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Theorem efgrelexlema 15074
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 4827 . . 3  |-  Rel  L
3 brrelex12 4742 . . 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 3473 . . . . . 6  |-  ( a  e.  ( `' S " { A } )  ->  -.  ( `' S " { A }
)  =  (/) )
6 snprc 3708 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
7 imaeq2 5024 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
86, 7sylbi 187 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
9 ima0 5046 . . . . . . 7  |-  ( `' S " (/) )  =  (/)
108, 9syl6eq 2344 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  (/) )
115, 10nsyl2 119 . . . . 5  |-  ( a  e.  ( `' S " { A } )  ->  A  e.  _V )
12 n0i 3473 . . . . . 6  |-  ( b  e.  ( `' S " { B } )  ->  -.  ( `' S " { B }
)  =  (/) )
13 snprc 3708 . . . . . . . 8  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
14 imaeq2 5024 . . . . . . . 8  |-  ( { B }  =  (/)  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1513, 14sylbi 187 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1615, 9syl6eq 2344 . . . . . 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 2685 . 2  |-  ( E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21 fveq1 5540 . . . . . 6  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
2221eqeq1d 2304 . . . . 5  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
23 fveq1 5540 . . . . . 6  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
2423eqeq2d 2307 . . . . 5  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
2522, 24cbvrex2v 2786 . . . 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 3664 . . . . . 6  |-  ( i  =  A  ->  { i }  =  { A } )
2726imaeq2d 5028 . . . . 5  |-  ( i  =  A  ->  ( `' S " { i } )  =  ( `' S " { A } ) )
2827rexeqdv 2756 . . . 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 3664 . . . . . 6  |-  ( j  =  B  ->  { j }  =  { B } )
3130imaeq2d 5028 . . . . 5  |-  ( j  =  B  ->  ( `' S " { j } )  =  ( `' S " { B } ) )
3231rexeqdv 2756 . . . 4  |-  ( j  =  B  ->  ( E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
)  <->  E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) ) )
3332rexbidv 2577 . . 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 4300 . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {crab 2560   _Vcvv 2801    \ cdif 3162   (/)c0 3468   {csn 3653   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093    _I cid 4320    X. cxp 4703   `'ccnv 4704   ran crn 4706   "cima 4708   Rel wrel 4710   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1oc1o 6488   2oc2o 6489   0cc0 8753   1c1 8754    - cmin 9053   ...cfz 10798  ..^cfzo 10886   #chash 11353  Word cword 11419   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgrelexlemb  15075  efgrelex  15076
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279
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