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Theorem efgsdmi 15356
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 15355 . . 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 10513 . . . . . . 7  |-  NN  =  ( ZZ>= `  1 )
119, 10syl6eleq 2525 . . . . . 6  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
( ( # `  F
)  -  1 )  e.  ( ZZ>= `  1
) )
12 eluzfz1 11056 . . . . . 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 15354 . . . . . . . . . 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 3324 . . . . . . 7  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  ->  F  e. Word  W )
18 lencl 11727 . . . . . . 7  |-  ( F  e. Word  W  ->  ( # `
 F )  e. 
NN0 )
19 nn0z 10296 . . . . . . 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 11133 . . . . . 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 2512 . . . 4  |-  ( ( F  e.  dom  S  /\  ( ( # `  F
)  -  1 )  e.  NN )  -> 
1  e.  ( 1..^ ( # `  F
) ) )
24 fzoend 11179 . . . 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 5720 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  i )  =  ( F `  ( (
# `  F )  -  1 ) ) )
29 oveq1 6080 . . . . . . . 8  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( i  -  1 )  =  ( ( ( # `  F )  -  1 )  -  1 ) )
3029fveq2d 5724 . . . . . . 7  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( F `  ( i  -  1 ) )  =  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) )
3130fveq2d 5724 . . . . . 6  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( T `  ( F `  (
i  -  1 ) ) )  =  ( T `  ( F `
 ( ( (
# `  F )  -  1 )  - 
1 ) ) ) )
3231rneqd 5089 . . . . 5  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ran  ( T `
 ( F `  ( i  -  1 ) ) )  =  ran  ( T `  ( F `  ( ( ( # `  F
)  -  1 )  -  1 ) ) ) )
3328, 32eleq12d 2503 . . . 4  |-  ( i  =  ( ( # `  F )  -  1 )  ->  ( ( F `  i )  e.  ran  ( T `  ( F `  ( i  -  1 ) ) )  <->  ( F `  ( ( # `  F
)  -  1 ) )  e.  ran  ( T `  ( F `  ( ( ( # `  F )  -  1 )  -  1 ) ) ) ) )
3433rspcv 3040 . . 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 2509 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 1652    e. wcel 1725   A.wral 2697   {crab 2701    \ cdif 3309   (/)c0 3620   {csn 3806   <.cop 3809   <.cotp 3810   U_ciun 4085    e. cmpt 4258    _I cid 4485    X. cxp 4868   dom cdm 4870   ran crn 4871   ` cfv 5446  (class class class)co 6073    e. cmpt2 6075   1oc1o 6709   2oc2o 6710   0cc0 8982   1c1 8983    - cmin 9283   NNcn 9992   NN0cn0 10213   ZZcz 10274   ZZ>=cuz 10480   ...cfz 11035  ..^cfzo 11127   #chash 11610  Word cword 11709   splice csplice 11713   <"cs2 11797   ~FG cefg 15330
This theorem is referenced by:  efgs1b  15360  efgredlemg  15366  efgredlemd  15368  efgredlem  15371
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-hash 11611  df-word 11715
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