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Theorem efgred2 16250
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 16236 . . . . . . 7  |-  S : dom  S -onto-> W
8 fof 5620 . . . . . . 7  |-  ( S : dom  S -onto-> W  ->  S : dom  S --> W )
97, 8ax-mp 5 . . . . . 6  |-  S : dom  S --> W
109ffvelrni 5842 . . . . 5  |-  ( B  e.  dom  S  -> 
( S `  B
)  e.  W )
1110ad2antlr 726 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  B )  e.  W
)
121, 2, 3, 4, 5, 6efgredeu 16249 . . . 4  |-  ( ( S `  B )  e.  W  ->  E! d  e.  D  d  .~  ( S `  B
) )
13 reurmo 2938 . . . 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 16227 . . . . 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 1004 . . . 4  |-  ( A  e.  dom  S  -> 
( A `  0
)  e.  D )
1716ad2antrr 725 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  e.  D )
181, 2efger 16215 . . . . 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 16231 . . . . 5  |-  ( A  e.  dom  S  -> 
( A `  0
)  .~  ( S `  A ) )
2120ad2antrr 725 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  A ) )
22 simpr 461 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( S `  A )  .~  ( S `  B )
)
2319, 21, 22ertrd 7117 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( A `  0 )  .~  ( S `  B ) )
241, 2, 3, 4, 5, 6efgsdm 16227 . . . . 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 1004 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  e.  D )
2625ad2antlr 726 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  e.  D )
271, 2, 3, 4, 5, 6efgsrel 16231 . . . 4  |-  ( B  e.  dom  S  -> 
( B `  0
)  .~  ( S `  B ) )
2827ad2antlr 726 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( S `
 A )  .~  ( S `  B ) )  ->  ( B `  0 )  .~  ( S `  B ) )
29 breq1 4295 . . . 4  |-  ( d  =  ( A ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( A `  0 )  .~  ( S `  B ) ) )
30 breq1 4295 . . . 4  |-  ( d  =  ( B ` 
0 )  ->  (
d  .~  ( S `  B )  <->  ( B `  0 )  .~  ( S `  B ) ) )
3129, 30rmoi 3287 . . 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 1227 . 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 725 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  A
) )
35 simpr 461 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  =  ( B ` 
0 ) )
3627ad2antlr 726 . . . 4  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( B `  0 )  .~  ( S `  B
) )
3735, 36eqbrtrd 4312 . . 3  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( A `  0 )  .~  ( S `  B
) )
3833, 34, 37ertr3d 7119 . 2  |-  ( ( ( A  e.  dom  S  /\  B  e.  dom  S )  /\  ( A `
 0 )  =  ( B `  0
) )  ->  ( S `  A )  .~  ( S `  B
) )
3932, 38impbida 828 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 369    = wceq 1369    e. wcel 1756   A.wral 2715   E!wreu 2717   E*wrmo 2718   {crab 2719    \ cdif 3325   (/)c0 3637   {csn 3877   <.cop 3883   <.cotp 3885   U_ciun 4171   class class class wbr 4292    e. cmpt 4350    _I cid 4631    X. cxp 4838   dom cdm 4840   ran crn 4841   -->wf 5414   -onto->wfo 5416   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   1oc1o 6913   2oc2o 6914    Er wer 7098   0cc0 9282   1c1 9283    - cmin 9595   ...cfz 11437  ..^cfzo 11548   #chash 12103  Word cword 12221   splice csplice 12226   <"cs2 12468   ~FG cefg 16203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-ot 3886  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-map 7216  df-pm 7217  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-card 8109  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-hash 12104  df-word 12229  df-concat 12231  df-s1 12232  df-substr 12233  df-splice 12234  df-s2 12475  df-efg 16206
This theorem is referenced by: (None)
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