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Theorem List for Metamath Proof Explorer - 14601-14700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempwsinvg 14601 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 14602 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 14603 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 14604* 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 14605* 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 14606 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 14607* 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 14608 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 14609 Extend class notation with all subgroups of a group.
 class SubGrp
 
Syntaxcnsg 14610 Extend class notation with all normal subgroups of a group.
 class NrmSGrp
 
Syntaxcqg 14611 Quotient group equivalence class.
 class ~QG
 
Definitiondf-subg 14612* Define a subgroup of a group as a set of elements that is a group in its own right. Equivalently (issubg2 14630), a subgroup is a subset of the group that is closed for the group internal operation (see subgcl 14625), contains the neutral element of the group (see subg0 14621) and contains the inverses for all of its elements (see subginvcl 14624). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |- SubGrp  =  ( w  e.  Grp  |->  { s  e.  ~P ( Base `  w )  |  ( ws  s )  e.  Grp } )
 
Definitiondf-nsg 14613* 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 14614* 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 14615 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 14616 A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  (SubGrp `  G )  ->  S  C_  B )
 
Theoremsubgid 14617 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 14618 A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( S  e.  (SubGrp `  G )  ->  H  e.  Grp )
 
Theoremsubgbas 14619 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 14620 Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  ( S  e.  (SubGrp `  G )  ->  G  e.  Grp )
 
Theoremsubg0 14621 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 14622 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 14623 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 14624 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 14625 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 14626 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 14627 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 14628 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 14629 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 14630* 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 14631* 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 14632* 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 14633 A subgroup is a submonoid. (Contributed by Mario Carneiro, 18-Jun-2015.)
 |-  ( S  e.  (SubGrp `  G )  ->  S  e.  (SubMnd `  G )
 )
 
Theoremsubsubg 14634 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 14635 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 14636 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 14637* 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 14638* 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 14639* 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 14640* 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 14641* Weaken the condition of isnsg 14640 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 14642 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 14643 A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.)
 |-  ( S  e.  (NrmSGrp `  G )  ->  S  e.  (SubGrp `  G )
 )
 
Theoremnsgconj 14644 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 14645* 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 14646 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 14647 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 14648* 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 14649 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 14650* 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 14651* 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 14652* 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 14653* 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 14654* 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 14655* 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 14656 The zero subgroup is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  {  .0.  }  e.  (NrmSGrp `  G ) )
 
Theoremnsgid 14657 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 14658 The left coset equivalence relation is a relation. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  R  =  ( G ~QG  S )   =>    |- 
 Rel  R
 
Theoremeqgfval 14659* 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 ) }
 )
 
Theoremeqgval 14660 Value of the subgroup left coset equivalence relation. (Contributed by Mario Carneiro, 15-Jan-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |- 
 .+  =  ( +g  `  G )   &    |-  R  =  ( G ~QG 
 S )   =>    |-  ( ( G  e.  V  /\  S  C_  X )  ->  ( A R B 
 <->  ( A  e.  X  /\  B  e.  X  /\  ( ( N `  A )  .+  B )  e.  S ) ) )
 
Theoremeqger 14661 The subgroup coset equivalence relation is an equivalence relation. (Contributed by Mario Carneiro, 13-Jan-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   =>    |-  ( Y  e.  (SubGrp `  G )  ->  .~  Er  X )
 
Theoremeqglact 14662* A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  Y  C_  X  /\  A  e.  X )  ->  [ A ]  .~  =  ( ( x  e.  X  |->  ( A 
 .+  x ) )
 " Y ) )
 
Theoremeqgid 14663 The left coset containing the identity is the original subgroup. (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( Y  e.  (SubGrp `  G )  ->  [  .0.  ] 
 .~  =  Y )
 
Theoremeqgen 14664 Each coset is equipotent to the subgroup itself (which is also the coset containing the identity). (Contributed by Mario Carneiro, 20-Sep-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  A  e.  ( X /.  .~  ) )  ->  Y  ~~  A )
 
Theoremeqgcpbl 14665 The subgroup coset equivalence relation is compatible with addition when the subgroup is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( Y  e.  (NrmSGrp `  G )  ->  ( ( A  .~  C  /\  B  .~  D )  ->  ( A  .+  B )  .~  ( C 
 .+  D ) ) )
 
Theoremdivsgrp 14666 If  Y is a normal subgroup of  G, then  H  =  G  /  Y is a group, called the quotient of  G by  Y. (Contributed by Mario Carneiro, 14-Jun-2015.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   =>    |-  ( S  e.  (NrmSGrp `  G )  ->  H  e.  Grp )
 
Theoremdivseccl 14667 Closure of the quotient map for a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  B  =  ( Base `  H )   =>    |-  (
 ( S  e.  (NrmSGrp `  G )  /\  X  e.  V )  ->  [ X ] ( G ~QG  S )  e.  B )
 
Theoremdivsadd 14668 Value of the group operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .+b  =  ( +g  `  H )   =>    |-  (
 ( S  e.  (NrmSGrp `  G )  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ] ( G ~QG  S )  .+b  [ Y ] ( G ~QG  S ) )  =  [
 ( X  .+  Y ) ] ( G ~QG  S )
 )
 
Theoremdivs0 14669 Value of the group identity operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( S  e.  (NrmSGrp `  G )  ->  [  .0.  ] ( G ~QG  S )  =  ( 0g `  H ) )
 
Theoremdivsinv 14670 Value of the group inverse operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  I  =  ( inv g `  G )   &    |-  N  =  ( inv g `  H )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  X  e.  V )  ->  ( N `  [ X ] ( G ~QG  S )
 )  =  [ ( I `  X ) ]
 ( G ~QG  S ) )
 
Theoremdivssub 14671 Value of the group subtraction operation in a quotient group. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   &    |-  V  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( -g `  H )   =>    |-  ( ( S  e.  (NrmSGrp `  G )  /\  X  e.  V  /\  Y  e.  V )  ->  ( [ X ]
 ( G ~QG  S ) N [ Y ] ( G ~QG  S )
 )  =  [ ( X  .-  Y ) ]
 ( G ~QG  S ) )
 
Theoremlagsubg2 14672 Lagrange's theorem for finite groups. Call the "order" of a group the cardinal number of the basic set of the group, and "index of a subgroup" the cardinal number of the set of left (or right, this is the same) cosets of this subgroup. Then the order of the group is the (cardinal) product of the order of any of its subgroups by the index of this subgroup. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  X  =  ( Base `  G )   &    |-  .~  =  ( G ~QG 
 Y )   &    |-  ( ph  ->  Y  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  X  e.  Fin )   =>    |-  ( ph  ->  ( # `  X )  =  ( ( # `  ( X /.  .~  ) )  x.  ( # `  Y ) ) )
 
Theoremlagsubg 14673 Lagrange theorem for Groups: the order of any subgroup of a finite group is a divisor of the order of the group. (Contributed by Mario Carneiro, 11-Jul-2014.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  X  =  ( Base `  G )   =>    |-  ( ( Y  e.  (SubGrp `  G )  /\  X  e.  Fin )  ->  ( # `  Y ) 
 ||  ( # `  X ) )
 
10.2.3  Elementary theory of group homomorphisms
 
Syntaxcghm 14674 Extend class notation with the generator of group hom-sets.
 class  GrpHom
 
Definitiondf-ghm 14675* A homomorphism of groups is a map between two structures which preserves the group operation. Requiring both sides to be groups simplifies most theorems at the cost of complicating the theorem which pushes forward a group structure. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  GrpHom  =  ( s  e. 
 Grp ,  t  e.  Grp  |->  { g  |  [. ( Base `  s )  /  w ]. ( g : w --> ( Base `  t )  /\  A. x  e.  w  A. y  e.  w  (
 g `  ( x ( +g  `  s )
 y ) )  =  ( ( g `  x ) ( +g  `  t ) ( g `
  y ) ) ) } )
 
Theoremreldmghm 14676 Lemma for group homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |- 
 Rel  dom  GrpHom
 
Theoremisghm 14677* Property of being a homomorphism of groups. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  ( F  e.  ( S  GrpHom  T )  <->  ( ( S  e.  Grp  /\  T  e.  Grp )  /\  ( F : X --> Y  /\  A. u  e.  X  A. v  e.  X  ( F `  ( u  .+  v ) )  =  ( ( F `  u )  .+^  ( F `
  v ) ) ) ) )
 
Theoremisghm3 14678* Property of a group homomorphism, similar to ismhm 14411. (Contributed by Mario Carneiro, 7-Mar-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( S  e.  Grp  /\  T  e.  Grp )  ->  ( F  e.  ( S  GrpHom  T )  <->  ( F : X
 --> Y  /\  A. u  e.  X  A. v  e.  X  ( F `  ( u  .+  v ) )  =  ( ( F `  u )  .+^  ( F `  v
 ) ) ) ) )
 
Theoremghmgrp1 14679 A group homomorphism is only defined when the domain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  S  e.  Grp )
 
Theoremghmgrp2 14680 A group homomorphism is only defined when the codomain is a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  T  e.  Grp )
 
Theoremghmf 14681 A group homomorphism is a function. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   =>    |-  ( F  e.  ( S  GrpHom  T ) 
 ->  F : X --> Y )
 
Theoremghmlin 14682 A homomorphism of groups is linear. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  X  =  ( Base `  S )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   =>    |-  (
 ( F  e.  ( S  GrpHom  T )  /\  U  e.  X  /\  V  e.  X )  ->  ( F `  ( U  .+  V ) )  =  ( ( F `
  U )  .+^  ( F `  V ) ) )
 
Theoremghmid 14683 A homomorphism of groups preserves the identity. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  Y  =  ( 0g
 `  S )   &    |-  .0.  =  ( 0g `  T )   =>    |-  ( F  e.  ( S  GrpHom  T )  ->  ( F `  Y )  =  .0.  )
 
Theoremghminv 14684 A homomorphism of groups preserves inverses. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  S )   &    |-  M  =  ( inv g `  S )   &    |-  N  =  ( inv
 g `  T )   =>    |-  (
 ( F  e.  ( S  GrpHom  T )  /\  X  e.  B )  ->  ( F `  ( M `  X ) )  =  ( N `  ( F `  X ) ) )
 
Theoremghmsub 14685 Linearity of subtraction through a group homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  S )   &    |-  .-  =  ( -g `  S )   &    |-  N  =  ( -g `  T )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  B  /\  V  e.  B ) 
 ->  ( F `  ( U  .-  V ) )  =  ( ( F `
  U ) N ( F `  V ) ) )
 
Theoremisghmd 14686* Deduction for a group homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
 |-  X  =  ( Base `  S )   &    |-  Y  =  (
 Base `  T )   &    |-  .+  =  ( +g  `  S )   &    |-  .+^  =  (
 +g  `  T )   &    |-  ( ph  ->  S  e.  Grp )   &    |-  ( ph  ->  T  e.  Grp )   &    |-  ( ph  ->  F : X --> Y )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X )
 )  ->  ( F `  ( x  .+  y
 ) )  =  ( ( F `  x )  .+^  ( F `  y ) ) )   =>    |-  ( ph  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremghmmhm 14687 A group homorphism is a monoid homorphism. (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  F  e.  ( S MndHom  T ) )
 
Theoremghmmhmb 14688 Group homorphisms and monoid homomorphisms coincide. (Thus,  GrpHom is somewhat redundant, although its stronger reverse closure properties are sometimes useful.) (Contributed by Stefan O'Rear, 7-Mar-2015.)
 |-  ( ( S  e.  Grp  /\  T  e.  Grp )  ->  ( S  GrpHom  T )  =  ( S MndHom  T ) )
 
Theoremghmmulg 14689 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  GrpHom  H ) 
 /\  N  e.  ZZ  /\  X  e.  B ) 
 ->  ( F `  ( N  .x.  X ) )  =  ( N  .X.  ( F `  X ) ) )
 
Theoremghmrn 14690 The range of a homomorphism is a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( F  e.  ( S  GrpHom  T )  ->  ran  F  e.  (SubGrp `  T ) )
 
Theorem0ghm 14691 The constant zero linear function between two groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .0.  =  ( 0g `  N )   &    |-  B  =  (
 Base `  M )   =>    |-  ( ( M  e.  Grp  /\  N  e.  Grp )  ->  ( B  X.  {  .0.  } )  e.  ( M  GrpHom  N ) )
 
Theoremidghm 14692 The identity homomorphism on a group. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  (  _I  |`  B )  e.  ( G  GrpHom  G ) )
 
Theoremresghm 14693 Restriction of a homomorphism to a subgroup. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  U  =  ( Ss  X )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  X  e.  (SubGrp `  S ) )  ->  ( F  |`  X )  e.  ( U  GrpHom  T ) )
 
Theoremresghm2 14694 One direction of resghm2b 14695. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( F  e.  ( S  GrpHom  U ) 
 /\  X  e.  (SubGrp `  T ) )  ->  F  e.  ( S  GrpHom  T ) )
 
Theoremresghm2b 14695 Restriction of a the codomain of a homomorphism. (Contributed by Mario Carneiro, 13-Jan-2015.) (Revised by Mario Carneiro, 18-Jun-2015.)
 |-  U  =  ( Ts  X )   =>    |-  ( ( X  e.  (SubGrp `  T )  /\  ran 
 F  C_  X )  ->  ( F  e.  ( S  GrpHom  T )  <->  F  e.  ( S  GrpHom  U ) ) )
 
Theoremghmco 14696 The composition of group homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( F  e.  ( T  GrpHom  U ) 
 /\  G  e.  ( S  GrpHom  T ) ) 
 ->  ( F  o.  G )  e.  ( S  GrpHom  U ) )
 
Theoremghmima 14697 The image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  (SubGrp `  S ) )  ->  ( F " U )  e.  (SubGrp `  T ) )
 
Theoremghmpreima 14698 The inverse image of a subgroup under a homomorphism. (Contributed by Stefan O'Rear, 31-Dec-2014.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  V  e.  (SubGrp `  T ) )  ->  ( `' F " V )  e.  (SubGrp `  S ) )
 
Theoremghmeql 14699 The equalizer of two group homomorphisms is a subgroup. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  G  e.  ( S  GrpHom  T ) ) 
 ->  dom  (  F  i^i  G )  e.  (SubGrp `  S ) )
 
Theoremghmnsgima 14700 The image of a normal subgroup under a surjective homomorphism is normal. (Contributed by Mario Carneiro, 4-Feb-2015.)
 |-  Y  =  ( Base `  T )   =>    |-  ( ( F  e.  ( S  GrpHom  T ) 
 /\  U  e.  (NrmSGrp `  S )  /\  ran  F  =  Y )  ->  ( F " U )  e.  (NrmSGrp `  T ) )
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