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Theorem List for Metamath Proof Explorer - 14401-14500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmhmco 14401 The composition of monoid homomorphisms is a homomorphism. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  ( ( F  e.  ( T MndHom  U )  /\  G  e.  ( S MndHom  T ) )  ->  ( F  o.  G )  e.  ( S MndHom  U )
 )
 
Theoremmhmima 14402 The homomorphic image of a submonoid is a submonoid. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ( F  e.  ( M MndHom  N )  /\  X  e.  (SubMnd `  M ) )  ->  ( F
 " X )  e.  (SubMnd `  N )
 )
 
Theoremmhmeql 14403 The equalizer of two monoid homomorphisms is a submonoid. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( ( F  e.  ( S MndHom  T )  /\  G  e.  ( S MndHom  T ) )  ->  dom  (  F  i^i  G )  e.  (SubMnd `  S )
 )
 
Theoremsubmacs 14404 Submonoids are an algebraic closure system. (Contributed by Stefan O'Rear, 22-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Mnd  ->  (SubMnd `  G )  e.  (ACS `  B )
 )
 
Theoremprdspjmhm 14405* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  S  e.  X )   &    |-  ( ph  ->  R : I --> Mnd )   &    |-  ( ph  ->  A  e.  I
 )   =>    |-  ( ph  ->  ( x  e.  B  |->  ( x `
  A ) )  e.  ( Y MndHom  ( R `  A ) ) )
 
Theorempwspjmhm 14406* A projection from a product of monoids to one of the factors is a monoid homomorphism. (Contributed by Mario Carneiro, 15-Jun-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  Y )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  V  /\  A  e.  I )  ->  ( x  e.  B  |->  ( x `
  A ) )  e.  ( Y MndHom  R ) )
 
Theorempwsdiagmhm 14407* Diagonal monoid homomorphism into a structure power. (Contributed by Stefan O'Rear, 12-Mar-2015.)
 |-  Y  =  ( R 
 ^s  I )   &    |-  B  =  (
 Base `  R )   &    |-  F  =  ( x  e.  B  |->  ( I  X.  { x } ) )   =>    |-  ( ( R  e.  Mnd  /\  I  e.  W )  ->  F  e.  ( R MndHom  Y )
 )
 
Theorempwsco1mhm 14408* Right composition with a function on the index sets yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( R  ^s  B )   &    |-  C  =  (
 Base `  Z )   &    |-  ( ph  ->  R  e.  Mnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  (
 g  e.  C  |->  ( g  o.  F ) )  e.  ( Z MndHom  Y ) )
 
Theorempwsco2mhm 14409* Left composition with a monoid homomorphism yields a monoid homomorphism of structure powers. (Contributed by Mario Carneiro, 12-Jun-2015.)
 |-  Y  =  ( R 
 ^s 
 A )   &    |-  Z  =  ( S  ^s  A )   &    |-  B  =  (
 Base `  Y )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F  e.  ( R MndHom  S )
 )   =>    |-  ( ph  ->  (
 g  e.  B  |->  ( F  o.  g ) )  e.  ( Y MndHom  Z ) )
 
10.1.3  Ordered group sum operation

One important use of words is as formal composites in cases where order is significant, using the general sum operator df-gsum 13367. If order is not significant, it is simpler to use families instead.

 
Theoremgsumvallem1 14410* Lemma for properties of the set of identities of  G. Either  G has no identities, and  O  =  (/), or it has one and this identity is unique and identified by the 
0g function. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .+  y )  =  y  /\  ( y 
 .+  x )  =  y ) }   =>    |-  ( G  e.  V  ->  O  C_  {  .0.  } )
 
Theoremgsumvallem2 14411* Lemma for properties of the set of identities of  G. The set of identities of a monoid is exactly the unique identity element. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .+  y )  =  y  /\  ( y 
 .+  x )  =  y ) }   =>    |-  ( G  e.  Mnd 
 ->  O  =  {  .0.  } )
 
Theoremfisuppfi 14412 A function on a finite set is finitely supported. (Contributed by Mario Carneiro, 20-Jun-2015.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( `' F " C )  e. 
 Fin )
 
Theoremgsumvalx 14413* Expand out the substitutions in df-gsum 13367. (Contributed by Mario Carneiro, 18-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s 
 .+  t )  =  t  /\  ( t 
 .+  s )  =  t ) }   &    |-  ( ph  ->  W  =  ( `' F " ( _V  \  O ) ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  F  e.  X )   &    |-  ( ph  ->  dom  F  =  A )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  ( ZZ>=
 `  m ) ( A  =  ( m
 ... n )  /\  x  =  (  seq  m (  .+  ,  F ) `  n ) ) ) ,  ( iota
 x E. f ( f : ( 1
 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq  1 ( 
 .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
 
Theoremgsumval 14414* Expand out the substitutions in df-gsum 13367. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { s  e.  B  |  A. t  e.  B  ( ( s 
 .+  t )  =  t  /\  ( t 
 .+  s )  =  t ) }   &    |-  ( ph  ->  W  =  ( `' F " ( _V  \  O ) ) )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  if ( ran  F  C_  O ,  .0.  ,  if ( A  e.  ran  ... ,  ( iota x E. m E. n  e.  ( ZZ>=
 `  m ) ( A  =  ( m
 ... n )  /\  x  =  (  seq  m (  .+  ,  F ) `  n ) ) ) ,  ( iota
 x E. f ( f : ( 1
 ... ( # `  W ) ) -1-1-onto-> W  /\  x  =  (  seq  1 ( 
 .+  ,  ( F  o.  f ) ) `  ( # `  W ) ) ) ) ) ) )
 
Theoremgsumpropd 14415 The group sum depends only on the base set and additive operation. Note that for entirely unrestricted functions, there can be dependency on out-of-domain values of the operation, so this is somewhat weaker than mndpropd 14360 etc. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by Mario Carneiro, 18-Sep-2015.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H )
 )   &    |-  ( ph  ->  ( +g  `  G )  =  ( +g  `  H ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumress 14416* The group sum in a substructure is the same as the group sum in the original structure. The only requirement on the substructure is that it contain the identity element; neither  G nor 
H need be groups. (Contributed by Mario Carneiro, 19-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  H  =  ( Gs  S )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  S  C_  B )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  .0.  e.  S )   &    |-  ( ( ph  /\  x  e.  B ) 
 ->  ( (  .0.  .+  x )  =  x  /\  ( x  .+  .0.  )  =  x )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumsubm 14417 Evaluate a group sum in a submonoid. (Contributed by Mario Carneiro, 19-Dec-2014.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  S  e.  (SubMnd `  G ) )   &    |-  ( ph  ->  F : A --> S )   &    |-  H  =  ( Gs  S )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumval1 14418* Value of the group sum operation when every element being summed is an identity of  G. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x 
 .+  y )  =  y  /\  ( y 
 .+  x )  =  y ) }   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  A  e.  W )   &    |-  ( ph  ->  F : A --> O )   =>    |-  ( ph  ->  ( G  gsumg  F )  =  .0.  )
 
Theoremgsum0 14419 Value of the empty group sum. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( G  gsumg  (/) )  =  .0.
 
Theoremgsumz 14420* Value of a group sum over the zero element. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  V ) 
 ->  ( G  gsumg  ( k  e.  A  |->  .0.  ) )  =  .0.  )
 
Theoremgsumval2a 14421* Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  F : ( M ... N ) --> B )   &    |-  O  =  { x  e.  B  |  A. y  e.  B  ( ( x  .+  y )  =  y  /\  ( y  .+  x )  =  y ) }   &    |-  ( ph  ->  -.  ran  F 
 C_  O )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( 
 seq  M (  .+  ,  F ) `  N ) )
 
Theoremgsumval2 14422 Value of the group sum operation over a finite set of sequential integers. (Contributed by Mario Carneiro, 7-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  V )   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( ph  ->  F : ( M ... N ) --> B )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( 
 seq  M (  .+  ,  F ) `  N ) )
 
Theoremgsumwsubmcl 14423 Closure of the composite in any submonoid. (Contributed by Stefan O'Rear, 15-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  ( ( S  e.  (SubMnd `  G )  /\  W  e. Word  S )  ->  ( G  gsumg 
 W )  e.  S )
 
Theoremgsumws1 14424 A singleton composite recovers the initial symbol. (Contributed by Stefan O'Rear, 16-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( S  e.  B  ->  ( G  gsumg 
 <" S "> )  =  S )
 
Theoremgsumwcl 14425 Closure of the composite of a word in a structure  G. (Contributed by Stefan O'Rear, 15-Aug-2015.)
 |-  B  =  ( Base `  G )   =>    |-  ( ( G  e.  Mnd  /\  W  e. Word  B )  ->  ( G  gsumg 
 W )  e.  B )
 
Theoremgsumccat 14426 Homomorphic property of composites. (Contributed by Stefan O'Rear, 16-Aug-2015.) (Revised by Mario Carneiro, 1-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  W  e. Word  B 
 /\  X  e. Word  B )  ->  ( G  gsumg  ( W concat  X ) )  =  ( ( G  gsumg  W ) 
 .+  ( G  gsumg  X ) ) )
 
Theoremgsumws2 14427 Valuation of a pair in a monoid. (Contributed by Stefan O'Rear, 23-Aug-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Mnd  /\  S  e.  B  /\  T  e.  B )  ->  ( G  gsumg  <" S T "> )  =  ( S  .+  T ) )
 
Theoremgsumspl 14428 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 14429 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 14430* 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 14431 Extend class definition with the free monoid construction.
 class freeMnd
 
Syntaxcvrmd 14432 Extend class notation with free monoid injection.
 class varFMnd
 
Definitiondf-frmd 14433 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 14434* 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 14435 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 14436 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 14437 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 14438 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 14439 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 14440* 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 14441 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 14442 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 14443 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 14444 The identity of the free monoid is the empty word. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  (freeMnd `  I
 )   =>    |-  (/)  =  ( 0g `  M )
 
Theoremfrmdsssubm 14445 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 14446 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 14447 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 14448* 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 14449* 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 14450* 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 14451* Define class of all groups. A group is a monoid (df-mnd 14329) 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 13115) and an internal group operation (notated  ( +g  `  G
) per df-plusg 13183). 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 14457), associativity (so  ( (
a +g  b ) +g  c )  =  ( a +g  ( b +g  c ) ) for any a, b, c, see grpass 14458), 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 14460). Groups need not be commutative; a commutative group is an Abelian group (see df-abl 15054). Subgroups can often be formed from groups, see df-subg 14580. An example of an (Abelian) group is the set of complex numbers  CC over the group operation  + (addition), as proven in cnaddablx 15120; an Abelian group is a group as proven in ablgrp 15056. Other structures include groups, including unital rings (df-ring 15302) and fields (df-field 15477). (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 14452* 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 14453* 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 14454* 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 14455* 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 14456 A group is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Grp  ->  G  e.  Mnd )
 
Theoremgrpcl 14457 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 14458 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 14459* 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 14460* 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 14461 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 14462* 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 14463 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 14464 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 14465 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 15308, 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 14466* Deduce a group from its properties. In this version of isgrpd2 14467, 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 14467* 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 2258, but we make an exception for theorems such as isgrpd2 14467, ismndd 14358, and islmodd 15595 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 14468* Deduce a group from its properties. In this version of isgrpd 14469, 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 14469* Deduce a group from its properties. Unlike isgrpd2 14467, 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 14470* 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 14471* 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 14472 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 14473 The base set of a group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  B  =  ( Base `  G )   =>    |-  ( G  e.  Grp  ->  B  =/=  (/) )
 
Theoremgrplid 14474 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 14475 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 14476 A group is not empty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.) (Revised by Mario Carneiro, 2-Dec-2014.)
 |-  ( G  e.  Grp  ->  G  =/=  (/) )
 
Theoremgrprcan 14477 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 14478* 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 14479 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 14480 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 14481* Deduce the identity element of a group from its properties. Useful in conjunction with isgrpd 14469. (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 14482* 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 14483* 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 14484 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 14485 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 14486* 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 14487 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 14488 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 14489 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 14490 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 14491 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 14492 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 14493 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 14494* 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 14495 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 14496 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 14497 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 14498 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 14499 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 14500 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 )
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