MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  efgred2 Structured version   Unicode version

Theorem efgred2 17093
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 17079 . . . . . . 7  |-  S : dom  S -onto-> W
8 fof 5777 . . . . . . 7  |-  ( S : dom  S -onto-> W  ->  S : dom  S --> W )
97, 8ax-mp 5 . . . . . 6  |-  S : dom  S --> W
109ffvelrni 6007 . . . . 5  |-  ( B  e.  dom  S  -> 
( S `  B
)  e.  W )
1110ad2antlr 725 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  B )  e.  W
)
121, 2, 3, 4, 5, 6efgredeu 17092 . . . 4  |-  ( ( S `  B )  e.  W  ->  E! d  e.  D  d  .~  ( S `  B
) )
13 reurmo 3024 . . . 4  |-  ( E! d  e.  D  d  .~  ( S `  B )  ->  E* d  e.  D  d  .~  ( S `  B
) )
1411, 12, 133syl 20 . . 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 17070 . . . . 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 1013 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
1716ad2antrr 724 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  e.  D )
181, 2efger 17058 . . . . 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 17074 . . . . 5  |-  ( A  e.  dom  S  -> 
( A `  0
)  .~  ( S `  A ) )
2120ad2antrr 724 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  A ) )
22 simpr 459 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  A )  .~  ( S `  B )
)
2319, 21, 22ertrd 7363 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  B ) )
241, 2, 3, 4, 5, 6efgsdm 17070 . . . . 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 1013 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  e.  D )
2625ad2antlr 725 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  e.  D )
271, 2, 3, 4, 5, 6efgsrel 17074 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  .~  ( S `  B ) )
2827ad2antlr 725 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  .~  ( S `  B ) )
29 breq1 4397 . . . 4  |-  ( d  =  ( A ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( A `  0 )  .~  ( S `  B ) ) )
30 breq1 4397 . . . 4  |-  ( d  =  ( B ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( B `  0 )  .~  ( S `  B ) ) )
3129, 30rmoi 3369 . . 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 1239 . 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 724 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  A
) )
35 simpr 459 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  =  ( B ` 
0 ) )
3627ad2antlr 725 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( B `  0 )  .~  ( S `  B
) )
3735, 36eqbrtrd 4414 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  B
) )
3833, 34, 37ertr3d 7365 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( S `  A )  .~  ( S `  B
) )
3932, 38impbida 833 1  |-  ( ( A  e.  dom  S  /\  B  e.  dom  S )  ->  ( ( S `  A )  .~  ( S `  B
)  <->  ( A ` 
0 )  =  ( B `  0 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2753   E!wreu 2755   E*wrmo 2756   {crab 2757    \ cdif 3410   (/)c0 3737   {csn 3971   <.cop 3977   <.cotp 3979   U_ciun 4270   class class class wbr 4394    |-> cmpt 4452    _I cid 4732    X. cxp 4820   dom cdm 4822   ran crn 4823   -->wf 5564   -onto->wfo 5566   ` cfv 5568  (class class class)co 6277    |-> cmpt2 6279   1oc1o 7159   2oc2o 7160    Er wer 7344   0cc0 9521   1c1 9522    - cmin 9840   ...cfz 11724  ..^cfzo 11852   #chash 12450  Word cword 12581   splice csplice 12586   <"cs2 12860   ~FG cefg 17046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-ot 3980  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-2o 7167  df-oadd 7170  df-er 7347  df-ec 7349  df-map 7458  df-pm 7459  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-rp 11265  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-concat 12591  df-s1 12592  df-substr 12593  df-splice 12594  df-s2 12867  df-efg 17049
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator