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Theorem efgsdmi 15292
Description: Property of the last link in the chain of extensions. (Contributed by Mario Carneiro, 29-Sep-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
efgsdmi  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
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:    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)    F( x, y, z, w, v, t, k, m, n)    I( k)    M( y, z, k)

Proof of Theorem efgsdmi
Dummy variable  i is distinct from all other variables.
StepHypRef Expression
1 efgval.w . . . 4  |-  W  =  (  _I  ` Word  ( I  X.  2o ) )
2 efgval.r . . . 4  |-  .~  =  ( ~FG  `  I )
3 efgval2.m . . . 4  |-  M  =  ( y  e.  I ,  z  e.  2o  |->  <. y ,  ( 1o 
\  z ) >.
)
4 efgval2.t . . . 4  |-  T  =  ( v  e.  W  |->  ( n  e.  ( 0 ... ( # `  v ) ) ,  w  e.  ( I  X.  2o )  |->  ( v splice  <. n ,  n ,  <" w ( M `  w ) "> >. )
) )
5 efgred.d . . . 4  |-  D  =  ( W  \  U_ x  e.  W  ran  ( T `  x ) )
6 efgred.s . . . 4  |-  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, 6efgsval 15291 . . 3  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
87adantr 452 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
9 simpr 448 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  NN )
10 nnuz 10454 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
119, 10syl6eleq 2478 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
) )
12 eluzfz1 10997 . . . . . 6  |-  ( ( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... (
( # `  F )  -  1 ) ) )
1311, 12syl 16 . . . . 5  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1 ... ( ( # `  F )  -  1 ) ) )
141, 2, 3, 4, 5, 6efgsdm 15290 . . . . . . . . . 10  |-  ( F  e.  dom  S  <->  ( F  e.  (Word  W  \  { (/)
} )  /\  ( F `  0 )  e.  D  /\  A. i  e.  ( 1..^ ( # `  F ) ) ( F `  i )  e.  ran  ( T `
 ( F `  ( i  -  1 ) ) ) ) )
1514simp1bi 972 . . . . . . . . 9  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
1615adantr 452 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e.  (Word  W  \  { (/) } ) )
1716eldifad 3276 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e. Word  W )
18 lencl 11663 . . . . . . 7  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
19 nn0z 10237 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ZZ )
2017, 18, 193syl 19 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( # `  F )  e.  ZZ )
21 fzoval 11072 . . . . . 6  |-  ( (
# `  F )  e.  ZZ  ->  ( 1..^ ( # `  F
) )  =  ( 1 ... ( (
# `  F )  -  1 ) ) )
2220, 21syl 16 . . . . 5  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( 1..^ ( # `  F ) )  =  ( 1 ... (
( # `  F )  -  1 ) ) )
2313, 22eleqtrrd 2465 . . . 4  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1..^ ( # `  F
) ) )
24 fzoend 11115 . . . 4  |-  ( 1  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ( 1..^ ( # `  F
) ) )
2523, 24syl 16 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( 1..^ (
# `  F )
) )
2614simp3bi 974 . . . 4  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
2726adantr 452 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
28 fveq2 5669 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  i )  =  ( F `  ( (
# `  F )  -  1 ) ) )
29 oveq1 6028 . . . . . . . 8  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( i  -  1 )  =  ( ( ( # `  F )  -  1 )  -  1 ) )
3029fveq2d 5673 . . . . . . 7  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  ( i  -  1 ) )  =  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) )
3130fveq2d 5673 . . . . . 6  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( T `  ( F `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( ( (
# `  F )  -  1 )  - 
1 ) ) ) )
3231rneqd 5038 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ran  ( T `
 ( F `  ( i  -  1 ) ) )  =  ran  ( T `  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) ) )
3328, 32eleq12d 2456 . . . 4  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( ( F `  i )  e.  ran  ( T `  ( F `  ( i  -  1 ) ) )  <->  ( F `  ( ( # `  F
)  -  1 ) )  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) ) )
3433rspcv 2992 . . 3  |-  ( ( ( # `  F
)  -  1 )  e.  ( 1..^ (
# `  F )
)  ->  ( A. i  e.  ( 1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) )  ->  ( F `  ( ( # `  F
)  -  1 ) )  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) ) )
3525, 27, 34sylc 58 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( F `  (
( # `  F )  -  1 ) )  e.  ran  ( T `
 ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
368, 35eqeltrd 2462 1  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   A.wral 2650   {crab 2654    \ cdif 3261   (/)c0 3572   {csn 3758   <.cop 3761   <.cotp 3762   U_ciun 4036    e. cmpt 4208    _I cid 4435    X. cxp 4817   dom cdm 4819   ran crn 4820   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1oc1o 6654   2oc2o 6655   0cc0 8924   1c1 8925    - cmin 9224   NNcn 9933   NN0cn0 10154   ZZcz 10215   ZZ>=cuz 10421   ...cfz 10976  ..^cfzo 11066   #chash 11546  Word cword 11645   splice csplice 11649   <"cs2 11733   ~FG cefg 15266
This theorem is referenced by:  efgs1b  15296  efgredlemg  15302  efgredlemd  15304  efgredlem  15307
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 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-fzo 11067  df-hash 11547  df-word 11651
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