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Theorem List for Metamath Proof Explorer - 15001-15100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempj2f 15001 The right projection function maps a direct subspace sum onto the right factor. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( U P T ) : ( T  .(+)  U ) --> U )
 
Theorempj1id 15002 Any element of a direct subspace sum can be decomposed into projections onto the left and right factors. (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  ( T 
 .(+)  U ) )  ->  X  =  ( (
 ( T P U ) `  X )  .+  ( ( U P T ) `  X ) ) )
 
Theorempj1eq 15003 Any element of a direct subspace sum can be decomposed uniquely into projections onto the left and right factors. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   &    |-  ( ph  ->  X  e.  ( T  .(+)  U ) )   &    |-  ( ph  ->  B  e.  T )   &    |-  ( ph  ->  C  e.  U )   =>    |-  ( ph  ->  ( X  =  ( B 
 .+  C )  <->  ( ( ( T P U ) `
  X )  =  B  /\  ( ( U P T ) `
  X )  =  C ) ) )
 
Theorempj1lid 15004 The left projection function is the identity on the left subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  T ) 
 ->  ( ( T P U ) `  X )  =  X )
 
Theorempj1rid 15005 The left projection function is the zero operator on the right subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ( ph  /\  X  e.  U ) 
 ->  ( ( T P U ) `  X )  =  .0.  )
 
Theorempj1ghm 15006 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U )
 )  GrpHom  G ) )
 
Theorempj1ghm2 15007 The left projection function is a group homomorphism. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |- 
 .+  =  ( +g  `  G )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  P  =  ( proj 1 `
  G )   =>    |-  ( ph  ->  ( T P U )  e.  ( ( Gs  ( T  .(+)  U )
 )  GrpHom  ( Gs  T ) ) )
 
Theoremlsmhash 15008 The order of the direct product of groups. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  }
 )   &    |-  ( ph  ->  T  C_  ( Z `  U ) )   &    |-  ( ph  ->  T  e.  Fin )   &    |-  ( ph  ->  U  e.  Fin )   =>    |-  ( ph  ->  ( # `
  ( T  .(+)  U ) )  =  ( ( # `  T )  x.  ( # `  U ) ) )
 
10.2.11  Free groups
 
Syntaxcefg 15009 Extend class notation with the free group equivalence relation.
 class ~FG
 
Syntaxcfrgp 15010 Extend class notation with the free group construction.
 class freeGrp
 
Syntaxcvrgp 15011 Extend class notation with free group injection.
 class varFGrp
 
Definitiondf-efg 15012* Define the free group equivalence relation, which is the smallest equivalence relation  ~~ such that for any words 
A ,  B and formal symbol  x with inverse  inv g x,  A B  ~~  A x ( inv g
x ) B. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |- ~FG  =  ( i  e.  _V  |->  |^|
 { r  |  ( r  Er Word  ( i  X.  2o )  /\  A. x  e. Word  ( i  X.  2o ) A. n  e.  ( 0 ... ( # `
  x ) )
 A. y  e.  i  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >.
 <. y ,  ( 1o  \  z ) >. "> >.
 ) ) } )
 
Definitiondf-frgp 15013 Define the free group on a set  I of generators, defined as the quotient of the free monoid on  I  X.  2o (representing the generator elements and their formal inverses) by the free group equivalence relation df-efg 15012. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
 ) ) )
 
Definitiondf-vrgp 15014* Define the canonical injection from the generating set  I into the base set of the free group. (Contributed by Mario Carneiro, 2-Oct-2015.)
 |- varFGrp  =  ( i  e.  _V  |->  ( j  e.  i  |->  [ <" <. j ,  (/) >. "> ] ( ~FG  `  i
 ) ) )
 
Theoremefgmval 15015* Value of the formal inverse operation for the generating set of a free group. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  ( ( A  e.  I  /\  B  e.  2o )  ->  ( A M B )  =  <. A ,  ( 1o  \  B ) >. )
 
Theoremefgmf 15016* The formal inverse operation is an endofunction on the generating set. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  M : ( I  X.  2o ) --> ( I  X.  2o )
 
Theoremefgmnvl 15017* The inversion function on the generators is an involution. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o  \  z ) >. )   =>    |-  ( A  e.  ( I  X.  2o )  ->  ( M `  ( M `
  A ) )  =  A )
 
Theoremefgrcl 15018 Lemma for efgval 15020. (Contributed by Mario Carneiro, 1-Oct-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   =>    |-  ( A  e.  W  ->  ( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
 
Theoremefglem 15019* Lemma for efgval 15020. (Contributed by Mario Carneiro, 27-Sep-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   =>    |- 
 E. r ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0 ... ( # `
  x ) )
 A. y  e.  I  A. z  e.  2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >.
 <. y ,  ( 1o  \  z ) >. "> >.
 ) )
 
Theoremefgval 15020* Value of the free group construction. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |- 
 .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  A. n  e.  ( 0
 ... ( # `  x ) ) A. y  e.  I  A. z  e. 
 2o  x r ( x splice  <. n ,  n ,  <" <. y ,  z >. <. y ,  ( 1o  \  z ) >. "> >. ) ) }
 
Theoremefger 15021 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |- 
 .~  Er  W
 
Theoremefgi 15022 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( ( A  e.  W  /\  N  e.  ( 0 ... ( # `
  A ) ) )  /\  ( J  e.  I  /\  K  e.  2o ) )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  K >. <. J ,  ( 1o  \  K )
 >. "> >. ) )
 
Theoremefgi0 15023 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( A  e.  W  /\  N  e.  (
 0 ... ( # `  A ) )  /\  J  e.  I )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  (/) >. <. J ,  1o >. "> >. ) )
 
Theoremefgi1 15024 Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-2015.) (Revised by Mario Carneiro, 27-Feb-2016.)
 |-  W  =  (  _I  ` Word  ( I  X.  2o ) )   &    |-  .~  =  ( ~FG  `  I )   =>    |-  ( ( A  e.  W  /\  N  e.  (
 0 ... ( # `  A ) )  /\  J  e.  I )  ->  A  .~  ( A splice  <. N ,  N ,  <" <. J ,  1o >. <. J ,  (/)
 >. "> >. ) )
 
Theoremefgtf 15025* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-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 ) "> >. ) ) )   =>    |-  ( X  e.  W  ->  ( ( T `  X )  =  (
 a  e.  ( 0
 ... ( # `  X ) ) ,  b  e.  ( I  X.  2o )  |->  ( X splice  <. a ,  a ,  <" b
 ( M `  b
 ) "> >. ) ) 
 /\  ( T `  X ) : ( ( 0 ... ( # `
  X ) )  X.  ( I  X.  2o ) ) --> W ) )
 
Theoremefgtval 15026* Value of the extension function, which maps a word (a representation of the group element as a sequence of elements and their inverses) to its direct extensions, defined as the original representation with an element and its inverse inserted somewhere in the string. (Contributed by Mario Carneiro, 29-Sep-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 ) "> >. ) ) )   =>    |-  ( ( X  e.  W  /\  N  e.  (
 0 ... ( # `  X ) )  /\  A  e.  ( I  X.  2o )
 )  ->  ( N ( T `  X ) A )  =  ( X splice  <. N ,  N ,  <" A ( M `  A ) "> >. ) )
 
Theoremefgval2 15027* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-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 ) "> >. ) ) )   =>    |- 
 .~  =  |^| { r  |  ( r  Er  W  /\  A. x  e.  W  ran  (  T `  x )  C_  [ x ]
 r ) }
 
Theoremefgi2 15028* Value of the free group construction. (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 ) "> >. ) ) )   =>    |-  ( ( A  e.  W  /\  B  e.  ran  (  T `  A ) )  ->  A  .~  B )
 
Theoremefgtlen 15029* Value of the free group construction. (Contributed by Mario Carneiro, 27-Sep-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 ) "> >. ) ) )   =>    |-  ( ( X  e.  W  /\  A  e.  ran  (  T `  X ) )  ->  ( # `  A )  =  ( ( # `
  X )  +  2 ) )
 
Theoremefginvrel2 15030* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (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 ) "> >. ) ) )   =>    |-  ( A  e.  W  ->  ( A concat  ( M  o.  (reverse `  A )
 ) )  .~  (/) )
 
Theoremefginvrel1 15031* The inverse of the reverse of a word composed with the word relates to the identity. (This provides an explicit expression for the representation of the group inverse, given a representative of the free group equivalence class.) (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 ) "> >. ) ) )   =>    |-  ( A  e.  W  ->  ( ( M  o.  (reverse `  A ) ) concat  A )  .~  (/) )
 
Theoremefgsf 15032* Value of the auxiliary function  S defining a sequence of extensions starting at some irreducible word. (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 ) ) )   =>    |-  S : { t  e.  (Word  W  \  { (/)
 } )  |  ( ( t `  0
 )  e.  D  /\  A. k  e.  ( 1..^ ( # `  t
 ) ) ( t `
  k )  e. 
 ran  (  T `  ( t `  (
 k  -  1 ) ) ) ) } --> W
 
Theoremefgsdm 15033* Elementhood in the domain of  S, the set of sequences of extensions starting at an irreducible word. (Contributed by Mario Carneiro, 27-Sep-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 ) ) )   =>    |-  ( F  e.  dom  S  <-> 
 ( F  e.  (Word  W 
 \  { (/) } )  /\  ( F `  0
 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `
  i )  e. 
 ran  (  T `  ( F `  ( i  -  1 ) ) ) ) )
 
Theoremefgsval 15034* Value of the auxiliary function  S defining a sequence of extensions (Contributed by Mario Carneiro, 27-Sep-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 ) ) )   =>    |-  ( F  e.  dom  S 
 ->  ( S `  F )  =  ( F `  ( ( # `  F )  -  1 ) ) )
 
Theoremefgsdmi 15035* Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-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 ) ) )   =>    |-  ( ( F  e.  dom 
 S  /\  ( ( # `
  F )  -  1 )  e.  NN )  ->  ( S `  F )  e.  ran  (  T `  ( F `
  ( ( ( # `  F )  -  1 )  -  1
 ) ) ) )
 
Theoremefgsval2 15036* Value of the auxiliary function  S defining a sequence of extensions (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  e. Word  W 
 /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  B )
 
Theoremefgsrel 15037* The start and end of any extension sequence are related (i.e. evaluate to the same element of the quotient group to be created). (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 ) ) )   =>    |-  ( F  e.  dom  S 
 ->  ( F `  0
 )  .~  ( S `  F ) )
 
Theoremefgs1 15038* A singleton of an irreducible word is an extension sequence. (Contributed by Mario Carneiro, 27-Sep-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  e.  D  -> 
 <" A ">  e. 
 dom  S )
 
Theoremefgs1b 15039* Every extension sequence ending in an irreducible word is trivial. (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  e.  dom  S 
 ->  ( ( S `  A )  e.  D  <->  ( # `  A )  =  1 ) )
 
Theoremefgsp1 15040* If  F is an extension sequence and  A is an extension of the last element of  F, then  F  +  <" A "> is an extension sequence. (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 ) ) )   =>    |-  ( ( F  e.  dom 
 S  /\  A  e.  ran  (  T `  ( S `  F ) ) )  ->  ( F concat  <" A "> )  e.  dom  S )
 
Theoremefgsres 15041* An initial segment of an extension sequence is an extension sequence. (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 ) ) )   =>    |-  ( ( F  e.  dom 
 S  /\  N  e.  ( 1 ... ( # `
  F ) ) )  ->  ( F  |`  ( 0..^ N ) )  e.  dom  S )
 
Theoremefgsfo 15042* For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indexes somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-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 ) ) )   =>    |-  S : dom  S -onto-> W
 
Theoremefgredlema 15043* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the ending representation. (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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   =>    |-  ( ph  ->  (
 ( ( # `  A )  -  1 )  e. 
 NN  /\  ( ( # `
  B )  -  1 )  e.  NN ) )
 
Theoremefgredlemf 15044* Lemma for efgredleme 15046. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   =>    |-  ( ph  ->  (
 ( A `  K )  e.  W  /\  ( B `  L )  e.  W ) )
 
Theoremefgredlemg 15045* Lemma for efgred 15051. (Contributed by Mario Carneiro, 4-Jun-2016.)
 |-  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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   &    |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K ) ) ) )   &    |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L ) ) ) )   &    |-  ( ph  ->  U  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  V  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  ( S `  A )  =  ( P ( T `  ( A `
  K ) ) U ) )   &    |-  ( ph  ->  ( S `  B )  =  ( Q ( T `  ( B `  L ) ) V ) )   =>    |-  ( ph  ->  ( # `  ( A `  K ) )  =  ( # `  ( B `  L ) ) )
 
Theoremefgredleme 15046* Lemma for efgred 15051. (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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   &    |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K ) ) ) )   &    |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L ) ) ) )   &    |-  ( ph  ->  U  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  V  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  ( S `  A )  =  ( P ( T `  ( A `
  K ) ) U ) )   &    |-  ( ph  ->  ( S `  B )  =  ( Q ( T `  ( B `  L ) ) V ) )   &    |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )   &    |-  ( ph  ->  P  e.  ( ZZ>= `  ( Q  +  2 )
 ) )   &    |-  ( ph  ->  C  e.  dom  S )   &    |-  ( ph  ->  ( S `  C )  =  (
 ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K ) )
 >. ) ) )   =>    |-  ( ph  ->  ( ( A `  K )  e.  ran  (  T `
  ( S `  C ) )  /\  ( B `  L )  e.  ran  (  T `  ( S `  C ) ) ) )
 
Theoremefgredlemd 15047* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the ending representation. (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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   &    |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K ) ) ) )   &    |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L ) ) ) )   &    |-  ( ph  ->  U  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  V  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  ( S `  A )  =  ( P ( T `  ( A `
  K ) ) U ) )   &    |-  ( ph  ->  ( S `  B )  =  ( Q ( T `  ( B `  L ) ) V ) )   &    |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )   &    |-  ( ph  ->  P  e.  ( ZZ>= `  ( Q  +  2 )
 ) )   &    |-  ( ph  ->  C  e.  dom  S )   &    |-  ( ph  ->  ( S `  C )  =  (
 ( ( B `  L ) substr  <. 0 ,  Q >. ) concat  ( ( A `  K ) substr  <. ( Q  +  2 ) ,  ( # `  ( A `  K ) )
 >. ) ) )   =>    |-  ( ph  ->  ( A `  0 )  =  ( B `  0 ) )
 
Theoremefgredlemc 15048* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the ending representation. (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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   &    |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K ) ) ) )   &    |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L ) ) ) )   &    |-  ( ph  ->  U  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  V  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  ( S `  A )  =  ( P ( T `  ( A `
  K ) ) U ) )   &    |-  ( ph  ->  ( S `  B )  =  ( Q ( T `  ( B `  L ) ) V ) )   &    |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )   =>    |-  ( ph  ->  ( P  e.  ( ZZ>= `  Q )  ->  ( A `
  0 )  =  ( B `  0
 ) ) )
 
Theoremefgredlemb 15049* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   &    |-  K  =  ( ( ( # `  A )  -  1 )  -  1 )   &    |-  L  =  ( ( ( # `  B )  -  1 )  -  1 )   &    |-  ( ph  ->  P  e.  ( 0 ... ( # `  ( A `  K ) ) ) )   &    |-  ( ph  ->  Q  e.  ( 0 ... ( # `  ( B `  L ) ) ) )   &    |-  ( ph  ->  U  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  V  e.  ( I  X.  2o ) )   &    |-  ( ph  ->  ( S `  A )  =  ( P ( T `  ( A `
  K ) ) U ) )   &    |-  ( ph  ->  ( S `  B )  =  ( Q ( T `  ( B `  L ) ) V ) )   &    |-  ( ph  ->  -.  ( A `  K )  =  ( B `  L ) )   =>    |- 
 -.  ph
 
Theoremefgredlem 15050* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the ending representation. (Contributed by Mario Carneiro, 30-Sep-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 ) ) )   &    |-  ( ph  ->  A. a  e.  dom  S A. b  e.  dom  S ( ( # `  ( S `  a ) )  <  ( # `  ( S `  A ) ) 
 ->  ( ( S `  a )  =  ( S `  b )  ->  ( a `  0
 )  =  ( b `
  0 ) ) ) )   &    |-  ( ph  ->  A  e.  dom  S )   &    |-  ( ph  ->  B  e.  dom  S )   &    |-  ( ph  ->  ( S `  A )  =  ( S `  B ) )   &    |-  ( ph  ->  -.  ( A `  0 )  =  ( B `  0 ) )   =>    |- 
 -.  ph
 
Theoremefgred 15051* The reduced word that forms the base of the sequence in efgsval 15034 is uniquely determined, given the terminal point. (Contributed by Mario Carneiro, 28-Sep-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  e.  dom 
 S  /\  B  e.  dom 
 S  /\  ( S `  A )  =  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0 ) )
 
Theoremefgrelexlema 15052* If two words  A ,  B are related under the free group equivalence, then there exist two extension sequences  a ,  b such that  a ends at  A,  b ends at  B, and  a and  B have the same starting point. (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 ) ) )   &    |-  L  =  { <. i ,  j >.  | 
 E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } )
 ( c `  0
 )  =  ( d `
  0 ) }   =>    |-  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b `  0 ) )
 
Theoremefgrelexlemb 15053* If two words  A ,  B are related under the free group equivalence, then there exist two extension sequences  a ,  b such that  a ends at  A,  b ends at  B, and  a and  B have the same starting point. (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 ) ) )   &    |-  L  =  { <. i ,  j >.  | 
 E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } )
 ( c `  0
 )  =  ( d `
  0 ) }   =>    |-  .~  C_  L
 
Theoremefgrelex 15054* If two words  A ,  B are related under the free group equivalence, then there exist two extension sequences  a ,  b such that  a ends at  A,  b ends at  B, and  a and  B have the same starting point. (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  .~  B  ->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b `  0 ) )
 
Theoremefgredeu 15055* There is a unique reduced word equivalent to a given word. (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  e.  W  ->  E! d  e.  D  d  .~  A )
 
Theoremefgred2 15056* 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  e.  dom 
 S  /\  B  e.  dom 
 S )  ->  (
 ( S `  A )  .~  ( S `  B )  <->  ( A `  0 )  =  ( B `  0 ) ) )
 
Theoremefgcpbllema 15057* Lemma for efgrelex 15054. Define an auxiliary equivalence relation  L such that  A L B if there are sequences from  A to  B passing through the same reduced word. (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 ) ) )   &    |-  L  =  { <. i ,  j >.  |  ( { i ,  j }  C_  W  /\  ( ( A concat  i
 ) concat  B )  .~  (
 ( A concat  j ) concat  B ) ) }   =>    |-  ( X L Y 
 <->  ( X  e.  W  /\  Y  e.  W  /\  ( ( A concat  X ) concat  B )  .~  (
 ( A concat  Y ) concat  B ) ) )
 
Theoremefgcpbllemb 15058* Lemma for efgrelex 15054. Show that  L is an equivalence relation containing all direct extensions of a word, so is closed under  .~. (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 ) ) )   &    |-  L  =  { <. i ,  j >.  |  ( { i ,  j }  C_  W  /\  ( ( A concat  i
 ) concat  B )  .~  (
 ( A concat  j ) concat  B ) ) }   =>    |-  ( ( A  e.  W  /\  B  e.  W )  ->  .~  C_  L )
 
Theoremefgcpbl 15059* 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  e.  W  /\  B  e.  W  /\  X  .~  Y ) 
 ->  ( ( A concat  X ) concat  B )  .~  (
 ( A concat  Y ) concat  B ) )
 
Theoremefgcpbl2 15060* 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 15061 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 15062 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 15063 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 15064 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 15065 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  e.  V  ->  G  e.  Grp )
 
Theoremfrgpadd 15066 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 15067* 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 15068* 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 15069* 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 15070 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 15071 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 15072 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 15073* 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 15074* 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 15075* 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 15076* 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 15077* 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 15078* 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 15079* 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 15080* 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 15081* 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 15082 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 15083 Extend class notation with class of all commutative monoids.
 class CMnd
 
Syntaxcabel 15084 Extend class notation with class of all Abelian groups.
 class  Abel
 
Definitiondf-cmn 15085* 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 15086 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 15087 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
 |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  G  e. CMnd ) )
 
Theoremablgrp 15088 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
 |-  ( G  e.  Abel  ->  G  e.  Grp )
 
Theoremablcmn 15089 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Abel  ->  G  e. CMnd )
 
Theoremiscmn 15090* 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 15091* 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 15092* 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 15093* 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 15094 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 15095* 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 15096* 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 15097* 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 15098 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e. CMnd  ->  G  e.  Mnd )
 
Theoremcmncom 15099 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 15100 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 ) )
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