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Theorem List for Metamath Proof Explorer - 14901-15000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefgcpbl2 14901* Two extension sequences have related endpoints iff they have the same base. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z
 ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0
 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   &    |-  D  =  ( W  \  U_ x  e.  W  ran  (  T `
  x ) )   &    |-  S  =  ( m  e.  { t  e.  (Word  W 
 \  { (/) } )  |  ( ( t `  0 )  e.  D  /\  A. k  e.  (
 1..^ ( # `  t
 ) ) ( t `
  k )  e. 
 ran  (  T `  ( t `  (
 k  -  1 ) ) ) ) }  |->  ( m `  (
 ( # `  m )  -  1 ) ) )   =>    |-  ( ( A  .~  X  /\  B  .~  Y )  ->  ( A concat  B )  .~  ( X concat  Y ) )
 
Theoremfrgpval 14902 Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  G  =  (freeGrp `  I
 )   &    |-  M  =  (freeMnd `  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( I  e.  V  ->  G  =  ( M  /.s 
 .~  ) )
 
Theoremfrgpcpbl 14903 Compatibility of the group operation with the free group equivalence relation. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  M  =  (freeMnd `  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  M )   =>    |-  (
 ( A  .~  C  /\  B  .~  D ) 
 ->  ( A  .+  B )  .~  ( C  .+  D ) )
 
Theoremfrgp0 14904 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |- 
 .~  =  ( ~FG  `  I
 )   =>    |-  ( I  e.  V  ->  ( G  e.  Grp  /\ 
 [ (/) ]  .~  =  ( 0g `  G ) ) )
 
Theoremfrgpeccl 14905 Closure of the quotient map in a free group. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  G  =  (freeGrp `  I
 )   &    |- 
 .~  =  ( ~FG  `  I
 )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  B  =  (
 Base `  G )   =>    |-  ( X  e.  W  ->  [ X ]  .~  e.  B )
 
Theoremfrgpgrp 14906 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  e.  V  ->  G  e.  Grp )
 
Theoremfrgpadd 14907 Addition in the free group is given by concatenation. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  G  =  (freeGrp `  I )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( A  e.  W  /\  B  e.  W )  ->  ( [ A ]  .~  .+  [ B ]  .~  )  =  [ ( A concat  B ) ]  .~  )
 
Theoremfrgpinv 14908* The inverse of an element of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  G  =  (freeGrp `  I )   &    |-  .~  =  ( ~FG  `  I )   &    |-  N  =  ( inv g `  G )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  ( A  e.  W  ->  ( N `  [ A ]  .~  )  =  [
 ( M  o.  (reverse `  A ) ) ]  .~  )
 
Theoremfrgpmhm 14909* The "natural map" from words of the free monoid to their cosets in the free group is a surjective monoid homomorphism. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |-  M  =  (freeMnd `  ( I  X.  2o ) )   &    |-  W  =  ( Base `  M )   &    |-  G  =  (freeGrp `  I )   &    |-  .~  =  ( ~FG  `  I )   &    |-  F  =  ( x  e.  W  |->  [ x ]  .~  )   =>    |-  ( I  e.  V  ->  F  e.  ( M MndHom  G ) )
 
Theoremvrgpfval 14910* The canonical injection from the generating set  I to the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 .~  =  ( ~FG  `  I
 )   &    |-  U  =  (varFGrp `  I )   =>    |-  ( I  e.  V  ->  U  =  ( j  e.  I  |->  [ <"
 <. j ,  (/) >. "> ] 
 .~  ) )
 
Theoremvrgpval 14911 The value of the generating elements of a free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 .~  =  ( ~FG  `  I
 )   &    |-  U  =  (varFGrp `  I )   =>    |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( U `
  A )  =  [ <" <. A ,  (/)
 >. "> ]  .~  )
 
Theoremvrgpf 14912 The mapping from the index set to the generators is a function into the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 .~  =  ( ~FG  `  I
 )   &    |-  U  =  (varFGrp `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  (
 Base `  G )   =>    |-  ( I  e.  V  ->  U : I
 --> X )
 
Theoremvrgpinv 14913 The inverse of a generating element is represented by  <. A ,  1 >. instead of  <. A ,  0
>.. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- 
 .~  =  ( ~FG  `  I
 )   &    |-  U  =  (varFGrp `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( I  e.  V  /\  A  e.  I )  ->  ( N `
  ( U `  A ) )  =  [ <" <. A ,  1o >. "> ]  .~  )
 
Theoremfrgpuptf 14914* 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, 2-Oct-2015.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   =>    |-  ( ph  ->  T : ( I  X.  2o ) --> B )
 
Theoremfrgpuptinv 14915* 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, 2-Oct-2015.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <.
 y ,  ( 1o  \  z ) >. )   =>    |-  ( ( ph  /\  A  e.  ( I  X.  2o ) ) 
 ->  ( T `  ( M `  A ) )  =  ( N `  ( T `  A ) ) )
 
Theoremfrgpuplem 14916* 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, 2-Oct-2015.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( ph  /\  A  .~  C ) 
 ->  ( H  gsumg  ( T  o.  A ) )  =  ( H  gsumg  ( T  o.  C ) ) )
 
Theoremfrgpupf 14917* 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, 2-Oct-2015.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ran  (  g  e.  W  |->  <. [ g ]  .~  ,  ( H  gsumg  ( T  o.  g ) )
 >. )   =>    |-  ( ph  ->  E : X --> B )
 
Theoremfrgpupval 14918* 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, 2-Oct-2015.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ran  (  g  e.  W  |->  <. [ g ]  .~  ,  ( H  gsumg  ( T  o.  g ) )
 >. )   =>    |-  ( ( ph  /\  A  e.  W )  ->  ( E `  [ A ]  .~  )  =  ( H  gsumg  ( T  o.  A ) ) )
 
Theoremfrgpup1 14919* 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, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ran  (  g  e.  W  |->  <. [ g ]  .~  ,  ( H  gsumg  ( T  o.  g ) )
 >. )   =>    |-  ( ph  ->  E  e.  ( G  GrpHom  H ) )
 
Theoremfrgpup2 14920* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ran  (  g  e.  W  |->  <. [ g ]  .~  ,  ( H  gsumg  ( T  o.  g ) )
 >. )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  A  e.  I )   =>    |-  ( ph  ->  ( E `  ( U `
  A ) )  =  ( F `  A ) )
 
Theoremfrgpup3lem 14921* The evaluation map has the intended behavior on the generators. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  B  =  ( Base `  H )   &    |-  N  =  ( inv g `  H )   &    |-  T  =  ( y  e.  I ,  z  e.  2o  |->  if ( z  =  (/) ,  ( F `  y ) ,  ( N `  ( F `  y ) ) ) )   &    |-  ( ph  ->  H  e.  Grp )   &    |-  ( ph  ->  I  e.  V )   &    |-  ( ph  ->  F : I --> B )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  G  =  (freeGrp `  I )   &    |-  X  =  ( Base `  G )   &    |-  E  =  ran  (  g  e.  W  |->  <. [ g ]  .~  ,  ( H  gsumg  ( T  o.  g ) )
 >. )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  K  e.  ( G  GrpHom  H ) )   &    |-  ( ph  ->  ( K  o.  U )  =  F )   =>    |-  ( ph  ->  K  =  E )
 
Theoremfrgpup3 14922* Universal property of the free monoid by existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  B  =  ( Base `  H )   &    |-  U  =  (varFGrp `  I )   =>    |-  ( ( H  e.  Grp  /\  I  e.  V  /\  F : I --> B ) 
 ->  E! m  e.  ( G  GrpHom  H ) ( m  o.  U )  =  F )
 
Theorem0frgp 14923 The free group on zero generators is trivial. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  (/) )   &    |-  B  =  ( Base `  G )   =>    |-  B  ~~ 
 1o
 
10.3  Abelian groups
 
10.3.1  Definition and basic properties
 
Syntaxccmn 14924 Extend class notation with class of all commutative monoids.
 class CMnd
 
Syntaxcabel 14925 Extend class notation with class of all Abelian groups.
 class  Abel
 
Definitiondf-cmn 14926* Define class of all commutative monoids. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |- CMnd  =  { g  e.  Mnd  | 
 A. a  e.  ( Base `  g ) A. b  e.  ( Base `  g ) ( a ( +g  `  g
 ) b )  =  ( b ( +g  `  g ) a ) }
 
Definitiondf-abl 14927 Define class of all Abelian groups. (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |- 
 Abel  =  ( Grp  i^i CMnd )
 
Theoremisabl 14928 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
 |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  G  e. CMnd ) )
 
Theoremablgrp 14929 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
 |-  ( G  e.  Abel  ->  G  e.  Grp )
 
Theoremablcmn 14930 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Abel  ->  G  e. CMnd )
 
Theoremiscmn 14931* The predicate "is a commutative monoid." (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e. CMnd  <->  ( G  e.  Mnd  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  (
 y  .+  x )
 ) )
 
Theoremisabl2 14932* The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.) (Revised by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  A. x  e.  B  A. y  e.  B  ( x  .+  y )  =  ( y  .+  x ) ) )
 
Theoremcmnpropd 14933* If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-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. CMnd  <->  L  e. CMnd ) )
 
Theoremablpropd 14934* If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
 |-  ( 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.  Abel 
 <->  L  e.  Abel )
 )
 
Theoremablprop 14935 If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)
 |-  ( Base `  K )  =  ( Base `  L )   &    |-  ( +g  `  K )  =  ( +g  `  L )   =>    |-  ( K  e.  Abel  <->  L  e.  Abel )
 
Theoremiscmnd 14936* Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Mnd )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e. CMnd )
 
Theoremisabld 14937* Properties that determine an Abelian group. (Contributed by NM, 6-Aug-2013.)
 |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  .+  =  ( +g  `  G )
 )   &    |-  ( ph  ->  G  e.  Grp )   &    |-  ( ( ph  /\  x  e.  B  /\  y  e.  B )  ->  ( x  .+  y
 )  =  ( y 
 .+  x ) )   =>    |-  ( ph  ->  G  e.  Abel
 )
 
Theoremisabli 14938* Properties that determine an Abelian group. (Contributed by NM, 4-Sep-2011.)
 |-  G  e.  Grp   &    |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  (
 ( x  e.  B  /\  y  e.  B )  ->  ( x  .+  y )  =  (
 y  .+  x )
 )   =>    |-  G  e.  Abel
 
Theoremcmnmnd 14939 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e. CMnd  ->  G  e.  Mnd )
 
Theoremcmncom 14940 An commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremablcom 14941 An Abelian group operation is commutative. (Contributed by NM, 26-Aug-2011.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .+  Y )  =  ( Y  .+  X ) )
 
Theoremcmn32 14942 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( ( X  .+  Y ) 
 .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremcmn4 14943 Commutative/associative law for Abelian groups. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B )  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  .+  ( Z 
 .+  W ) )  =  ( ( X 
 .+  Z )  .+  ( Y  .+  W ) ) )
 
Theoremcmn12 14944 Commutative/associative law for Abelian monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )  ->  ( X 
 .+  ( Y  .+  Z ) )  =  ( Y  .+  ( X  .+  Z ) ) )
 
Theoremabl32 14945 Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .+  Z )  =  ( ( X  .+  Z )  .+  Y ) )
 
Theoremablinvadd 14946 The inverse of an Abelian group operation. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  N  =  ( inv g `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( N `  ( X  .+  Y ) )  =  ( ( N `
  X )  .+  ( N `  Y ) ) )
 
Theoremablsub2inv 14947 Abelian group subtraction of two inverses. (Contributed by Stefan O'Rear, 24-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  N  =  ( inv g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  (
 ( N `  X )  .-  ( N `  Y ) )  =  ( Y  .-  X ) )
 
Theoremablsubadd 14948 Relationship between Abelian group subtraction and addition. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .-  Y )  =  Z  <->  ( Y  .+  Z )  =  X ) )
 
Theoremablsub4 14949 Commutative/associative subtraction law for Abelian groups. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y ) 
 .-  ( Z  .+  W ) )  =  ( ( X  .-  Z )  .+  ( Y 
 .-  W ) ) )
 
Theoremabladdsub4 14950 Abelian group addition/subtraction law. (Contributed by NM, 31-Mar-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( Z  e.  B  /\  W  e.  B ) )  ->  ( ( X  .+  Y )  =  ( Z  .+  W )  <->  ( X  .-  Z )  =  ( W  .-  Y ) ) )
 
Theoremabladdsub 14951 Associative-type law for group subtraction and addition. (Contributed by NM, 19-Apr-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
 )  ->  ( ( X  .+  Y )  .-  Z )  =  (
 ( X  .-  Z )  .+  Y ) )
 
Theoremablpncan2 14952 Cancellation law for subtraction. (Contributed by NM, 2-Oct-2014.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  X  e.  B  /\  Y  e.  B )  ->  ( ( X  .+  Y )  .-  X )  =  Y )
 
Theoremablpncan3 14953 A cancellation law for commutative groups. (Contributed by NM, 23-Mar-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X  .+  ( Y  .-  X ) )  =  Y )
 
Theoremablsubsub 14954 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( Y  .-  Z ) )  =  ( ( X  .-  Y )  .+  Z ) )
 
Theoremablsubsub4 14955 Law for double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( X  .-  ( Y  .+  Z ) ) )
 
Theoremablpnpcan 14956 Cancellation law for mixed addition and subtraction. (pnpcan 8966 analog.) (Contributed by NM, 29-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .+  Y )  .-  ( X  .+  Z ) )  =  ( Y  .-  Z ) )
 
Theoremablnncan 14957 Cancellation law for group division. (nncan 8956 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   =>    |-  ( ph  ->  ( X  .-  ( X  .-  Y ) )  =  Y )
 
Theoremablsub32 14958 Swap the second and third terms in a double subtraction. (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  Z )  =  ( ( X  .-  Z )  .-  Y ) )
 
Theoremablnnncan1 14959 Cancellation law for subtraction. (nnncan1 8963 analog.) (Contributed by NM, 7-Apr-2015.)
 |-  B  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  B )   =>    |-  ( ph  ->  (
 ( X  .-  Y )  .-  ( X  .-  Z ) )  =  ( Z  .-  Y ) )
 
Theoremmulgnn0di 14960 Group multiple of a sum, for nonnegative multiples. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e. CMnd  /\  ( M  e.  NN0  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M 
 .x.  ( X  .+  Y ) )  =  ( ( M  .x.  X )  .+  ( M 
 .x.  Y ) ) )
 
Theoremmulgdi 14961 Group multiple of a sum. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M  .x.  ( X 
 .+  Y ) )  =  ( ( M 
 .x.  X )  .+  ( M  .x.  Y ) ) )
 
Theoremmulgmhm 14962* The map from  x to  n x for a fixed positive integer  n is a monoid homomorphism if the monoid is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e. CMnd  /\  M  e.  NN0 )  ->  ( x  e.  B  |->  ( M  .x.  x ) )  e.  ( G MndHom  G ) )
 
Theoremmulgghm 14963* The map from  x to  n x for a fixed integer  n is a group homomorphism if the group is commutative. (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Abel  /\  M  e.  ZZ )  ->  ( x  e.  B  |->  ( M  .x.  x ) )  e.  ( G 
 GrpHom  G ) )
 
Theoremmulgsubdi 14964 Group multiple of a difference. (Contributed by Mario Carneiro, 13-Dec-2014.)
 |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   &    |-  .-  =  ( -g `  G )   =>    |-  ( ( G  e.  Abel  /\  ( M  e.  ZZ  /\  X  e.  B  /\  Y  e.  B ) )  ->  ( M  .x.  ( X 
 .-  Y ) )  =  ( ( M 
 .x.  X )  .-  ( M  .x.  Y ) ) )
 
Theoreminvghm 14965 The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
 |-  B  =  ( Base `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( G  e.  Abel  <->  I  e.  ( G  GrpHom  G ) )
 
Theoremeqgabl 14966 Value of the subgroup coset equivalence relation on an abelian group. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  X  =  ( Base `  G )   &    |-  .-  =  ( -g `  G )   &    |-  .~  =  ( G ~QG  S )   =>    |-  ( ( G  e.  Abel  /\  S  C_  X )  ->  ( A  .~  B  <->  ( A  e.  X  /\  B  e.  X  /\  ( B  .-  A )  e.  S ) ) )
 
Theoremsubgabl 14967 A subgroup of an abelian group is also abelian. (Contributed by Mario Carneiro, 3-Dec-2014.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) )  ->  H  e.  Abel
 )
 
Theoremsubcmn 14968 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 10-Jan-2015.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. CMnd  /\  H  e.  Mnd )  ->  H  e. CMnd )
 
Theoremsubmcmn 14969 A submonoid of a commutative monoid is also commutative. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  H  =  ( Gs  S )   =>    |-  ( ( G  e. CMnd  /\  S  e.  (SubMnd `  G ) )  ->  H  e. CMnd )
 
Theoremsubmcmn2 14970 A submonoid is commutative iff it is a subset of its own centralizer. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  H  =  ( Gs  S )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( S  e.  (SubMnd `  G )  ->  ( H  e. CMnd  <->  S  C_  ( Z `
  S ) ) )
 
Theoremcntzcmn 14971 The centralizer of any subset in a commutative monoid is the whole monoid. (Contributed by Mario Carneiro, 3-Oct-2015.)
 |-  B  =  ( Base `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( G  e. CMnd  /\  S  C_  B )  ->  ( Z `  S )  =  B )
 
Theoremcntzspan 14972 If the generators commute, the generated monoid is commutative. (Contributed by Mario Carneiro, 25-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  K  =  (mrCls `  (SubMnd `  G )
 )   &    |-  H  =  ( Gs  ( K `  S ) )   =>    |-  ( ( G  e.  Mnd  /\  S  C_  ( Z `  S ) )  ->  H  e. CMnd )
 
Theoremghmplusg 14973 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |- 
 .+  =  ( +g  `  N )   =>    |-  ( ( N  e.  Abel  /\  F  e.  ( M 
 GrpHom  N )  /\  G  e.  ( M  GrpHom  N ) )  ->  ( F  o F  .+  G )  e.  ( M  GrpHom  N ) )
 
Theoremablnsg 14974 Every subgroup of an abelian group is normal. (Contributed by Mario Carneiro, 14-Jun-2015.)
 |-  ( G  e.  Abel  ->  (NrmSGrp `  G )  =  (SubGrp `  G )
 )
 
Theoremodadd1 14975 The order of a product in an abelian group divides the LCM of the orders of the factors. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  ( ( O `  ( A  .+  B ) )  x.  ( ( O `  A ) 
 gcd  ( O `  B ) ) ) 
 ||  ( ( O `
  A )  x.  ( O `  B ) ) )
 
Theoremodadd2 14976 The order of a product in an abelian group is divisible by the LCM of the orders of the factors divided by the GCD. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  ( ( O `  A )  x.  ( O `  B ) ) 
 ||  ( ( O `
  ( A  .+  B ) )  x.  ( ( ( O `
  A )  gcd  ( O `  B ) ) ^ 2 ) ) )
 
Theoremodadd 14977 The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   &    |-  X  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   =>    |-  (
 ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `
  B ) )  =  1 )  ->  ( O `  ( A 
 .+  B ) )  =  ( ( O `
  A )  x.  ( O `  B ) ) )
 
Theoremgex2abl 14978 A group with exponent 2 (or 1) is abelian. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   =>    |-  ( ( G  e.  Grp  /\  E  ||  2 )  ->  G  e.  Abel )
 
Theoremgexexlem 14979* Lemma for gexex 14980. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   &    |-  ( ph  ->  G  e.  Abel
 )   &    |-  ( ph  ->  E  e.  NN )   &    |-  ( ph  ->  A  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  ( O `  y )  <_  ( O `  A ) )   =>    |-  ( ph  ->  ( O `  A )  =  E )
 
Theoremgexex 14980* In an abelian group with finite exponent, there is an element in the group with order equal to the exponent. In other words, all orders of elements divide the largest order of an element of the group. This fails if  E  =  0, for example in an infinite p-group, where there are elements of arbitrarily large orders (so  E is zero) but no elements of infinite order. (Contributed by Mario Carneiro, 24-Apr-2016.)
 |-  X  =  ( Base `  G )   &    |-  E  =  (gEx `  G )   &    |-  O  =  ( od `  G )   =>    |-  ( ( G  e.  Abel  /\  E  e.  NN )  ->  E. x  e.  X  ( O `  x )  =  E )
 
Theoremtorsubg 14981 The set of all elements of finite order forms a subgroup of any abelian group, called the torsion subgroup. (Contributed by Mario Carneiro, 20-Oct-2015.)
 |-  O  =  ( od
 `  G )   =>    |-  ( G  e.  Abel 
 ->  ( `' O " NN )  e.  (SubGrp `  G ) )
 
Theoremoddvdssubg 14982* The set of all elements whose order divides a fixed integer is a subgroup of any abelian group. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  O  =  ( od
 `  G )   &    |-  B  =  ( Base `  G )   =>    |-  (
 ( G  e.  Abel  /\  N  e.  ZZ )  ->  { x  e.  B  |  ( O `  x )  ||  N }  e.  (SubGrp `  G ) )
 
Theoremlsmcomx 14983 Subgroup sum commutes (extended domain version). (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  C_  B  /\  U  C_  B )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremablcntzd 14984 All subgroups in an abelian group commute. (Contributed by Mario Carneiro, 19-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  G  e.  Abel )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   =>    |-  ( ph  ->  T  C_  ( Z `  U ) )
 
Theoremlsmcom 14985 Subgroup sum commutes. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremlsmsubg2 14986 The sum of two subgroups is a subgroup. (Contributed by NM, 4-Feb-2014.) (Proof shortened by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  .(+)  U )  e.  (SubGrp `  G ) )
 
Theoremlsm4 14987 Commutative/associative law for subgroup sum. (Contributed by NM, 26-Sep-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( G  e.  Abel  /\  ( Q  e.  (SubGrp `  G )  /\  R  e.  (SubGrp `  G )
 )  /\  ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) ) )  ->  ( ( Q  .(+)  R )  .(+)  ( T  .(+)  U ) )  =  ( ( Q  .(+)  T )  .(+)  ( R  .(+)  U ) ) )
 
Theoremprdscmnd 14988 The product of a family of commutative monoids is commutative. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I -->CMnd )   =>    |-  ( ph  ->  Y  e. CMnd )
 
Theoremprdsabld 14989 The product of a family of Abelian groups is an Abelian group. (Contributed by Stefan O'Rear, 10-Jan-2015.)
 |-  Y  =  ( S
 X_s
 R )   &    |-  ( ph  ->  I  e.  W )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  R : I --> Abel )   =>    |-  ( ph  ->  Y  e.  Abel )
 
Theorempwscmn 14990 The structure power on a commutative monoid is commutative. (Contributed by Mario Carneiro, 11-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e. CMnd  /\  I  e.  V ) 
 ->  Y  e. CMnd )
 
Theorempwsabl 14991 The structure power on an Abelian group is Abelian. (Contributed by Mario Carneiro, 21-Jan-2015.)
 |-  Y  =  ( R 
 ^s  I )   =>    |-  ( ( R  e.  Abel  /\  I  e.  V )  ->  Y  e.  Abel )
 
Theoremdivsabl 14992 If  Y is a subgroup of the abelian group  G, then  H  =  G  /  Y is an abelian group. (Contributed by Mario Carneiro, 26-Apr-2016.)
 |-  H  =  ( G 
 /.s 
 ( G ~QG  S ) )   =>    |-  ( ( G  e.  Abel  /\  S  e.  (SubGrp `  G ) ) 
 ->  H  e.  Abel )
 
Theoremcnaddablx 14993 The complex numbers are an Abelian group under addition. This version of cnaddabl 14994 shows the explicit structure "scaffold" we chose for the definition for Abelian groups. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use; use cnaddabl 14994 instead. (Contributed by NM, 18-Oct-2012.)
 |-  G  =  { <. 1 ,  CC >. ,  <. 2 ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremcnaddabl 14994 The complex numbers are an Abelian group under addition. This version of cnaddablx 14993 hides the explicit structure indices i.e. is "scaffold-independent". Note that the proof also does not reference explicit structure indices. The actual structure is dependent on how  Base and  +g is defined. This theorem should not be referenced in any proof. For the group/ring properties of the complex numbers, see cnrng 16228. (Contributed by NM, 20-Oct-2012.) (New usage is discouraged.)
 |-  G  =  { <. (
 Base `  ndx ) ,  CC >. ,  <. ( +g  ` 
 ndx ) ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremzaddablx 14995 The integers are an Abelian group under addition. Note: This theorem has hard-coded structure indices for demonstration purposes. It is not intended for general use. Use zsubrg 16257 instead. (Contributed by NM, 4-Sep-2011.)
 |-  G  =  { <. 1 ,  ZZ >. ,  <. 2 ,  +  >. }   =>    |-  G  e.  Abel
 
Theoremfrgpnabllem1 14996* Lemma for frgpnabl 14998. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  G )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <.
 y ,  ( 1o  \  z ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  (
 0 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   &    |-  D  =  ( W  \  U_ x  e.  W  ran  (  T `
  x ) )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  A  e.  I
 )   &    |-  ( ph  ->  B  e.  I )   =>    |-  ( ph  ->  <" <. A ,  (/) >. <. B ,  (/) >. ">  e.  ( D  i^i  (
 ( U `  A )  .+  ( U `  B ) ) ) )
 
Theoremfrgpnabllem2 14997* Lemma for frgpnabl 14998. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   &    |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   &    |-  .+  =  ( +g  `  G )   &    |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <.
 y ,  ( 1o  \  z ) >. )   &    |-  T  =  ( v  e.  W  |->  ( n  e.  (
 0 ... ( # `  v
 ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. ) ) )   &    |-  D  =  ( W  \  U_ x  e.  W  ran  (  T `
  x ) )   &    |-  U  =  (varFGrp `  I )   &    |-  ( ph  ->  I  e.  _V )   &    |-  ( ph  ->  A  e.  I
 )   &    |-  ( ph  ->  B  e.  I )   &    |-  ( ph  ->  ( ( U `  A )  .+  ( U `  B ) )  =  ( ( U `  B )  .+  ( U `
  A ) ) )   =>    |-  ( ph  ->  A  =  B )
 
Theoremfrgpnabl 14998 The free group on two or more generators is not abelian. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( 1o  ~<  I  ->  -.  G  e.  Abel )
 
10.3.2  Cyclic groups
 
Syntaxccyg 14999 Cyclic group.
 class CycGrp
 
Definitiondf-cyg 15000* Define a cyclic group, which is a group with an element  x, called the generator of the group, such that all elements in the group are multiples of  x. A generator is usually not unique. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- CycGrp  =  { g  e.  Grp  | 
 E. x  e.  ( Base `  g ) ran  (  n  e.  ZZ  |->  ( n (.g `  g ) x ) )  =  (
 Base `  g ) }
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