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Theorem efgrelexlema 16638
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 . . 3  |-  L  =  { <. i ,  j
>.  |  E. c  e.  ( `' S " { i } ) E. d  e.  ( `' S " { j } ) ( c `
 0 )  =  ( d `  0
) }
21bropaex12 5060 . 2  |-  ( A L B  ->  ( A  e.  _V  /\  B  e.  _V ) )
3 n0i 3773 . . . . . 6  |-  ( a  e.  ( `' S " { A } )  ->  -.  ( `' S " { A }
)  =  (/) )
4 snprc 4075 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
5 imaeq2 5320 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
64, 5sylbi 195 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
7 ima0 5339 . . . . . . 7  |-  ( `' S " (/) )  =  (/)
86, 7syl6eq 2498 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  (/) )
93, 8nsyl2 127 . . . . 5  |-  ( a  e.  ( `' S " { A } )  ->  A  e.  _V )
10 n0i 3773 . . . . . 6  |-  ( b  e.  ( `' S " { B } )  ->  -.  ( `' S " { B }
)  =  (/) )
11 snprc 4075 . . . . . . . 8  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
12 imaeq2 5320 . . . . . . . 8  |-  ( { B }  =  (/)  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1311, 12sylbi 195 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1413, 7syl6eq 2498 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  (/) )
1510, 14nsyl2 127 . . . . 5  |-  ( b  e.  ( `' S " { B } )  ->  B  e.  _V )
169, 15anim12i 566 . . . 4  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1716a1d 25 . . 3  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  ->  ( A  e.  _V  /\  B  e. 
_V ) ) )
1817rexlimivv 2938 . 2  |-  ( E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 )  ->  ( A  e.  _V  /\  B  e.  _V ) )
19 fveq1 5852 . . . . . 6  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
2019eqeq1d 2443 . . . . 5  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
21 fveq1 5852 . . . . . 6  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
2221eqeq2d 2455 . . . . 5  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
2320, 22cbvrex2v 3077 . . . 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
) )
24 sneq 4021 . . . . . 6  |-  ( i  =  A  ->  { i }  =  { A } )
2524imaeq2d 5324 . . . . 5  |-  ( i  =  A  ->  ( `' S " { i } )  =  ( `' S " { A } ) )
2625rexeqdv 3045 . . . 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
) ) )
2723, 26syl5bb 257 . . 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
) ) )
28 sneq 4021 . . . . . 6  |-  ( j  =  B  ->  { j }  =  { B } )
2928imaeq2d 5324 . . . . 5  |-  ( j  =  B  ->  ( `' S " { j } )  =  ( `' S " { B } ) )
3029rexeqdv 3045 . . . 4  |-  ( j  =  B  ->  ( E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
)  <->  E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) ) )
3130rexbidv 2952 . . 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
) ) )
3227, 31, 1brabg 4753 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 ) ) )
332, 18, 32pm5.21nii 353 1  |-  ( A L B  <->  E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   {crab 2795   _Vcvv 3093    \ cdif 3456   (/)c0 3768   {csn 4011   <.cop 4017   <.cotp 4019   U_ciun 4312   class class class wbr 4434   {copab 4491    |-> cmpt 4492    _I cid 4777    X. cxp 4984   `'ccnv 4985   ran crn 4987   "cima 4989   ` cfv 5575  (class class class)co 6278    |-> cmpt2 6280   1oc1o 7122   2oc2o 7123   0cc0 9492   1c1 9493    - cmin 9807   ...cfz 11678  ..^cfzo 11800   #chash 12381  Word cword 12510   splice csplice 12515   <"cs2 12782   ~FG cefg 16595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-uni 4232  df-br 4435  df-opab 4493  df-xp 4992  df-cnv 4994  df-dm 4996  df-rn 4997  df-res 4998  df-ima 4999  df-iota 5538  df-fv 5583
This theorem is referenced by:  efgrelexlemb  16639  efgrelex  16640
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