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Theorem efgcpbllema 15063
Description: Lemma for efgrelex 15060. 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.)
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 ) ) )
efgcpbllem.1  |-  L  =  { <. i ,  j
>.  |  ( {
i ,  j } 
C_  W  /\  (
( A concat  i ) concat  B )  .~  ( ( A concat  j ) concat  B
) ) }
Assertion
Ref Expression
efgcpbllema  |-  ( X L Y  <->  ( X  e.  W  /\  Y  e.  W  /\  ( ( A concat  X ) concat  B
)  .~  ( ( A concat  Y ) concat  B ) ) )
Distinct variable groups:    i, j, A    y, z    t, n, v, w, y, z   
i, m, n, t, v, w, x, M, j    i, k, T, j, m, t, x   
i, X, j    y,
i, z, W, j   
k, n, v, w, y, z, W, m, t, x    .~ , i, j, m, t, x, y, z    B, i, j    S, i, j    i, Y, j   
i, I, j, m, n, t, v, w, x, y, z    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)    I( k)    L( x, y, z, w, v, t, i, j, k, m, n)    M( y, z, k)    X( x, y, z, w, v, t, k, m, n)    Y( x, y, z, w, v, t, k, m, n)

Proof of Theorem efgcpbllema
StepHypRef Expression
1 oveq2 5866 . . . . 5  |-  ( i  =  X  ->  ( A concat  i )  =  ( A concat  X ) )
21oveq1d 5873 . . . 4  |-  ( i  =  X  ->  (
( A concat  i ) concat  B )  =  ( ( A concat  X ) concat  B
) )
3 oveq2 5866 . . . . 5  |-  ( j  =  Y  ->  ( A concat  j )  =  ( A concat  Y ) )
43oveq1d 5873 . . . 4  |-  ( j  =  Y  ->  (
( A concat  j ) concat  B )  =  ( ( A concat  Y ) concat  B
) )
52, 4breqan12d 4038 . . 3  |-  ( ( i  =  X  /\  j  =  Y )  ->  ( ( ( A concat 
i ) concat  B )  .~  ( ( A concat  j
) concat  B )  <->  ( ( A concat  X ) concat  B )  .~  ( ( A concat  Y ) concat  B ) ) )
6 efgcpbllem.1 . . . 4  |-  L  =  { <. i ,  j
>.  |  ( {
i ,  j } 
C_  W  /\  (
( A concat  i ) concat  B )  .~  ( ( A concat  j ) concat  B
) ) }
7 vex 2791 . . . . . . 7  |-  i  e. 
_V
8 vex 2791 . . . . . . 7  |-  j  e. 
_V
97, 8prss 3769 . . . . . 6  |-  ( ( i  e.  W  /\  j  e.  W )  <->  { i ,  j } 
C_  W )
109anbi1i 676 . . . . 5  |-  ( ( ( i  e.  W  /\  j  e.  W
)  /\  ( ( A concat  i ) concat  B )  .~  ( ( A concat 
j ) concat  B )
)  <->  ( { i ,  j }  C_  W  /\  ( ( A concat 
i ) concat  B )  .~  ( ( A concat  j
) concat  B ) ) )
1110opabbii 4083 . . . 4  |-  { <. i ,  j >.  |  ( ( i  e.  W  /\  j  e.  W
)  /\  ( ( A concat  i ) concat  B )  .~  ( ( A concat 
j ) concat  B )
) }  =  { <. i ,  j >.  |  ( { i ,  j }  C_  W  /\  ( ( A concat 
i ) concat  B )  .~  ( ( A concat  j
) concat  B ) ) }
126, 11eqtr4i 2306 . . 3  |-  L  =  { <. i ,  j
>.  |  ( (
i  e.  W  /\  j  e.  W )  /\  ( ( A concat  i
) concat  B )  .~  (
( A concat  j ) concat  B ) ) }
135, 12brab2ga 4763 . 2  |-  ( X L Y  <->  ( ( X  e.  W  /\  Y  e.  W )  /\  ( ( A concat  X
) concat  B )  .~  (
( A concat  Y ) concat  B ) ) )
14 df-3an 936 . 2  |-  ( ( X  e.  W  /\  Y  e.  W  /\  ( ( A concat  X
) concat  B )  .~  (
( A concat  Y ) concat  B ) )  <->  ( ( X  e.  W  /\  Y  e.  W )  /\  ( ( A concat  X
) concat  B )  .~  (
( A concat  Y ) concat  B ) ) )
1513, 14bitr4i 243 1  |-  ( X L Y  <->  ( X  e.  W  /\  Y  e.  W  /\  ( ( A concat  X ) concat  B
)  .~  ( ( A concat  Y ) concat  B ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149    C_ wss 3152   (/)c0 3455   {csn 3640   {cpr 3641   <.cop 3643   <.cotp 3644   U_ciun 3905   class class class wbr 4023   {copab 4076    e. cmpt 4077    _I cid 4304    X. cxp 4687   ran crn 4690   ` 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   concat cconcat 11404   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgcpbllemb  15064  efgcpbl  15065
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-iota 5219  df-fv 5263  df-ov 5861
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