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Theorem List for Metamath Proof Explorer - 14801-14900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlsmcom2 14801 Subgroup sum commutes. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  ( Z `  U ) ) 
 ->  ( T  .(+)  U )  =  ( U  .(+)  T ) )
 
Theoremlsmub1 14802 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  T  C_  ( T  .(+)  U ) )
 
Theoremlsmub2 14803 Subgroup sum is an upper bound of its arguments. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  U  C_  ( T  .(+)  U ) )
 
Theoremlsmunss 14804 Union of subgroups is a subset of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Proof shortened by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T  u.  U )  C_  ( T  .(+)  U ) )
 
Theoremlsmless1 14805 Subset implies subgroup sum subset. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  S  C_  T )  ->  ( S  .(+)  U )  C_  ( T  .(+)  U ) )
 
Theoremlsmless2 14806 Subset implies subgroup sum subset. (Contributed by NM, 25-Feb-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( S  .(+)  T )  C_  ( S  .(+)  U ) )
 
Theoremlsmless12 14807 Subset implies subgroup sum subset. (Contributed by NM, 14-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  /\  ( R  C_  S  /\  T  C_  U ) ) 
 ->  ( R  .(+)  T ) 
 C_  ( S  .(+)  U ) )
 
Theoremlsmidm 14808 Subgroup sum is idempotent. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( U  e.  (SubGrp `  G )  ->  ( U  .(+)  U )  =  U )
 
Theoremlsmlub 14809 Least upper bound property of subgroup sum. (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 21-Jun-2014.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( ( S  C_  U  /\  T  C_  U ) 
 <->  ( S  .(+)  T ) 
 C_  U ) )
 
Theoremlsmss1 14810 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  T  C_  U )  ->  ( T  .(+)  U )  =  U )
 
Theoremlsmss1b 14811 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( T 
 C_  U  <->  ( T  .(+)  U )  =  U ) )
 
Theoremlsmss2 14812 Subgroup sum with a subset. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )  /\  U  C_  T )  ->  ( T  .(+)  U )  =  T )
 
Theoremlsmss2b 14813 Subgroup sum with a subset. (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( U 
 C_  T  <->  ( T  .(+)  U )  =  T ) )
 
Theoremlsmass 14814 Subgroup sum is associative. (Contributed by NM, 2-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( R  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G ) )  ->  ( ( R  .(+)  T )  .(+)  U )  =  ( R  .(+)  ( T 
 .(+)  U ) ) )
 
Theoremlsm01 14815 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( X  e.  (SubGrp `  G )  ->  ( X  .(+)  {  .0.  }
 )  =  X )
 
Theoremlsm02 14816 Subgroup sum with the zero subgroup. (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .0.  =  ( 0g `  G )   &    |-  .(+)  =  ( LSSum `  G )   =>    |-  ( X  e.  (SubGrp `  G )  ->  ( {  .0.  }  .(+)  X )  =  X )
 
Theoremsubglsm 14817 The subgroup sum evaluated within a subgroup. (Contributed by Mario Carneiro, 27-Apr-2016.)
 |-  H  =  ( Gs  S )   &    |-  .(+)  =  ( LSSum `  G )   &    |-  A  =  (
 LSSum `  H )   =>    |-  ( ( S  e.  (SubGrp `  G )  /\  T  C_  S  /\  U  C_  S )  ->  ( T  .(+)  U )  =  ( T A U ) )
 
Theoremlssnle 14818 Equivalent expressions for "not less than". (chnlei 21894 analog.) (Contributed by NM, 10-Jan-2015.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   =>    |-  ( ph  ->  ( -.  U  C_  T  <->  T  C.  ( T 
 .(+)  U ) ) )
 
Theoremlsmmod 14819 The modular law holds for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
 )  /\  S  C_  U )  ->  ( S  .(+)  ( T  i^i  U ) )  =  ( ( S  .(+)  T )  i^i  U ) )
 
Theoremlsmmod2 14820 Modular law dual for subgroup sum. Similar to part of Theorem 16.9 of [MaedaMaeda] p. 70. (Contributed by NM, 8-Jan-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   =>    |-  ( ( ( S  e.  (SubGrp `  G )  /\  T  e.  (SubGrp `  G )  /\  U  e.  (SubGrp `  G )
 )  /\  U  C_  S )  ->  ( S  i^i  ( T  .(+)  U ) )  =  ( ( S  i^i  T ) 
 .(+)  U ) )
 
Theoremlsmpropd 14821* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 29-Jun-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.  _V )   &    |-  ( ph  ->  L  e.  _V )   =>    |-  ( ph  ->  (
 LSSum `  K )  =  ( LSSum `  L )
 )
 
Theoremcntzrecd 14822 Commute the "subgroups commute" predicate. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  U  e.  (SubGrp `  G ) )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  U 
 C_  ( Z `  T ) )
 
Theoremlsmcntz 14823 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  (
 ( S  .(+)  T ) 
 C_  ( Z `  U )  <->  ( S  C_  ( Z `  U ) 
 /\  T  C_  ( Z `  U ) ) ) )
 
Theoremlsmcntzr 14824 The "subgroups commute" predicate applied to a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |-  Z  =  (Cntz `  G )   =>    |-  ( ph  ->  ( S  C_  ( Z `  ( T  .(+)  U ) )  <->  ( S  C_  ( Z `  T ) 
 /\  S  C_  ( Z `  U ) ) ) )
 
Theoremlsmdisj 14825 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  ( ( S  i^i  U )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  }
 ) )
 
Theoremlsmdisj2 14826 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  } )   =>    |-  ( ph  ->  ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  } )
 
Theoremlsmdisj3 14827 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )   &    |-  ( ph  ->  ( S  i^i  T )  =  {  .0.  } )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  S 
 C_  ( Z `  T ) )   =>    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )
 
Theoremlsmdisjr 14828 Disjointness from a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   =>    |-  ( ph  ->  ( ( S  i^i  T )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  }
 ) )
 
Theoremlsmdisj2r 14829 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   =>    |-  ( ph  ->  (
 ( S  .(+)  U )  i^i  T )  =  {  .0.  } )
 
Theoremlsmdisj3r 14830 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 22-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  ( ph  ->  ( S  i^i  ( T 
 .(+)  U ) )  =  {  .0.  } )   &    |-  ( ph  ->  ( T  i^i  U )  =  {  .0.  } )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  ( ( S  .(+)  T )  i^i  U )  =  {  .0.  } )
 
Theoremlsmdisj2a 14831 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  (
 ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( T  i^i  ( S  .(+)  U ) )  =  {  .0.  }  /\  ( S  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj2b 14832 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   =>    |-  ( ph  ->  (
 ( ( ( S 
 .(+)  U )  i^i  T )  =  {  .0.  } 
 /\  ( S  i^i  U )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj3a 14833 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  S 
 C_  ( Z `  T ) )   =>    |-  ( ph  ->  ( ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremlsmdisj3b 14834 Association of the disjointness constraint in a subgroup sum. (Contributed by Mario Carneiro, 21-Apr-2016.)
 |-  .(+)  =  ( LSSum `  G )   &    |-  ( ph  ->  S  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  T  e.  (SubGrp `  G )
 )   &    |-  ( ph  ->  U  e.  (SubGrp `  G )
 )   &    |- 
 .0.  =  ( 0g `  G )   &    |-  Z  =  (Cntz `  G )   &    |-  ( ph  ->  T 
 C_  ( Z `  U ) )   =>    |-  ( ph  ->  ( ( ( ( S 
 .(+)  T )  i^i  U )  =  {  .0.  } 
 /\  ( S  i^i  T )  =  {  .0.  } )  <->  ( ( S  i^i  ( T  .(+)  U ) )  =  {  .0.  }  /\  ( T  i^i  U )  =  {  .0.  } )
 ) )
 
Theoremsubgdisj1 14835 Vectors belonging to disjoint commuting subgroups are uniquely determined by their sum. (Contributed by NM, 2-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  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  ->  A  e.  T )   &    |-  ( ph  ->  C  e.  T )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( A  .+  B )  =  ( C  .+  D ) )   =>    |-  ( ph  ->  A  =  C )
 
Theoremsubgdisj2 14836 Vectors belonging to disjoint subgroups are uniquely determined by their sum. (Contributed by NM, 12-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  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  ->  A  e.  T )   &    |-  ( ph  ->  C  e.  T )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e.  U )   &    |-  ( ph  ->  ( A  .+  B )  =  ( C  .+  D ) )   =>    |-  ( ph  ->  B  =  D )
 
Theoremsubgdisjb 14837 Vectors belonging to disjoint subgroups are uniquely determined by their sum. Analogous to opth 4138, this theorem shows a way of representing a pair of vectors. (Contributed by NM, 5-Jul-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
 |- 
 .+  =  ( +g  `  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  ->  A  e.  T )   &    |-  ( ph  ->  C  e.  T )   &    |-  ( ph  ->  B  e.  U )   &    |-  ( ph  ->  D  e.  U )   =>    |-  ( ph  ->  (
 ( A  .+  B )  =  ( C  .+  D )  <->  ( A  =  C  /\  B  =  D ) ) )
 
Theorempj1fval 14838* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  P  =  ( proj 1 `  G )   =>    |-  ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  ->  ( T P U )  =  ( z  e.  ( T  .(+)  U ) 
 |->  ( iota_ x  e.  T E. y  e.  U  z  =  ( x  .+  y ) ) ) )
 
Theorempj1val 14839* The left projection function (for a direct product of group subspaces). (Contributed by Mario Carneiro, 15-Oct-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)
 |-  B  =  ( Base `  G )   &    |-  .+  =  ( +g  `  G )   &    |-  .(+)  =  (
 LSSum `  G )   &    |-  P  =  ( proj 1 `  G )   =>    |-  ( ( ( G  e.  V  /\  T  C_  B  /\  U  C_  B )  /\  X  e.  ( T  .(+)  U ) )  ->  ( ( T P U ) `  X )  =  ( iota_ x  e.  T E. y  e.  U  X  =  ( x  .+  y
 ) ) )
 
Theorempj1eu 14840* Uniqueness of a left projection. (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 ) )   =>    |-  ( ( ph  /\  X  e.  ( T  .(+)  U ) )  ->  E! x  e.  T  E. y  e.  U  X  =  ( x  .+  y ) )
 
Theorempj1f 14841 The left projection function maps a direct subspace sum onto the left 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  ->  ( T P U ) : ( T  .(+)  U ) --> T )
 
Theorempj2f 14842 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 14843 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 14844 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 14845 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 14846 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 14847 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 14848 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 14849 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 14850 Extend class notation with the free group equivalence relation.
 class ~FG
 
Syntaxcfrgp 14851 Extend class notation with the free group construction.
 class freeGrp
 
Syntaxcvrgp 14852 Extend class notation with free group injection.
 class varFGrp
 
Definitiondf-efg 14853* 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 14854 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 14853. (Contributed by Mario Carneiro, 1-Oct-2015.)
 |- freeGrp  =  ( i  e.  _V  |->  ( (freeMnd `  ( i  X.  2o ) )  /.s  ( ~FG  `  i
 ) ) )
 
Definitiondf-vrgp 14855* 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 14856* 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 14857* 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 14858* 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 14859 Lemma for efgval 14861. (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 14860* Lemma for efgval 14861. (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 14861* 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 14862 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 14863 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 14864 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 14865 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 14866* 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 14867* 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 14868* 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 14869* 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 14870* 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 14871* 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 14872* 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 14873* 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 14874* 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 14875* 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 14876* 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 14877* 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 14878* 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 14879* 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 14880* 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 14881* 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 14882* 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 14883* 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 14884* The reduced word that forms the base of the sequence in efgsval 14875 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 14885* Lemma for efgredleme 14887. (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 14886* Lemma for efgred 14892. (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 14887* Lemma for efgred 14892. (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 14888* The reduced word that forms the base of the sequence in efgsval 14875 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 14889* The reduced word that forms the base of the sequence in efgsval 14875 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 14890* The reduced word that forms the base of the sequence in efgsval 14875 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 14891* The reduced word that forms the base of the sequence in efgsval 14875 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 14892* The reduced word that forms the base of the sequence in efgsval 14875 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 14893* 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 14894* 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 14895* 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 14896* 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 14897* 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 14898* Lemma for efgrelex 14895. 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 14899* Lemma for efgrelex 14895. 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 14900* 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 ) )
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