MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgrelexlema Structured version   Unicode version

Theorem efgrelexlema 16640
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 5134 . . 3  |-  Rel  L
3 brrelex12 5043 . . 3  |-  ( ( Rel  L  /\  A L B )  ->  ( A  e.  _V  /\  B  e.  _V ) )
42, 3mpan 670 . 2  |-  ( A L B  ->  ( A  e.  _V  /\  B  e.  _V ) )
5 n0i 3795 . . . . . 6  |-  ( a  e.  ( `' S " { A } )  ->  -.  ( `' S " { A }
)  =  (/) )
6 snprc 4097 . . . . . . . 8  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
7 imaeq2 5339 . . . . . . . 8  |-  ( { A }  =  (/)  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
86, 7sylbi 195 . . . . . . 7  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  ( `' S " (/) ) )
9 ima0 5358 . . . . . . 7  |-  ( `' S " (/) )  =  (/)
108, 9syl6eq 2524 . . . . . 6  |-  ( -.  A  e.  _V  ->  ( `' S " { A } )  =  (/) )
115, 10nsyl2 127 . . . . 5  |-  ( a  e.  ( `' S " { A } )  ->  A  e.  _V )
12 n0i 3795 . . . . . 6  |-  ( b  e.  ( `' S " { B } )  ->  -.  ( `' S " { B }
)  =  (/) )
13 snprc 4097 . . . . . . . 8  |-  ( -.  B  e.  _V  <->  { B }  =  (/) )
14 imaeq2 5339 . . . . . . . 8  |-  ( { B }  =  (/)  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1513, 14sylbi 195 . . . . . . 7  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  ( `' S " (/) ) )
1615, 9syl6eq 2524 . . . . . 6  |-  ( -.  B  e.  _V  ->  ( `' S " { B } )  =  (/) )
1712, 16nsyl2 127 . . . . 5  |-  ( b  e.  ( `' S " { B } )  ->  B  e.  _V )
1811, 17anim12i 566 . . . 4  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( A  e. 
_V  /\  B  e.  _V ) )
1918a1d 25 . . 3  |-  ( ( a  e.  ( `' S " { A } )  /\  b  e.  ( `' S " { B } ) )  ->  ( ( a `
 0 )  =  ( b `  0
)  ->  ( A  e.  _V  /\  B  e. 
_V ) ) )
2019rexlimivv 2964 . 2  |-  ( E. a  e.  ( `' S " { A } ) E. b  e.  ( `' S " { B } ) ( a `  0 )  =  ( b ` 
0 )  ->  ( A  e.  _V  /\  B  e.  _V ) )
21 fveq1 5871 . . . . . 6  |-  ( c  =  a  ->  (
c `  0 )  =  ( a ` 
0 ) )
2221eqeq1d 2469 . . . . 5  |-  ( c  =  a  ->  (
( c `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( d `  0
) ) )
23 fveq1 5871 . . . . . 6  |-  ( d  =  b  ->  (
d `  0 )  =  ( b ` 
0 ) )
2423eqeq2d 2481 . . . . 5  |-  ( d  =  b  ->  (
( a `  0
)  =  ( d `
 0 )  <->  ( a `  0 )  =  ( b `  0
) ) )
2522, 24cbvrex2v 3102 . . . 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 4043 . . . . . 6  |-  ( i  =  A  ->  { i }  =  { A } )
2726imaeq2d 5343 . . . . 5  |-  ( i  =  A  ->  ( `' S " { i } )  =  ( `' S " { A } ) )
2827rexeqdv 3070 . . . 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 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
) ) )
30 sneq 4043 . . . . . 6  |-  ( j  =  B  ->  { j }  =  { B } )
3130imaeq2d 5343 . . . . 5  |-  ( j  =  B  ->  ( `' S " { j } )  =  ( `' S " { B } ) )
3231rexeqdv 3070 . . . 4  |-  ( j  =  B  ->  ( E. b  e.  ( `' S " { j } ) ( a `
 0 )  =  ( b `  0
)  <->  E. b  e.  ( `' S " { B } ) ( a `
 0 )  =  ( b `  0
) ) )
3332rexbidv 2978 . . 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 4772 . 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 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 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   {crab 2821   _Vcvv 3118    \ cdif 3478   (/)c0 3790   {csn 4033   <.cop 4039   <.cotp 4041   U_ciun 4331   class class class wbr 4453   {copab 4510    |-> cmpt 4511    _I cid 4796    X. cxp 5003   `'ccnv 5004   ran crn 5006   "cima 5008   Rel wrel 5010   ` cfv 5594  (class class class)co 6295    |-> cmpt2 6297   1oc1o 7135   2oc2o 7136   0cc0 9504   1c1 9505    - cmin 9817   ...cfz 11684  ..^cfzo 11804   #chash 12385  Word cword 12515   splice csplice 12520   <"cs2 12786   ~FG cefg 16597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fv 5602
This theorem is referenced by:  efgrelexlemb  16641  efgrelex  16642
  Copyright terms: Public domain W3C validator