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Theorem efgred2 15377
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 15363 . . . . . . 7  |-  S : dom  S -onto-> W
8 fof 5645 . . . . . . 7  |-  ( S : dom  S -onto-> W  ->  S : dom  S --> W )
97, 8ax-mp 8 . . . . . 6  |-  S : dom  S --> W
109ffvelrni 5861 . . . . 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 15376 . . . 4  |-  ( ( S `  B )  e.  W  ->  E! d  e.  D  d  .~  ( S `  B
) )
13 reurmo 2915 . . . 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 15354 . . . . 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 15342 . . . . 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 15358 . . . . 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 6913 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  B ) )
241, 2, 3, 4, 5, 6efgsdm 15354 . . . . 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 15358 . . . 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 4207 . . . 4  |-  ( d  =  ( A ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( A `  0 )  .~  ( S `  B ) ) )
30 breq1 4207 . . . 4  |-  ( d  =  ( B ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( B `  0 )  .~  ( S `  B ) ) )
3129, 30rmoi 3242 . . 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 4224 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  B
) )
3833, 34, 37ertr3d 6915 . 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 1652    e. wcel 1725   A.wral 2697   E!wreu 2699   E*wrmo 2700   {crab 2701    \ cdif 3309   (/)c0 3620   {csn 3806   <.cop 3809   <.cotp 3810   U_ciun 4085   class class class wbr 4204    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870   ran crn 4871   -->wf 5442   -onto->wfo 5444   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710    Er wer 6894   0cc0 8982   1c1 8983    - cmin 9283   ...cfz 11035  ..^cfzo 11127   #chash 11610  Word cword 11709   splice csplice 11713   <"cs2 11797   ~FG cefg 15330
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-ot 3816  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-ec 6899  df-map 7012  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715  df-concat 11716  df-s1 11717  df-substr 11718  df-splice 11719  df-s2 11804  df-efg 15333
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