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Theorem efgred2 15305
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
efgred2  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  .~  ( S `  B
)  <->  ( A ` 
0 )  =  ( B `  0 ) ) )
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)

Proof of Theorem efgred2
Dummy variables  d 
i are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . . . . . 8  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . . . . . 8  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . . . . . 8  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . . . . . 8  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . . . . . 8  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . . . . . 8  |-  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 ) ) )
71, 2, 3, 4, 5, 6efgsfo 15291 . . . . . . 7  |-  S : dom  S -onto-> W
8 fof 5586 . . . . . . 7  |-  ( S : dom  S -onto-> W  ->  S : dom  S --> W )
97, 8ax-mp 8 . . . . . 6  |-  S : dom  S --> W
109ffvelrni 5801 . . . . 5  |-  ( B  e.  dom  S  -> 
( S `  B
)  e.  W )
1110ad2antlr 708 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  B )  e.  W
)
121, 2, 3, 4, 5, 6efgredeu 15304 . . . 4  |-  ( ( S `  B )  e.  W  ->  E! d  e.  D  d  .~  ( S `  B
) )
13 reurmo 2859 . . . 4  |-  ( E! d  e.  D  d  .~  ( S `  B )  ->  E* d  e.  D d  .~  ( S `  B
) )
1411, 12, 133syl 19 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  E* d  e.  D d  .~  ( S `  B )
)
151, 2, 3, 4, 5, 6efgsdm 15282 . . . . 5  |-  ( A  e.  dom  S  <->  ( A  e.  (Word  W  \  { (/)
} )  /\  ( A `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  A ) ) ( A `  i )  e.  ran  ( T `
 ( A `  ( i  -  1 ) ) ) ) )
1615simp2bi 973 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
1716ad2antrr 707 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  e.  D )
181, 2efger 15270 . . . . 5  |-  .~  Er  W
1918a1i 11 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  .~  Er  W
)
201, 2, 3, 4, 5, 6efgsrel 15286 . . . . 5  |-  ( A  e.  dom  S  -> 
( A `  0
)  .~  ( S `  A ) )
2120ad2antrr 707 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  A ) )
22 simpr 448 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  A )  .~  ( S `  B )
)
2319, 21, 22ertrd 6850 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  B ) )
241, 2, 3, 4, 5, 6efgsdm 15282 . . . . 5  |-  ( B  e.  dom  S  <->  ( B  e.  (Word  W  \  { (/)
} )  /\  ( B `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  B ) ) ( B `  i )  e.  ran  ( T `
 ( B `  ( i  -  1 ) ) ) ) )
2524simp2bi 973 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  e.  D )
2625ad2antlr 708 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  e.  D )
271, 2, 3, 4, 5, 6efgsrel 15286 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  .~  ( S `  B ) )
2827ad2antlr 708 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  .~  ( S `  B ) )
29 breq1 4149 . . . 4  |-  ( d  =  ( A ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( A `  0 )  .~  ( S `  B ) ) )
30 breq1 4149 . . . 4  |-  ( d  =  ( B ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( B `  0 )  .~  ( S `  B ) ) )
3129, 30rmoi 3186 . . 3  |-  ( ( E* d  e.  D
d  .~  ( S `  B )  /\  (
( A `  0
)  e.  D  /\  ( A `  0 )  .~  ( S `  B ) )  /\  ( ( B ` 
0 )  e.  D  /\  ( B `  0
)  .~  ( S `  B ) ) )  ->  ( A ` 
0 )  =  ( B `  0 ) )
3214, 17, 23, 26, 28, 31syl122anc 1193 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  =  ( B `  0
) )
3318a1i 11 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  .~  Er  W )
3420ad2antrr 707 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  A
) )
35 simpr 448 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  =  ( B ` 
0 ) )
3627ad2antlr 708 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( B `  0 )  .~  ( S `  B
) )
3735, 36eqbrtrd 4166 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  B
) )
3833, 34, 37ertr3d 6852 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( S `  A )  .~  ( S `  B
) )
3932, 38impbida 806 1  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  .~  ( S `  B
)  <->  ( A ` 
0 )  =  ( B `  0 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2642   E!wreu 2644   E*wrmo 2645   {crab 2646    \ cdif 3253   (/)c0 3564   {csn 3750   <.cop 3753   <.cotp 3754   U_ciun 4028   class class class wbr 4146    e. cmpt 4200    _I cid 4427    X. cxp 4809   dom cdm 4811   ran crn 4812   -->wf 5383   -onto->wfo 5385   ` cfv 5387  (class class class)co 6013    e. cmpt2 6015   1oc1o 6646   2oc2o 6647    Er wer 6831   0cc0 8916   1c1 8917    - cmin 9216   ...cfz 10968  ..^cfzo 11058   #chash 11538  Word cword 11637   splice csplice 11641   <"cs2 11725   ~FG cefg 15258
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-pss 3272  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-tp 3758  df-op 3759  df-ot 3760  df-uni 3951  df-int 3986  df-iun 4030  df-iin 4031  df-br 4147  df-opab 4201  df-mpt 4202  df-tr 4237  df-eprel 4428  df-id 4432  df-po 4437  df-so 4438  df-fr 4475  df-we 4477  df-ord 4518  df-on 4519  df-lim 4520  df-suc 4521  df-om 4779  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-1st 6281  df-2nd 6282  df-riota 6478  df-recs 6562  df-rdg 6597  df-1o 6653  df-2o 6654  df-oadd 6657  df-er 6834  df-ec 6836  df-map 6949  df-pm 6950  df-en 7039  df-dom 7040  df-sdom 7041  df-fin 7042  df-card 7752  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219  df-nn 9926  df-2 9983  df-n0 10147  df-z 10208  df-uz 10414  df-rp 10538  df-fz 10969  df-fzo 11059  df-hash 11539  df-word 11643  df-concat 11644  df-s1 11645  df-substr 11646  df-splice 11647  df-s2 11732  df-efg 15261
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