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Theorem efgcpbl2 16252
Description: Two extension sequences have related endpoints iff they have the same base. (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 ) ) )
Assertion
Ref Expression
efgcpbl2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( X concat  Y ) )
Distinct variable groups:    y, z    t, n, v, w, y, z, m, x    m, M    x, n, M, t, v, w    k, m, t, x, T    k, n, v, w, y, z, W, m, t, x    .~ , m, t, x, y, z    m, I, n, t, v, w, x, y, z    D, 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)    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 efgcpbl2
StepHypRef Expression
1 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
31, 2efger 16213 . . 3  |-  .~  Er  W
43a1i 11 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  .~  Er  W )
5 simpl 457 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  .~  X )
64, 5ercl 7110 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e.  W )
7 wrd0 12250 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
81efgrcl 16210 . . . . . . 7  |-  ( A  e.  W  ->  (
I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
96, 8syl 16 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( I  e.  _V  /\  W  = Word  ( I  X.  2o ) ) )
109simprd 463 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  W  = Word  ( I  X.  2o ) )
117, 10syl5eleqr 2528 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  (/) 
e.  W )
12 simpr 461 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  .~  Y )
13 efgval2.m . . . . 5  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
14 efgval2.t . . . . 5  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
15 efgred.d . . . . 5  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
16 efgred.s . . . . 5  |-  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 ) ) )
171, 2, 13, 14, 15, 16efgcpbl 16251 . . . 4  |-  ( ( A  e.  W  /\  (/) 
e.  W  /\  B  .~  Y )  ->  (
( A concat  B ) concat  (/) )  .~  ( ( A concat  Y ) concat  (/) ) )
186, 11, 12, 17syl3anc 1218 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( A concat  B
) concat  (/) )  .~  (
( A concat  Y ) concat  (/) ) )
196, 10eleqtrd 2517 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e. Word  ( I  X.  2o ) )
204, 12ercl 7110 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e.  W )
2120, 10eleqtrd 2517 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e. Word  ( I  X.  2o ) )
22 ccatcl 12272 . . . . 5  |-  ( ( A  e. Word  ( I  X.  2o )  /\  B  e. Word  ( I  X.  2o ) )  -> 
( A concat  B )  e. Word  ( I  X.  2o ) )
2319, 21, 22syl2anc 661 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  e. Word  ( I  X.  2o ) )
24 ccatrid 12283 . . . 4  |-  ( ( A concat  B )  e. Word 
( I  X.  2o )  ->  ( ( A concat  B ) concat  (/) )  =  ( A concat  B ) )
2523, 24syl 16 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( A concat  B
) concat  (/) )  =  ( A concat  B ) )
264, 12ercl2 7112 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e.  W )
2726, 10eleqtrd 2517 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e. Word  ( I  X.  2o ) )
28 ccatcl 12272 . . . . 5  |-  ( ( A  e. Word  ( I  X.  2o )  /\  Y  e. Word  ( I  X.  2o ) )  -> 
( A concat  Y )  e. Word  ( I  X.  2o ) )
2919, 27, 28syl2anc 661 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  Y )  e. Word  ( I  X.  2o ) )
30 ccatrid 12283 . . . 4  |-  ( ( A concat  Y )  e. Word 
( I  X.  2o )  ->  ( ( A concat  Y ) concat  (/) )  =  ( A concat  Y ) )
3129, 30syl 16 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( A concat  Y
) concat  (/) )  =  ( A concat  Y ) )
3218, 25, 313brtr3d 4319 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( A concat  Y ) )
331, 2, 13, 14, 15, 16efgcpbl 16251 . . . 4  |-  ( (
(/)  e.  W  /\  Y  e.  W  /\  A  .~  X )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
3411, 26, 5, 33syl3anc 1218 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
35 ccatlid 12282 . . . . 5  |-  ( A  e. Word  ( I  X.  2o )  ->  ( (/) concat  A )  =  A )
3619, 35syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  A )  =  A )
3736oveq1d 6104 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  =  ( A concat  Y ) )
384, 5ercl2 7112 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e.  W )
3938, 10eleqtrd 2517 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e. Word  ( I  X.  2o ) )
40 ccatlid 12282 . . . . 5  |-  ( X  e. Word  ( I  X.  2o )  ->  ( (/) concat  X )  =  X )
4139, 40syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  X )  =  X )
4241oveq1d 6104 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  X ) concat  Y )  =  ( X concat  Y ) )
4334, 37, 423brtr3d 4319 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  Y )  .~  ( X concat  Y ) )
444, 32, 43ertrd 7115 1  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( X concat  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2713   {crab 2717   _Vcvv 2970    \ cdif 3323   (/)c0 3635   {csn 3875   <.cop 3881   <.cotp 3883   U_ciun 4169   class class class wbr 4290    e. cmpt 4348    _I cid 4629    X. cxp 4836   ran crn 4839   ` cfv 5416  (class class class)co 6089    e. cmpt2 6091   1oc1o 6911   2oc2o 6912    Er wer 7096   0cc0 9280   1c1 9281    - cmin 9593   ...cfz 11435  ..^cfzo 11546   #chash 12101  Word cword 12219   concat cconcat 12221   splice csplice 12224   <"cs2 12466   ~FG cefg 16201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-rep 4401  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-cnex 9336  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-pss 3342  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-tp 3880  df-op 3882  df-ot 3884  df-uni 4090  df-int 4127  df-iun 4171  df-iin 4172  df-br 4291  df-opab 4349  df-mpt 4350  df-tr 4384  df-eprel 4630  df-id 4634  df-po 4639  df-so 4640  df-fr 4677  df-we 4679  df-ord 4720  df-on 4721  df-lim 4722  df-suc 4723  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-om 6475  df-1st 6575  df-2nd 6576  df-recs 6830  df-rdg 6864  df-1o 6918  df-2o 6919  df-oadd 6922  df-er 7099  df-ec 7101  df-map 7214  df-pm 7215  df-en 7309  df-dom 7310  df-sdom 7311  df-fin 7312  df-card 8107  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-nn 10321  df-n0 10578  df-z 10645  df-uz 10860  df-fz 11436  df-fzo 11547  df-hash 12102  df-word 12227  df-concat 12229  df-s1 12230  df-substr 12231  df-splice 12232  df-s2 12473  df-efg 16204
This theorem is referenced by:  frgpcpbl  16254
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