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Theorem efgsval2 17324
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 ++  <" B "> )  e.  dom  S )  ->  ( S `  ( A ++  <" 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 17322 . . 3  |-  ( ( A ++  <" B "> )  e.  dom  S  ->  ( S `  ( A ++  <" B "> ) )  =  ( ( A ++  <" B "> ) `  ( ( # `  ( A ++  <" B "> ) )  -  1 ) ) )
873ad2ant3 1028 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( S `  ( A ++  <" B "> ) )  =  ( ( A ++  <" B "> ) `  ( ( # `  ( A ++  <" B "> ) )  -  1 ) ) )
9 lencl 12663 . . . . . . 7  |-  ( A  e. Word  W  ->  ( # `
 A )  e. 
NN0 )
1093ad2ant1 1026 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  NN0 )
1110nn0cnd 10916 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( # `  A
)  e.  CC )
12 ax-1cn 9586 . . . . 5  |-  1  e.  CC
13 pncan 9870 . . . . 5  |-  ( ( ( # `  A
)  e.  CC  /\  1  e.  CC )  ->  ( ( ( # `  A )  +  1 )  -  1 )  =  ( # `  A
) )
1411, 12, 13sylancl 666 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( (
( # `  A )  +  1 )  - 
1 )  =  (
# `  A )
)
15 simp1 1005 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  A  e. Word  W )
16 simp2 1006 . . . . . . . 8  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  B  e.  W )
1716s1cld 12718 . . . . . . 7  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  <" B ">  e. Word  W )
18 ccatlen 12697 . . . . . . 7  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W )  ->  ( # `
 ( A ++  <" B "> )
)  =  ( (
# `  A )  +  ( # `  <" B "> )
) )
1915, 17, 18syl2anc 665 . . . . . 6  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( # `  ( A ++  <" B "> ) )  =  ( ( # `  A
)  +  ( # `  <" B "> ) ) )
20 s1len 12720 . . . . . . 7  |-  ( # `  <" B "> )  =  1
2120oveq2i 6307 . . . . . 6  |-  ( (
# `  A )  +  ( # `  <" B "> )
)  =  ( (
# `  A )  +  1 )
2219, 21syl6eq 2477 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( # `  ( A ++  <" B "> ) )  =  ( ( # `  A
)  +  1 ) )
2322oveq1d 6311 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A ++  <" B "> )
)  -  1 )  =  ( ( (
# `  A )  +  1 )  - 
1 ) )
2411addid2d 9823 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( 0  +  ( # `  A
) )  =  (
# `  A )
)
2514, 23, 243eqtr4d 2471 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( ( # `
 ( A ++  <" B "> )
)  -  1 )  =  ( 0  +  ( # `  A
) ) )
2625fveq2d 5876 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( ( A ++  <" B "> ) `  ( (
# `  ( A ++  <" B "> ) )  -  1 ) )  =  ( ( A ++  <" B "> ) `  (
0  +  ( # `  A ) ) ) )
27 1nn 10609 . . . . . . 7  |-  1  e.  NN
2820, 27eqeltri 2504 . . . . . 6  |-  ( # `  <" B "> )  e.  NN
2928a1i 11 . . . . 5  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( # `  <" B "> )  e.  NN )
30 lbfzo0 11942 . . . . 5  |-  ( 0  e.  ( 0..^ (
# `  <" B "> ) )  <->  ( # `  <" B "> )  e.  NN )
3129, 30sylibr 215 . . . 4  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  0  e.  ( 0..^ ( # `  <" B "> )
) )
32 ccatval3 12700 . . . 4  |-  ( ( A  e. Word  W  /\  <" B ">  e. Word  W  /\  0  e.  ( 0..^ ( # `  <" B "> ) ) )  -> 
( ( A ++  <" B "> ) `  ( 0  +  (
# `  A )
) )  =  (
<" B "> `  0 ) )
3315, 17, 31, 32syl3anc 1264 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( ( A ++  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  ( <" B "> `  0 )
)
34 s1fv 12722 . . . 4  |-  ( B  e.  W  ->  ( <" B "> `  0 )  =  B )
35343ad2ant2 1027 . . 3  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( <" B "> `  0
)  =  B )
3633, 35eqtrd 2461 . 2  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( ( A ++  <" B "> ) `  ( 0  +  ( # `  A
) ) )  =  B )
378, 26, 363eqtrd 2465 1  |-  ( ( A  e. Word  W  /\  B  e.  W  /\  ( A ++  <" B "> )  e.  dom  S )  ->  ( S `  ( A ++  <" B "> ) )  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1867   A.wral 2773   {crab 2777    \ cdif 3430   (/)c0 3758   {csn 3993   <.cop 3999   <.cotp 4001   U_ciun 4293    |-> cmpt 4475    _I cid 4755    X. cxp 4843   dom cdm 4845   ran crn 4846   ` cfv 5592  (class class class)co 6296    |-> cmpt2 6298   1oc1o 7174   2oc2o 7175   CCcc 9526   0cc0 9528   1c1 9529    + caddc 9531    - cmin 9849   NNcn 10598   NN0cn0 10858   ...cfz 11771  ..^cfzo 11902   #chash 12501  Word cword 12632   ++ cconcat 12634   <"cs1 12635   splice csplice 12637   <"cs2 12911   ~FG cefg 17297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-pss 3449  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-tp 3998  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-br 4418  df-opab 4476  df-mpt 4477  df-tr 4512  df-eprel 4756  df-id 4760  df-po 4766  df-so 4767  df-fr 4804  df-we 4806  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-pred 5390  df-ord 5436  df-on 5437  df-lim 5438  df-suc 5439  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7027  df-recs 7089  df-rdg 7127  df-1o 7181  df-oadd 7185  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-fin 7572  df-card 8363  df-cda 8587  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852  df-nn 10599  df-2 10657  df-n0 10859  df-z 10927  df-uz 11149  df-fz 11772  df-fzo 11903  df-hash 12502  df-word 12640  df-concat 12642  df-s1 12643
This theorem is referenced by:  efgsfo  17330  efgredlemd  17335  efgrelexlemb  17341
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