HomeHome Metamath Proof Explorer
Theorem List (p. 154 of 325)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-22374)
  Hilbert Space Explorer  Hilbert Space Explorer
(22375-23897)
  Users' Mathboxes  Users' Mathboxes
(23898-32447)
 

Theorem List for Metamath Proof Explorer - 15301-15400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremefgmnvl 15301* 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 15302 Lemma for efgval 15304. (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 15303* Lemma for efgval 15304. (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 15304* 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 15305 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 15306 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 15307 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 15308 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 15309* 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 15310* 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 15311* 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 15312* 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 15313* 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 15314* 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 15315* 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 15316* 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 15317* 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 15318* 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 15319* 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 15320* 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 15321* 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 15322* 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 15323* 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 15324* 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 15325* 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 15326* 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 15327* The reduced word that forms the base of the sequence in efgsval 15318 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 15328* Lemma for efgredleme 15330. (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 15329* Lemma for efgred 15335. (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 15330* Lemma for efgred 15335. (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 15331* The reduced word that forms the base of the sequence in efgsval 15318 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 15332* The reduced word that forms the base of the sequence in efgsval 15318 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 15333* The reduced word that forms the base of the sequence in efgsval 15318 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 15334* The reduced word that forms the base of the sequence in efgsval 15318 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 15335* The reduced word that forms the base of the sequence in efgsval 15318 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 15336* 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 15337* 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 15338* 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 15339* 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 15340* 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 15341* Lemma for efgrelex 15338. 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 15342* Lemma for efgrelex 15338. 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 15343* 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 15344* 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 15345 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 15346 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 15347 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 15348 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 15349 The free group is a group. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |-  G  =  (freeGrp `  I
 )   =>    |-  ( I  e.  V  ->  G  e.  Grp )
 
Theoremfrgpadd 15350 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 15351* 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 15352* 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 15353* 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 15354 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 15355 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 15356 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 15357* 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 15358* 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 15359* 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 15360* 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 15361* 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 15362* 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 15363* 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 15364* 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 15365* 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 15366 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 15367 Extend class notation with class of all commutative monoids.
 class CMnd
 
Syntaxcabel 15368 Extend class notation with class of all Abelian groups.
 class  Abel
 
Definitiondf-cmn 15369* 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 15370 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 15371 The predicate "is an Abelian (commutative) group." (Contributed by NM, 17-Oct-2011.)
 |-  ( G  e.  Abel  <->  ( G  e.  Grp  /\  G  e. CMnd ) )
 
Theoremablgrp 15372 An Abelian group is a group. (Contributed by NM, 26-Aug-2011.)
 |-  ( G  e.  Abel  ->  G  e.  Grp )
 
Theoremablcmn 15373 An Abelian group is a commutative monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e.  Abel  ->  G  e. CMnd )
 
Theoremiscmn 15374* 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 15375* 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 15376* 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 15377* 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 15378 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 15379* 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 15380* 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 15381* 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 15382 A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
 |-  ( G  e. CMnd  ->  G  e.  Mnd )
 
Theoremcmncom 15383 A 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 15384 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 15385 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 15386 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 15387 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 15388 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 15389 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 15390 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 15391 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 15392 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 15393 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 15394 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 15395 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 15396 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 15397 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 15398 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 15399 Cancellation law for mixed addition and subtraction. (pnpcan 9296 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 15400 Cancellation law for group division. (nncan 9286 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 )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32447
  Copyright terms: Public domain < Previous  Next >