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Theorem efgrelexlema 15058
Description: 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.)
Hypotheses
Ref Expression
efgval.w  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
efgval.r  |-  .~  =  ( ~FG  `  I )
efgval2.m  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
efgval2.t  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
efgred.d  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
efgred.s  |-  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 ) ) )
efgrelexlem.1  |-  L  =  { <. i ,  j
>.  |  E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `
 0 )  =  ( d `  0
) }
Assertion
Ref Expression
efgrelexlema  |-  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) )
Distinct variable groups:    a, b,
c, d, i, j, A    y, a, z, b    L, a, b    n, c, t, v, w, y, z    m, a, n, t, v, w, x, M, b, c, i, j    k, a, T, b, c, i, j, m, t, x    W, a, b, c    k, d, m, n, t, v, w, x, y, z, W, i, j    .~ , a, b, c, d, i, j, m, t, x, y, z    B, a, b, c, d, i, j    S, a, b, c, d, i, j    I,
a, b, c, i, j, m, n, t, v, w, x, y, z    D, a, b, c, d, i, j, m, t
Allowed substitution hints:    A( x, y, z, w, v, t, k, m, n)    B( x, y, z, w, v, t, k, m, n)    D( x, y, z, w, v, k, n)    .~ ( w, v, k, n)    S( x, y, z, w, v, t, k, m, n)    T( y, z, w, v, n, d)    I( k, d)    L( x, y, z, w, v, t, i, j, k, m, n, c, d)    M( y, z, k, d)

Proof of Theorem efgrelexlema
StepHypRef Expression
1 efgrelexlem.1 . . . 4  |-  L  =  { <. i ,  j
>.  |  E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `
 0 )  =  ( d `  0
) }
21relopabi 4811 . . 3  |-  Rel  L
3 brrelex12 4726 . . 3  |-  ( ( Rel  L  /\  A L B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
42, 3mpan 651 . 2  |-  ( A L B  ->  ( A  e.  _V  /\  B  e.  _V ) )
5 n0i 3460 . . . . . 6  |-  ( a  e.  ( `' S " { A } )  ->  -.  ( `' S " { A }
)  =  (/) )
6 snprc 3695 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
7 imaeq2 5008 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
86, 7sylbi 187 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
9 ima0 5030 . . . . . . 7  |-  ( `' S " (/) )  =  (/)
108, 9syl6eq 2331 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  (/) )
115, 10nsyl2 119 . . . . 5  |-  ( a  e.  ( `' S " { A } )  ->  A  e.  _V )
12 n0i 3460 . . . . . 6  |-  ( b  e.  ( `' S " { B } )  ->  -.  ( `' S " { B }
)  =  (/) )
13 snprc 3695 . . . . . . . 8  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
14 imaeq2 5008 . . . . . . . 8  |-  ( { B }  =  (/)  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1513, 14sylbi 187 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1615, 9syl6eq 2331 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  (/) )
1712, 16nsyl2 119 . . . . 5  |-  ( b  e.  ( `' S " { B } )  ->  B  e.  _V )
1811, 17anim12i 549 . . . 4  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1918a1d 22 . . 3  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  ->  ( A  e.  _V  /\  B  e. 
_V ) ) )
2019rexlimivv 2672 . 2  |-  ( E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21 fveq1 5524 . . . . . 6  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
2221eqeq1d 2291 . . . . 5  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
23 fveq1 5524 . . . . . 6  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
2423eqeq2d 2294 . . . . 5  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
2522, 24cbvrex2v 2773 . . . 4  |-  ( E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `  0
)  =  ( d `
 0 )  <->  E. a  e.  ( `' S " { i } ) E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
) )
26 sneq 3651 . . . . . 6  |-  ( i  =  A  ->  { i }  =  { A } )
2726imaeq2d 5012 . . . . 5  |-  ( i  =  A  ->  ( `' S " { i } )  =  ( `' S " { A } ) )
2827rexeqdv 2743 . . . 4  |-  ( i  =  A  ->  ( E. a  e.  ( `' S " { i } ) E. b  e.  ( `' S " { j } ) ( a `  0
)  =  ( b `
 0 )  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
) ) )
2925, 28syl5bb 248 . . 3  |-  ( i  =  A  ->  ( E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `  0
)  =  ( d `
 0 )  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
) ) )
30 sneq 3651 . . . . . 6  |-  ( j  =  B  ->  { j }  =  { B } )
3130imaeq2d 5012 . . . . 5  |-  ( j  =  B  ->  ( `' S " { j } )  =  ( `' S " { B } ) )
3231rexeqdv 2743 . . . 4  |-  ( j  =  B  ->  ( E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
)  <->  E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) ) )
3332rexbidv 2564 . . 3  |-  ( j  =  B  ->  ( E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { j } ) ( a `  0
)  =  ( b `
 0 )  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) ) )
3429, 33, 1brabg 4284 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 ) ) )
354, 20, 34pm5.21nii 342 1  |-  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023   {copab 4076    e. cmpt 4077    _I cid 4304    X. cxp 4687   `'ccnv 4688   ran crn 4690   "cima 4692   Rel wrel 4694   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    - cmin 9037   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgrelexlemb  15059  efgrelex  15060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263
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