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Theorem efgsdmi 16546
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 16545 . . 3  |-  ( F  e.  dom  S  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
87adantr 465 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  =  ( F `
 ( ( # `  F )  -  1 ) ) )
9 simpr 461 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  NN )
10 nnuz 11113 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
119, 10syl6eleq 2565 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
) )
12 eluzfz1 11689 . . . . . 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 16544 . . . . . . . . 9  |-  ( 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 1011 . . . . . . . 8  |-  ( F  e.  dom  S  ->  F  e.  (Word  W  \  { (/) } ) )
1615adantr 465 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e.  (Word  W  \  { (/) } ) )
1716eldifad 3488 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e. Word  W )
18 lencl 12524 . . . . . 6  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
19 nn0z 10883 . . . . . 6  |-  ( (
# `  F )  e.  NN0  ->  ( # `  F
)  e.  ZZ )
20 fzoval 11794 . . . . . 6  |-  ( (
# `  F )  e.  ZZ  ->  ( 1..^ ( # `  F
) )  =  ( 1 ... ( (
# `  F )  -  1 ) ) )
2117, 18, 19, 204syl 21 . . . . 5  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( 1..^ ( # `  F ) )  =  ( 1 ... (
( # `  F )  -  1 ) ) )
2213, 21eleqtrrd 2558 . . . 4  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1..^ ( # `  F
) ) )
23 fzoend 11867 . . . 4  |-  ( 1  e.  ( 1..^ (
# `  F )
)  ->  ( ( # `
 F )  - 
1 )  e.  ( 1..^ ( # `  F
) ) )
2422, 23syl 16 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( 1..^ (
# `  F )
) )
2514simp3bi 1013 . . . 4  |-  ( F  e.  dom  S  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
2625adantr 465 . . 3  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  A. i  e.  (
1..^ ( # `  F
) ) ( F `
 i )  e. 
ran  ( T `  ( F `  ( i  -  1 ) ) ) )
27 fveq2 5864 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  i )  =  ( F `  ( (
# `  F )  -  1 ) ) )
28 oveq1 6289 . . . . . . . 8  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( i  -  1 )  =  ( ( ( # `  F )  -  1 )  -  1 ) )
2928fveq2d 5868 . . . . . . 7  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  ( i  -  1 ) )  =  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) )
3029fveq2d 5868 . . . . . 6  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( T `  ( F `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( ( (
# `  F )  -  1 )  - 
1 ) ) ) )
3130rneqd 5228 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ran  ( T `
 ( F `  ( i  -  1 ) ) )  =  ran  ( T `  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) ) )
3227, 31eleq12d 2549 . . . 4  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( ( F `  i )  e.  ran  ( T `  ( F `  ( i  -  1 ) ) )  <->  ( F `  ( ( # `  F
)  -  1 ) )  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) ) )
3332rspcv 3210 . . 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 ) ) ) ) )
3424, 26, 33sylc 60 . 2  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( F `  (
( # `  F )  -  1 ) )  e.  ran  ( T `
 ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
358, 34eqeltrd 2555 1  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( S `  F
)  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   {crab 2818    \ cdif 3473   (/)c0 3785   {csn 4027   <.cop 4033   <.cotp 4035   U_ciun 4325    |-> cmpt 4505    _I cid 4790    X. cxp 4997   dom cdm 4999   ran crn 5000   ` cfv 5586  (class class class)co 6282    |-> cmpt2 6284   1oc1o 7120   2oc2o 7121   0cc0 9488   1c1 9489    - cmin 9801   NNcn 10532   NN0cn0 10791   ZZcz 10860   ZZ>=cuz 11078   ...cfz 11668  ..^cfzo 11788   #chash 12369  Word cword 12496   splice csplice 12501   <"cs2 12765   ~FG cefg 16520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12370  df-word 12504
This theorem is referenced by:  efgs1b  16550  efgredlemg  16556  efgredlemd  16558  efgredlem  16561
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