ,  ( NN  X.  { X }
 ) )   =>    |-  ( ( N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  =  if ( N  =  0 ,  .0.  ,  if ( 0  <  N ,  ( S `  N ) ,  ( I `  ( S `  -u N ) ) ) ) )
 
Theoremmulgfn 14405 Functionality of the group multiple function. (Contributed by Mario Carneiro, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |- 
 .x.  Fn  ( ZZ  X.  B )
 
Theoremmulgfvi 14406 The group multiple function is compatible with identity-function protection. (Contributed by Mario Carneiro, 21-Mar-2015.)
 |- 
 .x.  =  (.g `  G )   =>    |- 
 .x.  =  (.g `  (  _I  `  G ) )
 
Theoremmulg0 14407 Group multiple (exponentiation) operation at zero. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .x. 
 =  (.g `  G )   =>    |-  ( X  e.  B  ->  ( 0  .x.  X )  =  .0.  )
 
Theoremmulgnn 14408 Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .x.  =  (.g `  G )   &    |-  S  =  seq  1 (  .+  ,  ( NN  X.  { X }
 ) )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X )  =  ( S `
  N ) )
 
Theoremmulg1 14409 Group multiple (exponentiation) operation at one. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( X  e.  B  ->  ( 1  .x.  X )  =  X )
 
Theoremmulgnnp1 14410 Group multiple (exponentiation) operation at a successor. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  (
 ( N  +  1 )  .x.  X )  =  ( ( N  .x.  X )  .+  X ) )
 
Theoremmulg2 14411 Group multiple (exponentiation) operation at two. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( X  e.  B  ->  ( 2  .x.  X )  =  ( X 
 .+  X ) )
 
Theoremmulgnegnn 14412 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N 
 .x.  X ) ) )
 
Theoremmulgnn0p1 14413 Group multiple (exponentiation) operation at a successor, extended to  NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B ) 
 ->  ( ( N  +  1 )  .x.  X )  =  ( ( N 
 .x.  X )  .+  X ) )
 
Theoremmulgnnsubcl 14414* Closure of the group multiple (exponentiation) operation in a subsemigroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   =>    |-  ( ( ph  /\  N  e.  NN  /\  X  e.  S )  ->  ( N 
 .x.  X )  e.  S )
 
Theoremmulgnn0subcl 14415* Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  .0. 
 e.  S )   =>    |-  ( ( ph  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremmulgsubcl 14416* Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ( ph  /\  x  e.  S  /\  y  e.  S )  ->  ( x  .+  y
 )  e.  S )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  .0. 
 e.  S )   &    |-  I  =  ( inv g `  G )   &    |-  ( ( ph  /\  x  e.  S ) 
 ->  ( I `  x )  e.  S )   =>    |-  (
 ( ph  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremmulgnncl 14417 Closure of the group multiple (exponentiation) operation. TODO: This can be generalized to a magma if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgnn0cl 14418 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgcl 14419 Closure of the group multiple (exponentiation) operation. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( N  .x.  X )  e.  B )
 
Theoremmulgneg 14420 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( I `  ( N  .x.  X ) ) )
 
Theoremmulgm1 14421 Group multiple (exponentiation) operation at negative one. (Contributed by Mario Carneiro, 20-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( -u 1  .x.  X )  =  ( I `  X ) )
 
Theoremmulgnn0z 14422 A group multiple of the identity, for nonnegative multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  N  e.  NN0 )  ->  ( N  .x.  .0.  )  =  .0.  )
 
Theoremmulgz 14423 A group multiple of the identity, for integer multiple. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ )  ->  ( N  .x.  .0.  )  =  .0.  )
 
Theoremmulgnndir 14424 Sum of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M 
 .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgnn0dir 14425 Sum of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M  .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgdirlem 14426 Lemma for mulgdir 14427. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )  /\  ( M  +  N )  e. 
 NN0 )  ->  (
 ( M  +  N )  .x.  X )  =  ( ( M  .x.  X )  .+  ( N 
 .x.  X ) ) )
 
Theoremmulgdir 14427 Sum of group multiples, generalized to  ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  +  N )  .x.  X )  =  ( ( M 
 .x.  X )  .+  ( N  .x.  X ) ) )
 
Theoremmulgp1 14428 Group multiple (exponentiation) operation at a successor, extended to  ZZ. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( ( N  +  1 )  .x.  X )  =  ( ( N  .x.  X )  .+  X ) )
 
Theoremmulgneg2 14429 Group multiple (exponentiation) operation at a negative integer. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  N  e.  ZZ  /\  X  e.  B )  ->  ( -u N  .x.  X )  =  ( N  .x.  ( I `  X ) ) )
 
Theoremmulgnnass 14430 Product of group multiples, for positive multiples. TODO: This can be generalized to a semigroup if/when we introduce them. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN  /\  N  e.  NN  /\  X  e.  B )
 )  ->  ( ( M  x.  N )  .x.  X )  =  ( M 
 .x.  ( N  .x.  X ) ) )
 
Theoremmulgnn0ass 14431 Product of group multiples, generalized to  NN0. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  N  e.  NN0  /\  X  e.  B ) )  ->  ( ( M  x.  N )  .x.  X )  =  ( M  .x.  ( N  .x.  X ) ) )
 
Theoremmulgass 14432 Product of group multiples, generalized to  ZZ. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B )
 )  ->  ( ( M  x.  N )  .x.  X )  =  ( M 
 .x.  ( N  .x.  X ) ) )
 
Theoremmulgsubdir 14433 Subtraction of a group element from itself. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( M  e.  ZZ  /\  N  e.  ZZ  /\  X  e.  B ) )  ->  ( ( M  -  N )  .x.  X )  =  ( ( M 
 .x.  X )  .-  ( N  .x.  X ) ) )
 
Theoremmhmmulg 14434 A homomorphism of monoids preserves group multiples. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .X.  =  (.g `  H )   =>    |-  ( ( F  e.  ( G MndHom  H )  /\  N  e.  NN0  /\  X  e.  B )  ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
 
Theoremmulgpropd 14435* Two structures with the same group-nature have the same group multiple function.  K is expected to either be  _V (when strong equality is available) or  B (when closure is available). (Contributed by Stefan O'Rear, 21-Mar-2015.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |- 
 .x.  =  (.g `  G )   &    |- 
 .X.  =  (.g `  H )   &    |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  B  =  ( Base `  H )
 )   &    |-  ( ph  ->  B  C_  K )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  K ) )  ->  ( x ( +g  `  G ) y )  e.  K )   &    |-  ( ( ph  /\  ( x  e.  K  /\  y  e.  K ) )  ->  ( x ( +g  `  G ) y )  =  ( x ( +g  `  H ) y ) )   =>    |-  ( ph  ->  .x.  =  .X.  )
 
Theoremsubmmulgcl 14436 Closure of the group multiple (exponentiation) operation in a submonoid. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  .xb  =  (.g `  G )   =>    |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S ) 
 ->  ( N  .xb  X )  e.  S )
 
Theoremsubmmulg 14437 A group multiple is the same if evaluated in a submonoid. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  .xb  =  (.g `  G )   &    |-  H  =  ( Gs  S )   &    |-  .x.  =  (.g `  H )   =>    |-  ( ( S  e.  (SubMnd `  G )  /\  N  e.  NN0  /\  X  e.  S )  ->  ( N  .xb  X )  =  ( N  .x.  X ) )
 
Theoremprdsinvlem 14438* Characterization of inverses in a structure product. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  ( ph  ->  F  e.  B )   &    |- 
 .0.  =  ( 0g  o.  R )   &    |-  N  =  ( y  e.  I  |->  ( ( inv g `  ( R `  y ) ) `  ( F `
  y ) ) )   =>    |-  ( ph  ->  ( N  e.  B  /\  ( N  .+  F )  =  .0.  ) )
 
Theoremprdsgrpd 14439 The product of a family of groups is a group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   =>    |-  ( ph  ->  Y  e.  Grp )
 
Theoremprdsinvgd 14440* Negation in a product of groups. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Grp )   &    |-  B  =  (
 Base `  Y )   &    |-  N  =  ( inv g `  Y )   &    |-  ( ph  ->  X  e.  B )   =>    |-  ( ph  ->  ( N `  X )  =  ( x  e.  I  |->  ( ( inv
 g `  ( R `  x ) ) `  ( X `  x ) ) ) )
 
Theorempwsgrp 14441 The product of a family of groups is a group. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Grp  /\  I  e.  V )  ->  Y  e.  Grp )
 
Theorempwsinvg 14442 Negation in a group power. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  M  =  ( inv g `  R )   &    |-  N  =  ( inv g `  Y )   =>    |-  ( ( R  e.  Grp  /\  I  e.  V  /\  X  e.  B ) 
 ->  ( N `  X )  =  ( M  o.  X ) )
 
Theorempwssub 14443 Subtraction in a group power. (Contributed by Mario Carneiro, 12-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  M  =  ( -g `  R )   &    |-  .-  =  ( -g `  Y )   =>    |-  ( ( ( R  e.  Grp  /\  I  e.  V )  /\  ( F  e.  B  /\  G  e.  B )
 )  ->  ( F  .-  G )  =  ( F  o F M G ) )
 
Theorempwsmulg 14444 Value of a group multiple in a structure power. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   &    |-  .xb  =  (.g `  Y )   &    |-  .x.  =  (.g `  R )   =>    |-  ( ( ( R  e.  Mnd  /\  I  e.  V )  /\  ( N  e.  NN0  /\  X  e.  B  /\  A  e.  I ) )  ->  ( ( N  .xb  X ) `  A )  =  ( N  .x.  ( X `  A ) ) )
 
Theoremimasgrp2 14445* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  W )   &    |-  (
 ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x  .+  y )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( F `
  ( ( x 
 .+  y )  .+  z ) )  =  ( F `  ( x  .+  ( y  .+  z ) ) ) )   &    |-  ( ph  ->  .0. 
 e.  V )   &    |-  (
 ( ph  /\  x  e.  V )  ->  ( F `  (  .0.  .+  x ) )  =  ( F `  x ) )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  N  e.  V )   &    |-  ( ( ph  /\  x  e.  V )  ->  ( F `  ( N  .+  x ) )  =  ( F `  .0.  ) )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasgrp 14446* The image structure of a group is a group. (Contributed by Mario Carneiro, 24-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( ph  ->  U  =  ( F  "s  R )
 )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  F : V -onto-> B )   &    |-  ( ( ph  /\  ( a  e.  V  /\  b  e.  V )  /\  ( p  e.  V  /\  q  e.  V ) )  ->  ( ( ( F `
  a )  =  ( F `  p )  /\  ( F `  b )  =  ( F `  q ) ) 
 ->  ( F `  (
 a  .+  b )
 )  =  ( F `
  ( p  .+  q ) ) ) )   &    |-  ( ph  ->  R  e.  Grp )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  ( F `  .0.  )  =  ( 0g `  U ) ) )
 
Theoremimasgrpf1 14447 The image of a group under an injection is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  U  =  ( F 
 "s 
 R )   &    |-  V  =  (
 Base `  R )   =>    |-  ( ( F : V -1-1-> B  /\  R  e.  Grp )  ->  U  e.  Grp )
 
Theoremdivsgrp2 14448* Prove that a quotient structure is a group. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ( ph  ->  U  =  ( R  /.s  .~  ) )   &    |-  ( ph  ->  V  =  ( Base `  R )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  R )
 )   &    |-  ( ph  ->  .~  Er  V )   &    |-  ( ph  ->  R  e.  X )   &    |-  ( ph  ->  ( ( a 
 .~  p  /\  b  .~  q )  ->  (
 a  .+  b )  .~  ( p  .+  q
 ) ) )   &    |-  (
 ( ph  /\  x  e.  V  /\  y  e.  V )  ->  ( x  .+  y )  e.  V )   &    |-  ( ( ph  /\  ( x  e.  V  /\  y  e.  V  /\  z  e.  V ) )  ->  ( ( x  .+  y ) 
 .+  z )  .~  ( x  .+  ( y 
 .+  z ) ) )   &    |-  ( ph  ->  .0. 
 e.  V )   &    |-  (
 ( ph  /\  x  e.  V )  ->  (  .0.  .+  x )  .~  x )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  N  e.  V )   &    |-  ( ( ph  /\  x  e.  V )  ->  ( N  .+  x )  .~  .0.  )   =>    |-  ( ph  ->  ( U  e.  Grp  /\  [  .0.  ]  .~  =  ( 0g `  U ) ) )
 
Theoremxpsgrp 14449 The binary product of groups is a group. (Contributed by Mario Carneiro, 20-Aug-2015.)
 |-  T  =  ( R  X.s  S )   =>    |-  ( ( R  e.  Grp  /\  S  e.  Grp )  ->  T  e.  Grp )
 
10.2.2  Subgroups and Quotient groups
 
Syntaxcsubg 14450 Extend class notation with all subgroups of a group.
 class SubGrp
 
Syntaxcnsg 14451 Extend class notation with all normal subgroups of a group.
 class NrmSGrp
 
Syntaxcqg 14452 Quotient group equivalence class.
 class ~QG
 
Definitiondf-subg 14453* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14471), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14466), contains the neutral element of the group (see subg0 14462) and contains the inverses for all of its elements (see subginvcl 14465). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
 
Definitiondf-nsg 14454* Define the equivalence relation in a quotient ring or quotient group (where  i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- NrmSGrp  =  ( w  e.  Grp  |->  { s  e.  (SubGrp `  w )  |  [. ( Base `  w )  /  b ]. [. ( +g  `  w )  /  p ]. A. x  e.  b  A. y  e.  b  ( ( x p y )  e.  s  <->  ( y p x )  e.  s
 ) } )
 
Definitiondf-eqg 14455* Define the equivalence relation in a quotient ring or quotient group (where  i is a two-sided ideal or a normal subgroup). For non-normal subgroups this generates the left cosets. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |- ~QG  =  ( r  e.  _V ,  i  e.  _V  |->  {
 <. x ,  y >.  |  ( { x ,  y }  C_  ( Base `  r )  /\  (
 ( ( inv g `  r ) `  x ) ( +g  `  r
 ) y )  e.  i ) } )
 
Theoremissubg 14456 The subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  (SubGrp `  G )  <->  ( G  e.  Grp  /\  S  C_  B  /\  ( Gs  S )  e.  Grp ) )
 
Theoremsubgss 14457 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  C_  B )
 
Theoremsubgid 14458 A group is a subgroup of itself. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  e.  (SubGrp `  G ) )
 
Theoremsubggrp 14459 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  H  e.  Grp )
 
Theoremsubgbas 14460 The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  =  ( Base `  H )
 )
 
Theoremsubgrcl 14461 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( S  e.  (SubGrp `  G )  ->  G  e.  Grp )
 
Theoremsubg0 14462 A subgroup of a group must have the same identity as the group. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  H  =  ( Gs  S )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  .0.  =  ( 0g `  H ) )
 
Theoremsubginv 14463 The inverse of an element in a subgroup is the same as the inverse in the larger group. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   &    |-  I  =  ( inv g `  G )   &    |-  J  =  ( inv
 g `  H )   =>    |-  (
 ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( I `  X )  =  ( J `  X ) )
 
Theoremsubg0cl 14464 The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  .0.  e.  S )
 
Theoremsubginvcl 14465 The inverse of an element is closed in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  I  =  ( inv
 g `  G )   =>    |-  (
 ( S  e.  (SubGrp `  G )  /\  X  e.  S )  ->  ( I `  X )  e.  S )
 
Theoremsubgcl 14466 A subgroup is closed under group operation. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- 
 .+  =  ( +g  `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .+  Y )  e.  S )
 
Theoremsubgsubcl 14467 A subgroup is closed under group subtraction. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  .-  =  ( -g `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X  .-  Y )  e.  S )
 
Theoremsubgsub 14468 The subtraction of elements in a subgroup is the same as subtraction in the group. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  .-  =  ( -g `  G )   &    |-  H  =  ( Gs  S )   &    |-  N  =  (
 -g `  H )   =>    |-  (
 ( S  e.  (SubGrp `  G )  /\  X  e.  S  /\  Y  e.  S )  ->  ( X 
 .-  Y )  =  ( X N Y ) )
 
Theoremsubgmulgcl 14469 Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |- 
 .x.  =  (.g `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  e.  S )
 
Theoremsubgmulg 14470 A group multiple is the same if evaluated in a subgroup. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |- 
 .x.  =  (.g `  G )   &    |-  H  =  ( Gs  S )   &    |-  .xb  =  (.g `  H )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  N  e.  ZZ  /\  X  e.  S )  ->  ( N  .x.  X )  =  ( N  .xb  X ) )
 
Theoremissubg2 14471* Characterize the subgroups of a group by closure properties. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G )  <->  ( S  C_  B  /\  S  =/=  (/)  /\  A. x  e.  S  ( A. y  e.  S  ( x  .+  y )  e.  S  /\  ( I `  x )  e.  S ) ) ) )
 
Theoremissubg3 14472* A subgroup is a symmetric submonoid. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  I  =  ( inv
 g `  G )   =>    |-  ( G  e.  Grp  ->  ( S  e.  (SubGrp `  G ) 
 <->  ( S  e.  (SubMnd `  G )  /\  A. x  e.  S  ( I `  x )  e.  S ) ) )
 
Theoremissubg4 14473* A subgroup is a nonempty subset of the group closed under subtraction. (Contributed by Mario Carneiro, 17-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  Grp 
 ->  ( S  e.  (SubGrp `  G )  <->  ( S  C_  B  /\  S  =/=  (/)  /\  A. x  e.  S  A. y  e.  S  ( x  .-  y )  e.  S ) ) )
 
Theoremsubgsubm 14474 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( S  e.  (SubGrp `  G )  ->  S  e.  (SubMnd `  G )
 )
 
Theoremsubsubg 14475 A subgroup of a subgroup is a subgroup. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  ( A  e.  (SubGrp `  H ) 
 <->  ( A  e.  (SubGrp `  G )  /\  A  C_  S ) ) )
 
Theoremsubgint 14476 The intersection of a nonempty collection of subgroups is a subgroup. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  ( ( S  C_  (SubGrp `  G )  /\  S  =/=  (/) )  ->  |^| S  e.  (SubGrp `  G )
 )
 
Theorem0subg 14477 The zero subgroup of an arbitrary group. (Contributed by Stefan O'Rear, 10-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  {  .0.  }  e.  (SubGrp `  G ) )
 
Theoremcycsubgcl 14478* The set of integer powers of an element  A of a group forms a subgroup containing  A, called the cyclic group generated by the element  A. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A )
 )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ran  F  e.  (SubGrp `  G )  /\  A  e.  ran  F ) )
 
Theoremcycsubgss 14479* The cyclic subgroup generated by an element  A is a subset of any subgroup containing  A. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A )
 )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  A  e.  S )  ->  ran  F  C_  S )
 
Theoremcycsubg 14480* The cyclic group generated by  A is the smallest subgroup containing  A. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A )
 )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ran  F  =  |^| { s  e.  (SubGrp `  G )  |  A  e.  s } )
 
Theoremisnsg 14481* Property of being a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( S  e.  (NrmSGrp `  G )  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  X  (
 ( x  .+  y
 )  e.  S  <->  ( y  .+  x )  e.  S ) ) )
 
Theoremisnsg2 14482* Weaken the condition of isnsg 14481 to only one side of the implication. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( S  e.  (NrmSGrp `  G )  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  X  (
 ( x  .+  y
 )  e.  S  ->  ( y  .+  x )  e.  S ) ) )
 
Theoremnsgbi 14483 Defining property of a normal subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  A  e.  X  /\  B  e.  X ) 
 ->  ( ( A  .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) )
 
Theoremnsgsubg 14484 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  ( S  e.  (NrmSGrp `  G )  ->  S  e.  (SubGrp `  G )
 )
 
Theoremnsgconj 14485 The conjugation of an element of a normal subgroup is in the subgroup. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  A  e.  X  /\  B  e.  S )  ->  ( ( A  .+  B )  .-  A )  e.  S )
 
Theoremisnsg3 14486* A subgroup is normal iff the conjugation of all the elements of the subgroup is in the subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( S  e.  (NrmSGrp `  G )  <->  ( S  e.  (SubGrp `  G )  /\  A. x  e.  X  A. y  e.  S  (
 ( x  .+  y
 )  .-  x )  e.  S ) )
 
Theoremsubgacs 14487 Subgroups are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.) (Revised by Mario Carneiro, 22-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  (SubGrp `  G )  e.  (ACS `  B )
 )
 
Theoremnsgacs 14488 Normal subgroups form an algebraic closure system. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  (NrmSGrp `  G )  e.  (ACS `  B )
 )
 
Theoremcycsubg2 14489* The subgroup generated by an element is exhausted by its multiples. (Contributed by Stefan O'Rear, 6-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  F  =  ( x  e.  ZZ  |->  ( x  .x.  A )
 )   &    |-  K  =  (mrCls `  (SubGrp `  G ) )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( K `  { A } )  =  ran  F )
 
Theoremcycsubg2cl 14490 Any multiple of an element is contained in the generated cyclic subgroup. (Contributed by Stefan O'Rear, 12-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  K  =  (mrCls `  (SubGrp `  G )
 )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X  /\  N  e.  ZZ )  ->  ( N  .x.  A )  e.  ( K ` 
 { A } )
 )
 
Theoremelnmz 14491* Elementhood in the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   =>    |-  ( A  e.  N  <->  ( A  e.  X  /\  A. z  e.  X  ( ( A  .+  z
 )  e.  S  <->  ( z  .+  A )  e.  S ) ) )
 
Theoremnmzbi 14492* Defining property of the normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   =>    |-  ( ( A  e.  N  /\  B  e.  X )  ->  ( ( A 
 .+  B )  e.  S  <->  ( B  .+  A )  e.  S ) )
 
Theoremnmzsubg 14493* The normalizer NG(S) of a subset  S of the group is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Grp  ->  N  e.  (SubGrp `  G )
 )
 
Theoremssnmz 14494* A subgroup is a subset of its normalizer. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  C_  N )
 
Theoremisnsg4 14495* A subgroup is normal iff its normalizer is the entire group. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( S  e.  (NrmSGrp `  G ) 
 <->  ( S  e.  (SubGrp `  G )  /\  N  =  X ) )
 
Theoremnmznsg 14496* Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.)
 |-  N  =  { x  e.  X  |  A. y  e.  X  ( ( x 
 .+  y )  e.  S  <->  ( y  .+  x )  e.  S ) }   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  H  =  ( Gs  N )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  e.  (NrmSGrp `  H )
 )
 
Theorem0nsg 14497 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  {  .0.  }  e.  (NrmSGrp `  G ) )
 
Theoremnsgid 14498 The whole group is a normal subgroup of itself. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  e.  (NrmSGrp `  G ) )
 
Theoremreleqg 14499 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  R  =  ( G ~QG  S )   =>    |- 
 Rel  R
 
Theoremeqgfval 14500* Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |- 
 .+  =  ( +g  `  G )   &    |-  R  =  ( G ~QG 
 S )   =>    |-  ( ( G  e.  V  /\  S  C_  X )  ->  R  =  { <. x ,  y >.  |  ( { x ,  y }  C_  X  /\  ( ( N `  x )  .+  y )  e.  S ) }
 )
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