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Theorem efgsval2 15294
Description: Value of the auxiliary function  S defining a sequence of extensions (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
efgsval2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  B )
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 efgsval2
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 15292 . . 3  |-  ( ( A concat  <" B "> )  e.  dom  S  ->  ( S `  ( A concat  <" B "> ) )  =  ( ( A concat  <" B "> ) `  (
( # `  ( A concat  <" B "> ) )  -  1 ) ) )
873ad2ant3 980 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  ( ( A concat  <" B "> ) `  (
( # `  ( A concat  <" B "> ) )  -  1 ) ) )
9 lencl 11664 . . . . . . 7  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1093ad2ant1 978 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  NN0 )
1110nn0cnd 10210 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  CC )
12 ax-1cn 8983 . . . . 5  |-  1  e.  CC
13 pncan 9245 . . . . 5  |-  ( ( ( # `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  A )  +  1 )  -  1 )  =  ( # `  A
) )
1411, 12, 13sylancl 644 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( (
( # `  A )  +  1 )  - 
1 )  =  (
# `  A )
)
15 simp1 957 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  A  e. Word  W )
16 simp2 958 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  B  e.  W )
1716s1cld 11685 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  <" B ">  e. Word  W )
18 ccatlen 11673 . . . . . . 7  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W )  ->  ( # `
 ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
1915, 17, 18syl2anc 643 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
20 s1len 11687 . . . . . . 7  |-  ( # `  <" B "> )  =  1
2120oveq2i 6033 . . . . . 6  |-  ( (
# `  A )  +  ( # `  <" B "> )
)  =  ( (
# `  A )  +  1 )
2219, 21syl6eq 2437 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  ( A concat  <" B "> ) )  =  ( ( # `  A
)  +  1 ) )
2322oveq1d 6037 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( ( ( # `  A
)  +  1 )  -  1 ) )
2411addid2d 9201 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( 0  +  ( # `  A
) )  =  (
# `  A )
)
2514, 23, 243eqtr4d 2431 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A concat  <" B "> ) )  - 
1 )  =  ( 0  +  ( # `  A ) ) )
2625fveq2d 5674 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( (
# `  ( A concat  <" B "> ) )  -  1 ) )  =  ( ( A concat  <" B "> ) `  (
0  +  ( # `  A ) ) ) )
27 1nn 9945 . . . . . . 7  |-  1  e.  NN
2820, 27eqeltri 2459 . . . . . 6  |-  ( # `  <" B "> )  e.  NN
2928a1i 11 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( # `  <" B "> )  e.  NN )
30 lbfzo0 11102 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  <" B "> ) )  <->  ( # `  <" B "> )  e.  NN )
3129, 30sylibr 204 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  0  e.  ( 0..^ ( # `  <" B "> )
) )
32 ccatval3 11676 . . . 4  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" B "> ) ) )  -> 
( ( A concat  <" B "> ) `  (
0  +  ( # `  A ) ) )  =  ( <" B "> `  0 )
)
3315, 17, 31, 32syl3anc 1184 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  ( <" B "> `  0 )
)
34 s1fv 11689 . . . 4  |-  ( B  e.  W  ->  ( <" B "> `  0 )  =  B )
35343ad2ant2 979 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( <" B "> `  0
)  =  B )
3633, 35eqtrd 2421 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( ( A concat  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  B )
378, 26, 363eqtrd 2425 1  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A concat  <" B "> )  e.  dom  S )  ->  ( S `  ( A concat  <" B "> ) )  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2651   {crab 2655    \ cdif 3262   (/)c0 3573   {csn 3759   <.cop 3762   <.cotp 3763   U_ciun 4037    e. cmpt 4209    _I cid 4436    X. cxp 4818   dom cdm 4820   ran crn 4821   ` cfv 5396  (class class class)co 6022    e. cmpt2 6024   1oc1o 6655   2oc2o 6656   CCcc 8923   0cc0 8925   1c1 8926    + caddc 8928    - cmin 9225   NNcn 9934   NN0cn0 10155   ...cfz 10977  ..^cfzo 11067   #chash 11547  Word cword 11646   concat cconcat 11647   <"cs1 11648   splice csplice 11650   <"cs2 11734   ~FG cefg 15267
This theorem is referenced by:  efgsfo  15300  efgredlemd  15305  efgrelexlemb  15311
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 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-oadd 6666  df-er 6843  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-hash 11548  df-word 11652  df-concat 11653  df-s1 11654
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