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Theorem efgcpbl2 15389
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 15350 . . 3  |-  .~  Er  W
43a1i 11 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  .~  Er  W )
5 simpl 444 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  .~  X )
64, 5ercl 6916 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e.  W )
7 wrd0 11732 . . . . 5  |-  (/)  e. Word  (
I  X.  2o )
81efgrcl 15347 . . . . . . 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 450 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  W  = Word  ( I  X.  2o ) )
117, 10syl5eleqr 2523 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  (/) 
e.  W )
12 simpr 448 . . . 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 15388 . . . 4  |-  ( ( A  e.  W  /\  (/) 
e.  W  /\  B  .~  Y )  ->  (
( A concat  B ) concat  (/) )  .~  ( ( A concat  Y ) concat  (/) ) )
186, 11, 12, 17syl3anc 1184 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( A concat  B
) concat  (/) )  .~  (
( A concat  Y ) concat  (/) ) )
196, 10eleqtrd 2512 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  A  e. Word  ( I  X.  2o ) )
204, 12ercl 6916 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e.  W )
2120, 10eleqtrd 2512 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  B  e. Word  ( I  X.  2o ) )
22 ccatcl 11743 . . . . 5  |-  ( ( A  e. Word  ( I  X.  2o )  /\  B  e. Word  ( I  X.  2o ) )  -> 
( A concat  B )  e. Word  ( I  X.  2o ) )
2319, 21, 22syl2anc 643 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  e. Word  ( I  X.  2o ) )
24 ccatrid 11749 . . . 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 6918 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e.  W )
2726, 10eleqtrd 2512 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  Y  e. Word  ( I  X.  2o ) )
28 ccatcl 11743 . . . . 5  |-  ( ( A  e. Word  ( I  X.  2o )  /\  Y  e. Word  ( I  X.  2o ) )  -> 
( A concat  Y )  e. Word  ( I  X.  2o ) )
2919, 27, 28syl2anc 643 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  Y )  e. Word  ( I  X.  2o ) )
30 ccatrid 11749 . . . 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 4241 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( A concat  Y ) )
331, 2, 13, 14, 15, 16efgcpbl 15388 . . . 4  |-  ( (
(/)  e.  W  /\  Y  e.  W  /\  A  .~  X )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
3411, 26, 5, 33syl3anc 1184 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  .~  (
( (/) concat  X ) concat  Y ) )
35 ccatlid 11748 . . . . 5  |-  ( A  e. Word  ( I  X.  2o )  ->  ( (/) concat  A )  =  A )
3619, 35syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  A )  =  A )
3736oveq1d 6096 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  A ) concat  Y )  =  ( A concat  Y ) )
384, 5ercl2 6918 . . . . . 6  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e.  W )
3938, 10eleqtrd 2512 . . . . 5  |-  ( ( A  .~  X  /\  B  .~  Y )  ->  X  e. Word  ( I  X.  2o ) )
40 ccatlid 11748 . . . . 5  |-  ( X  e. Word  ( I  X.  2o )  ->  ( (/) concat  X )  =  X )
4139, 40syl 16 . . . 4  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( (/) concat  X )  =  X )
4241oveq1d 6096 . . 3  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( ( (/) concat  X ) concat  Y )  =  ( X concat  Y ) )
4334, 37, 423brtr3d 4241 . 2  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  Y )  .~  ( X concat  Y ) )
444, 32, 43ertrd 6921 1  |-  ( ( A  .~  X  /\  B  .~  Y )  -> 
( A concat  B )  .~  ( X concat  Y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   {crab 2709   _Vcvv 2956    \ cdif 3317   (/)c0 3628   {csn 3814   <.cop 3817   <.cotp 3818   U_ciun 4093   class class class wbr 4212    e. cmpt 4266    _I cid 4493    X. cxp 4876   ran crn 4879   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1oc1o 6717   2oc2o 6718    Er wer 6902   0cc0 8990   1c1 8991    - cmin 9291   ...cfz 11043  ..^cfzo 11135   #chash 11618  Word cword 11717   concat cconcat 11718   splice csplice 11721   <"cs2 11805   ~FG cefg 15338
This theorem is referenced by:  frgpcpbl  15391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-ot 3824  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-ec 6907  df-map 7020  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-fzo 11136  df-hash 11619  df-word 11723  df-concat 11724  df-s1 11725  df-substr 11726  df-splice 11727  df-s2 11812  df-efg 15341
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