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Theorem efgcpbl2 16649
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 16610 . . 3  |-  .~  Er  W
43a1i 11 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  .~  Er  W )
5 simpl 457 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  .~  X )
64, 5ercl 7324 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e.  W )
7 wrd0 12544 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
81efgrcl 16607 . . . . . . 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 2538 . . . 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 16648 . . . 4  |-  ( ( A  e.  W  /\  (/) 
e.  W  /\  B  .~  Y )  ->  (
( A concat  B ) concat  (/) )  .~  ( ( A concat  Y ) concat  (/) ) )
186, 11, 12, 17syl3anc 1229 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( A concat  B
) concat  (/) )  .~  (
( A concat  Y ) concat  (/) ) )
196, 10eleqtrd 2533 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e. Word  ( I  X.  2o ) )
204, 12ercl 7324 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e.  W )
2120, 10eleqtrd 2533 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e. Word  ( I  X.  2o ) )
22 ccatcl 12572 . . . . 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 12583 . . . 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 7326 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e.  W )
2726, 10eleqtrd 2533 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e. Word  ( I  X.  2o ) )
28 ccatcl 12572 . . . . 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 12583 . . . 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 4466 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( A concat  Y ) )
331, 2, 13, 14, 15, 16efgcpbl 16648 . . . 4  |-  ( (
(/)  e.  W  /\  Y  e.  W  /\  A  .~  X )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
3411, 26, 5, 33syl3anc 1229 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
35 ccatlid 12582 . . . . 5  |-  ( A  e. Word  ( I  X.  2o )  ->  ( (/) concat  A )  =  A )
3619, 35syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  A )  =  A )
3736oveq1d 6296 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  =  ( A concat  Y ) )
384, 5ercl2 7326 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e.  W )
3938, 10eleqtrd 2533 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e. Word  ( I  X.  2o ) )
40 ccatlid 12582 . . . . 5  |-  ( X  e. Word  ( I  X.  2o )  ->  ( (/) concat  X )  =  X )
4139, 40syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  X )  =  X )
4241oveq1d 6296 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  X ) concat  Y )  =  ( X concat  Y ) )
4334, 37, 423brtr3d 4466 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  Y )  .~  ( X concat  Y ) )
444, 32, 43ertrd 7329 1  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( X concat  Y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1383    e. wcel 1804   A.wral 2793   {crab 2797   _Vcvv 3095    \ cdif 3458   (/)c0 3770   {csn 4014   <.cop 4020   <.cotp 4022   U_ciun 4315   class class class wbr 4437    |-> cmpt 4495    _I cid 4780    X. cxp 4987   ran crn 4990   ` cfv 5578  (class class class)co 6281    |-> cmpt2 6283   1oc1o 7125   2oc2o 7126    Er wer 7310   0cc0 9495   1c1 9496    - cmin 9810   ...cfz 11681  ..^cfzo 11803   #chash 12384  Word cword 12513   concat cconcat 12515   splice csplice 12518   <"cs2 12785   ~FG cefg 16598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-ot 4023  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-ec 7315  df-map 7424  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-card 8323  df-cda 8551  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-hash 12385  df-word 12521  df-concat 12523  df-s1 12524  df-substr 12525  df-splice 12526  df-s2 12792  df-efg 16601
This theorem is referenced by:  frgpcpbl  16651
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