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Theorem efgsdmi 15041
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 15040 . . 3  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
87adantr 451 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
9 simpr 447 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  NN )
10 nnuz 10263 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
119, 10syl6eleq 2373 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
) )
12 eluzfz1 10803 . . . . . 6  |-  ( ( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... (
( # `  F )  -  1 ) ) )
1311, 12syl 15 . . . . 5  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1 ... ( ( # `  F )  -  1 ) ) )
141, 2, 3, 4, 5, 6efgsdm 15039 . . . . . . . . . 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 970 . . . . . . . . 9  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
1615adantr 451 . . . . . . . 8  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e.  (Word  W  \  { (/) } ) )
17 eldifi 3298 . . . . . . . 8  |-  ( F  e.  (Word  W  \  { (/) } )  ->  F  e. Word  W )
1816, 17syl 15 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e. Word  W )
19 lencl 11421 . . . . . . 7  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
20 nn0z 10046 . . . . . . 7  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ZZ )
2118, 19, 203syl 18 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( # `  F )  e.  ZZ )
22 fzoval 10876 . . . . . 6  |-  ( (
# `  F )  e.  ZZ  ->  ( 1..^ ( # `  F
) )  =  ( 1 ... ( (
# `  F )  -  1 ) ) )
2321, 22syl 15 . . . . 5  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( 1..^ ( # `  F ) )  =  ( 1 ... (
( # `  F )  -  1 ) ) )
2413, 23eleqtrrd 2360 . . . 4  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1..^ ( # `  F
) ) )
25 fzoend 10914 . . . 4  |-  ( 1  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ( 1..^ ( # `  F
) ) )
2624, 25syl 15 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( 1..^ (
# `  F )
) )
2714simp3bi 972 . . . 4  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
2827adantr 451 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
29 fveq2 5525 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  i )  =  ( F `  ( (
# `  F )  -  1 ) ) )
30 oveq1 5865 . . . . . . . 8  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( i  -  1 )  =  ( ( ( # `  F )  -  1 )  -  1 ) )
3130fveq2d 5529 . . . . . . 7  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  ( i  -  1 ) )  =  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) )
3231fveq2d 5529 . . . . . 6  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( T `  ( F `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( ( (
# `  F )  -  1 )  - 
1 ) ) ) )
3332rneqd 4906 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ran  ( T `
 ( F `  ( i  -  1 ) ) )  =  ran  ( T `  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) ) )
3429, 33eleq12d 2351 . . . 4  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( ( F `  i )  e.  ran  ( T `  ( F `  ( i  -  1 ) ) )  <->  ( F `  ( ( # `  F
)  -  1 ) )  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) ) )
3534rspcv 2880 . . 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 ) ) ) ) )
3626, 28, 35sylc 56 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( F `  (
( # `  F )  -  1 ) )  e.  ran  ( T `
 ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
378, 36eqeltrd 2357 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 358    = wceq 1623    e. wcel 1684   A.wral 2543   {crab 2547    \ cdif 3149   (/)c0 3455   {csn 3640   <.cop 3643   <.cotp 3644   U_ciun 3905    e. cmpt 4077    _I cid 4304    X. cxp 4687   dom cdm 4689   ran crn 4690   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1oc1o 6472   2oc2o 6473   0cc0 8737   1c1 8738    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782  ..^cfzo 10870   #chash 11337  Word cword 11403   splice csplice 11407   <"cs2 11491   ~FG cefg 15015
This theorem is referenced by:  efgs1b  15045  efgredlemg  15051  efgredlemd  15053  efgredlem  15056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-hash 11338  df-word 11409
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