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Theorem efgcpbllema 15079
Description: Lemma for efgrelex 15076. 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 5882 . . . . 5  |-  ( i  =  X  ->  ( A concat  i )  =  ( A concat  X ) )
21oveq1d 5889 . . . 4  |-  ( i  =  X  ->  (
( A concat  i ) concat  B )  =  ( ( A concat  X ) concat  B
) )
3 oveq2 5882 . . . . 5  |-  ( j  =  Y  ->  ( A concat  j )  =  ( A concat  Y ) )
43oveq1d 5889 . . . 4  |-  ( j  =  Y  ->  (
( A concat  j ) concat  B )  =  ( ( A concat  Y ) concat  B
) )
52, 4breqan12d 4054 . . 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 2804 . . . . . . 7  |-  i  e. 
_V
8 vex 2804 . . . . . . 7  |-  j  e. 
_V
97, 8prss 3785 . . . . . 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 4099 . . . 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 2319 . . 3  |-  L  =  { <. i ,  j
>.  |  ( (
i  e.  W  /\  j  e.  W )  /\  ( ( A concat  i
) concat  B )  .~  (
( A concat  j ) concat  B ) ) }
135, 12brab2ga 4779 . 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 1632    e. wcel 1696   A.wral 2556   {crab 2560    \ cdif 3162    C_ wss 3165   (/)c0 3468   {csn 3653   {cpr 3654   <.cop 3656   <.cotp 3657   U_ciun 3921   class class class wbr 4039   {copab 4092    e. cmpt 4093    _I cid 4320    X. cxp 4703   ran crn 4706   ` 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   concat cconcat 11420   splice csplice 11423   <"cs2 11507   ~FG cefg 15031
This theorem is referenced by:  efgcpbllemb  15080  efgcpbl  15081
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-iota 5235  df-fv 5279  df-ov 5877
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