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Theorem List for Metamath Proof Explorer - 14301-14400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsumspl 14301 The primary purpose of the splice construction is to enable local rewrites. Thus, in any monoidal valuation, if a splice does not cause a local change it does not cause a global change. (Contributed by Stefan O'Rear, 23-Aug-2015.)
 |-  B  =  ( Base `  M )   &    |-  ( ph  ->  M  e.  Mnd )   &    |-  ( ph  ->  S  e. Word  B )   &    |-  ( ph  ->  F  e.  ( 0 ... T ) )   &    |-  ( ph  ->  T  e.  ( 0 ... ( # `  S ) ) )   &    |-  ( ph  ->  X  e. Word  B )   &    |-  ( ph  ->  Y  e. Word  B )   &    |-  ( ph  ->  ( M  gsumg 
 X )  =  ( M  gsumg 
 Y ) )   =>    |-  ( ph  ->  ( M  gsumg  ( S splice  <. F ,  T ,  X >. ) )  =  ( M 
 gsumg  ( S splice  <. F ,  T ,  Y >. ) ) )
 
Theoremgsumwmhm 14302 Behavior of homomorphisms on finite monoidal sums. (Contributed by Stefan O'Rear, 27-Aug-2015.)
 |-  B  =  ( Base `  M )   =>    |-  ( ( H  e.  ( M MndHom  N )  /\  W  e. Word  B )  ->  ( H `  ( M 
 gsumg  W ) )  =  ( N  gsumg  ( H  o.  W ) ) )
 
Theoremgsumwspan 14303* The submonoid generated by a set of elements is precisely the set of elements which can be expressed as finite products of the generator. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  B  =  ( Base `  M )   &    |-  K  =  (mrCls `  (SubMnd `  M )
 )   =>    |-  ( ( M  e.  Mnd  /\  G  C_  B )  ->  ( K `  G )  =  ran  (  w  e. Word  G  |->  ( M 
 gsumg  w ) ) )
 
10.1.4  Free monoids
 
Syntaxcfrmd 14304 Extend class definition with the free monoid construction.
 class freeMnd
 
Syntaxcvrmd 14305 Extend class notation with free monoid injection.
 class varFMnd
 
Definitiondf-frmd 14306 Define a free monoid over a set  i of generators, defined as the set of finite strings on  I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- freeMnd  =  ( i  e.  _V  |->  {
 <. ( Base `  ndx ) , Word 
 i >. ,  <. ( +g  ` 
 ndx ) ,  ( concat  |`  (Word  i  X. Word  i )
 ) >. } )
 
Definitiondf-vrmd 14307* Define a free monoid over a set  i of generators, defined as the set of finite strings on  I with the operation of concatenation. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |- varFMnd  =  ( i  e.  _V  |->  ( j  e.  i  |-> 
 <" j "> ) )
 
Theoremfrmdval 14308 Value of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  ( I  e.  V  ->  B  = Word  I )   &    |-  .+  =  ( concat  |`  ( B  X.  B ) )   =>    |-  ( I  e.  V  ->  M  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. } )
 
Theoremfrmdbas 14309 The base set of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   =>    |-  ( I  e.  V  ->  B  = Word  I )
 
Theoremfrmdelbas 14310 An element of the base set of a free monoid is a string on the generators. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   =>    |-  ( X  e.  B  ->  X  e. Word  I )
 
Theoremfrmdplusg 14311 The monoid operation of a free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  .+  =  ( concat  |`  ( B  X.  B ) )
 
Theoremfrmdadd 14312 Value of the monoid operation of the free monoid construction. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  M )   &    |-  .+  =  ( +g  `  M )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( X concat  Y )
 )
 
Theoremvrmdfval 14313* The canonical injection from the generating set  I to the base set of the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  <" j "> ) )
 
Theoremvrmdval 14314 The value of the generating elements of a free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `
  A )  = 
 <" A "> )
 
Theoremvrmdf 14315 The mapping from the index set to the generators is a function into the free monoid. (Contributed by Mario Carneiro, 27-Feb-2016.)
 |-  U  =  (varFMnd `  I )   =>    |-  ( I  e.  V  ->  U : I -->Word  I )
 
Theoremfrmdmnd 14316 A free monoid is a monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  ( I  e.  V  ->  M  e.  Mnd )
 
Theoremfrmd0 14317 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  (/)  =  ( 0g `  M )
 
Theoremfrmdsssubm 14318 The set of words taking values in a subset is a (free) submonoid of the free monoid. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  ( ( I  e.  V  /\  J  C_  I )  -> Word  J  e.  (SubMnd `  M ) )
 
Theoremfrmdgsum 14319 Any word in a free monoid can be expressed as the sum of the singletons composing it. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  W  e. Word  I )  ->  ( M  gsumg  ( U  o.  W ) )  =  W )
 
Theoremfrmdss2 14320 A subset of generators is contained in a submonoid iff the set of words on the generators is in the submonoid. This can be viewed as an elementary way of saying "the monoidal closure of  J is Word  J". (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( I  e.  V  /\  J  C_  I  /\  A  e.  (SubMnd `  M ) )  ->  ( ( U " J )  C_  A  <-> Word  J  C_  A )
 )
 
Theoremfrmdup1 14321* Any assignment of the generators to target elements can be extended (uniquely) to a homomorphism from a free monoid to an arbitrary other monoid. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  A : I --> B )   =>    |-  ( ph  ->  E  e.  ( M MndHom  G ) )
 
Theoremfrmdup2 14322* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  E  =  ( x  e. Word  I  |->  ( G  gsumg  ( A  o.  x ) ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ph  ->  I  e.  X )   &    |-  ( ph  ->  A : I --> B )   &    |-  U  =  (varFMnd `  I )   &    |-  ( ph  ->  Y  e.  I )   =>    |-  ( ph  ->  ( E `  ( U `
  Y ) )  =  ( A `  Y ) )
 
Theoremfrmdup3 14323* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  M  =  (freeMnd `  I
 )   &    |-  B  =  ( Base `  G )   &    |-  U  =  (varFMnd `  I )   =>    |-  ( ( G  e.  Mnd  /\  I  e.  V  /\  A : I --> B ) 
 ->  E! m  e.  ( M MndHom  G ) ( m  o.  U )  =  A )
 
10.2  Groups
 
10.2.1  Definition and basic properties
 
Definitiondf-grp 14324* Define class of all groups. A group is a monoid (df-mnd 14202) whose internal operation is such that every element admits a left inverse (which can be proven to be a two-sided inverse). Thus, a group  G is an algebraic structure formed from a base set of elements (notated  ( Base `  G
) per df-base 13027) and an internal group operation (notated  ( +g  `  G
) per df-plusg 13095). The operation combines any two elements of the group base set and must satisfy the 4 group axioms: closure (the result of the group operation must always be a member of the base set, see grpcl 14330), associativity (so  ( (
a +g  b ) +g  c )  =  ( a +g  ( b +g  c ) ) for any a, b, c, see grpass 14331), identity (there must be an element  e  =  ( 0g `  G
) such that  e +g  a  =  e +g  a  =  a for any a), and inverse (for each element a in the base set, there must be an element  b  =  inv g a in the base set such that  a +g  b  =  b +g  a  =  e). It can be proven that the identity element is unique (grpideu 14333). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 14927). Subgroups can often be formed from groups, see df-subg 14453. An example of an (Abelian) group is the set of complex numbers  CC over the group operation  + (addition), as proven in cnaddablx 14993; an Abelian group is a group as proven in ablgrp 14929. Other structures include groups, including unital rings (df-ring 15175) and fields (df-field 15350). (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |- 
 Grp  =  { g  e.  Mnd  |  A. a  e.  ( Base `  g ) E. m  e.  ( Base `  g ) ( m ( +g  `  g
 ) a )  =  ( 0g `  g
 ) }
 
Definitiondf-minusg 14325* Define inverse of group element. (Contributed by NM, 24-Aug-2011.)
 |- 
 inv g  =  (
 g  e.  _V  |->  ( x  e.  ( Base `  g )  |->  ( iota_ w  e.  ( Base `  g
 ) ( w (
 +g  `  g ) x )  =  ( 0g `  g ) ) ) )
 
Definitiondf-sbg 14326* Define group subtraction (also called division for multiplicative groups). (Contributed by NM, 31-Mar-2014.)
 |-  -g  =  ( g  e.  _V  |->  ( x  e.  ( Base `  g ) ,  y  e.  ( Base `  g )  |->  ( x ( +g  `  g
 ) ( ( inv
 g `  g ) `  y ) ) ) )
 
Definitiondf-mulg 14327* Define the group multiple function, also known as group exponentiation when viewed multiplicatively. (Contributed by Mario Carneiro, 11-Dec-2014.)
 |- .g  =  ( g  e.  _V  |->  ( n  e.  ZZ ,  x  e.  ( Base `  g )  |->  if ( n  =  0 ,  ( 0g `  g ) ,  [_  seq  1 ( ( +g  `  g ) ,  ( NN  X.  { x }
 ) )  /  s ]_ if ( 0  < 
 n ,  ( s `
  n ) ,  ( ( inv g `  g ) `  (
 s `  -u n ) ) ) ) ) )
 
Theoremisgrp 14328* The predicate "is a group." (This theorem demonstrates the use of symbols as variable names, first proposed by FL in 2010.) (Contributed by NM, 17-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  <->  ( G  e.  Mnd  /\  A. a  e.  B  E. m  e.  B  ( m  .+  a )  =  .0.  ) )
 
Theoremgrpmnd 14329 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Grp  ->  G  e.  Mnd )
 
Theoremgrpcl 14330 Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  e.  B )
 
Theoremgrpass 14331 A group operation is associative. (Contributed by NM, 14-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y )  .+  Z )  =  ( X  .+  ( Y  .+  Z ) ) )
 
Theoremgrpinvex 14332* Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  E. y  e.  B  ( y  .+  X )  =  .0.  )
 
Theoremgrpideu 14333* The two-sided identity element of a group is unique. Lemma 2.2.1(a) of [Herstein] p. 55. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 8-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  E! u  e.  B  A. x  e.  B  ( ( u  .+  x )  =  x  /\  ( x  .+  u )  =  x ) )
 
Theoremgrpplusf 14334 The group addition operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  F  =  ( + f `  G )   =>    |-  ( G  e.  Grp  ->  F : ( B  X.  B ) --> B )
 
Theoremgrppropd 14335* If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( K  e.  Grp  <->  L  e.  Grp ) )
 
Theoremgrpprop 14336 If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Grp  <->  L  e.  Grp )
 
Theoremgrppropstr 14337 Generalize a specific 2-element group  L to show that any set  K with the same (relevant) properties is also a group. (Contributed by NM, 28-Oct-2012.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( Base `  K )  =  B   &    |-  ( +g  `  K )  =  .+   &    |-  L  =  { <. ( Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   =>    |-  ( K  e.  Grp  <->  L  e.  Grp )
 
Theoremgrpss 14338 Show that a structure extending a constructed group (e.g. a ring) is also a group. This allows us to prove that a constructed potential ring  R is a group before we know that it is also a ring. (Theorem rnggrp 15181, on the other hand, requires that we know in advance that  R is a ring.) (Contributed by NM, 11-Oct-2013.)
 |-  G  =  { <. (
 Base `  ndx ) ,  B >. ,  <. ( +g  ` 
 ndx ) ,  .+  >. }   &    |-  R  e.  _V   &    |-  G  C_  R   &    |-  Fun  R   =>    |-  ( G  e.  Grp  <->  R  e.  Grp )
 
Theoremisgrpd2e 14339* Deduce a group from its properties. In this version of isgrpd2 14340, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd2 14340* Deduce a group from its properties. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). Note: normally we don't use a  ph antecedent on hypotheses that name structure components, since they can be eliminated with eqid 2253, but we make an exception for theorems such as isgrpd2 14340, ismndd 14231, and islmodd 15468 since theorems using them often rewrite the structure components. (Contributed by NM, 10-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  =  ( 0g `  G ) )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  (
 ( ph  /\  x  e.  B )  ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpde 14341* Deduce a group from its properties. In this version of isgrpd 14342, we don't assume there is an expression for the inverse of  x. (Contributed by NM, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  E. y  e.  B  ( y  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpd 14342* Deduce a group from its properties. Unlike isgrpd2 14340, this one goes straight from the base properties rather than going through  Mnd.  N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 6-Jun-2013.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B  /\  z  e.  B ) )  ->  ( ( x  .+  y ) 
 .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  ( ph  ->  .0. 
 e.  B )   &    |-  (
 ( ph  /\  x  e.  B )  ->  (  .0.  .+  x )  =  x )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  N  e.  B )   &    |-  ( ( ph  /\  x  e.  B )  ->  ( N  .+  x )  =  .0.  )   =>    |-  ( ph  ->  G  e.  Grp )
 
Theoremisgrpi 14343* Properties that determine a group. 
N (negative) is normally dependent on  x i.e. read it as  N ( x ). (Contributed by NM, 3-Sep-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  .0.  e.  B   &    |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )   &    |-  ( x  e.  B  ->  N  e.  B )   &    |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )   =>    |-  G  e.  Grp
 
Theoremisgrpix 14344* Properties that determine a group. Read  N as  N ( x ). Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. (Contributed by NM, 4-Sep-2011.)
 |-  B  e.  _V   &    |-  .+  e.  _V   &    |-  G  =  { <. 1 ,  B >. ,  <. 2 ,  .+  >. }   &    |-  ( ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  e.  B )   &    |-  ( ( x  e.  B  /\  y  e.  B  /\  z  e.  B )  ->  (
 ( x  .+  y
 )  .+  z )  =  ( x  .+  (
 y  .+  z )
 ) )   &    |-  .0.  e.  B   &    |-  ( x  e.  B  ->  (  .0.  .+  x )  =  x )   &    |-  ( x  e.  B  ->  N  e.  B )   &    |-  ( x  e.  B  ->  ( N  .+  x )  =  .0.  )   =>    |-  G  e.  Grp
 
Theoremgrpidcl 14345 The identity element of a group belongs to the group. (Contributed by NM, 27-Aug-2011.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  .0.  e.  B )
 
Theoremgrpbn0 14346 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  =/=  (/) )
 
Theoremgrplid 14347 The identity element of a group is a left identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  (  .0.  .+  X )  =  X )
 
Theoremgrprid 14348 The identity element of a group is a right identity. (Contributed by NM, 18-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .+  .0.  )  =  X )
 
Theoremgrpn0 14349 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  ( G  e.  Grp  ->  G  =/=  (/) )
 
Theoremgrprcan 14350 Right cancellation law for groups. (Contributed by NM, 24-Aug-2011.) (Proof shortened by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Z )  =  ( Y  .+  Z )  <->  X  =  Y ) )
 
Theoremgrpinveu 14351* The left inverse element of a group is unique. Lemma 2.2.1(b) of [Herstein] p. 55. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  E! y  e.  B  ( y  .+  X )  =  .0.  )
 
Theoremgrpid 14352 Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( ( X  .+  X )  =  X  <->  .0. 
 =  X ) )
 
Theoremisgrpid2 14353 Properties showing that an element 
Z is the identity element of a group. (Contributed by NM, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( G  e.  Grp  ->  ( ( Z  e.  B  /\  ( Z  .+  Z )  =  Z ) 
 <->  .0.  =  Z ) )
 
Theoremgrpidd2 14354* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14342. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  .0.  e.  B )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  (  .0.  .+  x )  =  x )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  .0.  =  ( 0g `  G ) )
 
Theoremgrpinvfval 14355* The inverse function of a group. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  N  =  ( x  e.  B  |->  ( iota_ y  e.  B ( y  .+  x )  =  .0.  ) )
 
Theoremgrpinvval 14356* The inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 7-Aug-2013.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  ( X  e.  B  ->  ( N `  X )  =  ( iota_ y  e.  B ( y  .+  X )  =  .0.  ) )
 
Theoremgrpinvfn 14357 Functionality of the group inverse function. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  N  Fn  B
 
Theoremgrpinvfvi 14358 The group inverse function is compatible with identity-function protection. (Contributed by Stefan O'Rear, 21-Mar-2015.)
 |-  N  =  ( inv
 g `  G )   =>    |-  N  =  ( inv g `  (  _I  `  G )
 )
 
Theoremgrpsubfval 14359* Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Stefan O'Rear, 27-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  .-  =  ( x  e.  B ,  y  e.  B  |->  ( x 
 .+  ( I `  y ) ) )
 
Theoremgrpsubval 14360 Group subtraction (division) operation. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y )  =  ( X  .+  ( I `  Y ) ) )
 
Theoremgrpinvf 14361 The group inversion operation is a function on the base set. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  N : B --> B )
 
Theoremgrpinvcl 14362 A group element's inverse is a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  X )  e.  B )
 
Theoremgrplinv 14363 The left inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( ( N `  X )  .+  X )  =  .0.  )
 
Theoremgrprinv 14364 The right inverse of a group element. (Contributed by NM, 24-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .+  ( N `  X ) )  =  .0.  )
 
Theoremgrpinvid1 14365 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <->  ( X  .+  Y )  =  .0.  ) )
 
Theoremgrpinvid2 14366 The inverse of a group element expressed in terms of the identity element. (Contributed by NM, 24-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  (
 ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( N `  X )  =  Y  <->  ( Y  .+  X )  =  .0.  ) )
 
Theoremisgrpinv 14367* Properties showing that a function 
M is the inverse function of a group. (Contributed by NM, 7-Aug-2013.) (Revised by Mario Carneiro, 2-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv
 g `  G )   =>    |-  ( G  e.  Grp  ->  (
 ( M : B --> B  /\  A. x  e.  B  ( ( M `
  x )  .+  x )  =  .0.  ) 
 <->  N  =  M ) )
 
Theoremgrpinvid 14368 The inverse of the identity element of a group. (Contributed by NM, 24-Aug-2011.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  ( N `  .0.  )  =  .0.  )
 
Theoremgrplcan 14369 Left cancellation law for groups. (Contributed by NM, 25-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( Z  .+  X )  =  ( Z  .+  Y )  <->  X  =  Y ) )
 
Theoremgrpinvinv 14370 Double inverse law for groups. Lemma 2.2.1(c) of [Herstein] p. 55. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  ( N `  X ) )  =  X )
 
Theoremgrpinvcnv 14371 The group inverse is its own inverse function. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( G  e.  Grp  ->  `' N  =  N )
 
Theoremgrpinv11 14372 The group inverse is one-to-one. (Contributed by NM, 22-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  =  ( N `  Y )  <->  X  =  Y ) )
 
Theoremgrpinvf1o 14373 The group inverse is a one-to-one onto function. (Contributed by NM, 22-Oct-2014.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  B  =  ( Base `  G )   &    |-  N  =  ( inv g `  G )   &    |-  ( ph  ->  G  e.  Grp )   =>    |-  ( ph  ->  N : B -1-1-onto-> B )
 
Theoremgrpinvnz 14374 The inverse of a nonzero group element is not zero. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv g `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  X  =/=  .0.  )  ->  ( N `  X )  =/=  .0.  )
 
Theoremgrpinvnzcl 14375 The inverse of a nonzero group element is a nonzero group element. (Contributed by Stefan O'Rear, 27-Feb-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  N  =  ( inv g `
  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  ( B  \  {  .0.  } ) )  ->  ( N `  X )  e.  ( B  \  {  .0.  } ) )
 
Theoremgrpsubinv 14376 Subtraction of an inverse. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `
  G )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( N `  Y ) )  =  ( X  .+  Y ) )
 
Theoremgrplmulf1o 14377* Left multiplication by a a group element is a bijection on any group. (Contributed by Mario Carneiro, 17-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  F  =  ( x  e.  B  |->  ( X  .+  x ) )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  F : B -1-1-onto-> B )
 
Theoremgrpinvpropd 14378* If two structures have the same group components (properties), they have the same group inversion function. (Contributed by Mario Carneiro, 27-Nov-2014.) (Revised by Stefan O'Rear, 21-Mar-2015.)
 |-  ( ph  ->  B  =  ( Base `  K )
 )   &    |-  ( ph  ->  B  =  ( Base `  L )
 )   &    |-  ( ( ph  /\  ( x  e.  B  /\  y  e.  B )
 )  ->  ( x ( +g  `  K )
 y )  =  ( x ( +g  `  L ) y ) )   =>    |-  ( ph  ->  ( inv g `
  K )  =  ( inv g `  L ) )
 
Theoremgrpinvadd 14379 The inverse of the group operation reverses the arguments. Lemma 2.2.1(d) of [Herstein] p. 55. (Contributed by NM, 27-Oct-2006.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  Y )  .+  ( N `  X ) ) )
 
Theoremgrpsubf 14380 Functionality of group subtraction. (Contributed by Mario Carneiro, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( G  e.  Grp 
 ->  .-  : ( B  X.  B ) --> B )
 
Theoremgrpsubcl 14381 Closure of group subtraction. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .-  Y )  e.  B )
 
Theoremgrpsubrcan 14382 Right cancellation law for group subtraction. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .-  Z )  =  ( Y  .-  Z )  <->  X  =  Y ) )
 
Theoremgrpinvsub 14383 Inverse of a group subtraction. (Contributed by NM, 9-Sep-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .-  Y ) )  =  ( Y  .-  X ) )
 
Theoremgrpinvval2 14384 A df-neg 8920-like equation for inverse in terms of group subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `  G )   &    |-  .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( N `  X )  =  (  .0.  .-  X ) )
 
Theoremgrpsubid 14385 Subtraction of a group element from itself. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .-  X )  =  .0.  )
 
Theoremgrpsubid1 14386 Subtraction of the identity from a group element. (Contributed by Mario Carneiro, 14-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B ) 
 ->  ( X  .-  .0.  )  =  X )
 
Theoremgrpsubeq0 14387 If the difference between two group elements is zero, they are equal. (subeq0 8953 analog.) (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  =  .0.  <->  X  =  Y ) )
 
Theoremgrpsubadd 14388 Relationship between group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  =  Z  <->  ( Z  .+  Y )  =  X ) )
 
Theoremgrpsubsub 14389 Double group subtraction. (Contributed by NM, 24-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( X  .-  ( Y  .-  Z ) )  =  ( X  .+  ( Z  .-  Y ) ) )
 
Theoremgrpaddsubass 14390 Associative-type law for group subtraction and addition. (Contributed by NM, 16-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .-  Z )  =  ( X  .+  ( Y  .-  Z ) ) )
 
Theoremgrppncan 14391 Cancellation law for subtraction (pncan 8937 analog). (Contributed by NM, 16-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  Y )  .-  Y )  =  X )
 
Theoremgrpnpcan 14392 Cancellation law for subtraction (npcan 8940 analog). . (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .-  Y )  .+  Y )  =  X )
 
Theoremgrpsubsub4 14393 Double group subtraction (subsub4 8960 analog). (Contributed by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  .-  Z )  =  ( X  .-  ( Z  .+  Y ) ) )
 
Theoremgrppnpcan2 14394 Cancellation law for mixed addition and subtraction. (pnpcan2 8967 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Z )  .-  ( Y  .+  Z ) )  =  ( X 
 .-  Y ) )
 
Theoremgrpnpncan 14395 Cancellation law for group subtraction. (npncan 8949 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  .+  ( Y  .-  Z ) )  =  ( X 
 .-  Z ) )
 
Theoremgrpnnncan2 14396 Cancellation law for group subtraction. (nnncan2 8964 analog.) (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .-  Z )  .-  ( Y 
 .-  Z ) )  =  ( X  .-  Y ) )
 
Theoremgrplactfval 14397* The left group action of element  A of group  G. (Contributed by Paul Chapman, 18-Mar-2008.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   =>    |-  ( A  e.  X  ->  ( F `  A )  =  (
 a  e.  X  |->  ( A  .+  a ) ) )
 
Theoremgrplactval 14398* The value of the left group action of element  A of group  G at  B. (Contributed by Paul Chapman, 18-Mar-2008.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( A  e.  X  /\  B  e.  X )  ->  (
 ( F `  A ) `  B )  =  ( A  .+  B ) )
 
Theoremgrplactcnv 14399* The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( ( F `  A ) : X -1-1-onto-> X  /\  `' ( F `  A )  =  ( F `  ( I `  A ) ) ) )
 
Theoremgrplactf1o 14400* The left group action of element  A of group  G maps the underlying set  X of  G one-to-one onto itself. (Contributed by Paul Chapman, 18-Mar-2008.) (Proof shortened by Mario Carneiro, 14-Aug-2015.)
 |-  F  =  ( g  e.  X  |->  ( a  e.  X  |->  ( g 
 .+  a ) ) )   &    |-  X  =  (
 Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Grp  /\  A  e.  X ) 
 ->  ( F `  A ) : X -1-1-onto-> X )
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