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Theorem coeid3 22929
Description: Reconstruct a polynomial as an explicit sum of the coefficient function up to at least the degree of the polynomial. (Contributed by Mario Carneiro, 22-Jul-2014.)
Hypotheses
Ref Expression
dgrub.1  |-  A  =  (coeff `  F )
dgrub.2  |-  N  =  (deg `  F )
Assertion
Ref Expression
coeid3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( X ^ k ) ) )
Distinct variable groups:    A, k    k, F    S, k    k, M   
k, N    k, X

Proof of Theorem coeid3
StepHypRef Expression
1 dgrub.1 . . . 4  |-  A  =  (coeff `  F )
2 dgrub.2 . . . 4  |-  N  =  (deg `  F )
31, 2coeid2 22928 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( X ^ k ) ) )
433adant2 1016 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( X ^ k ) ) )
5 fzss2 11778 . . . 4  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
653ad2ant2 1019 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... N )  C_  (
0 ... M ) )
7 elfznn0 11826 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
81coef3 22921 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
983ad2ant1 1018 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  A : NN0 --> CC )
109ffvelrnda 6009 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
11 expcl 12228 . . . . . 6  |-  ( ( X  e.  CC  /\  k  e.  NN0 )  -> 
( X ^ k
)  e.  CC )
12113ad2antl3 1161 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( X ^
k )  e.  CC )
1310, 12mulcld 9646 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  x.  ( X ^ k
) )  e.  CC )
147, 13sylan2 472 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( A `  k )  x.  ( X ^ k
) )  e.  CC )
15 eldifn 3566 . . . . . . 7  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
1615adantl 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
17 simpl1 1000 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  F  e.  (Poly `  S ) )
18 eldifi 3565 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
19 elfzuz 11738 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
2018, 19syl 17 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
2120adantl 464 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
22 nn0uz 11161 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2321, 22syl6eleqr 2501 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  NN0 )
241, 2dgrub 22923 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
25243expia 1199 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
2617, 23, 25syl2anc 659 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
27 simpl2 1001 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  M  e.  ( ZZ>= `  N )
)
28 eluzel2 11132 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
2927, 28syl 17 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  N  e.  ZZ )
30 elfz5 11734 . . . . . . . . 9  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
3121, 29, 30syl2anc 659 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
3226, 31sylibrd 234 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  e.  ( 0 ... N
) ) )
3332necon1bd 2621 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( -.  k  e.  ( 0 ... N )  -> 
( A `  k
)  =  0 ) )
3416, 33mpd 15 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( A `  k )  =  0 )
3534oveq1d 6293 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  x.  ( X ^ k
) )  =  ( 0  x.  ( X ^ k ) ) )
36 elfznn0 11826 . . . . . . 7  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
3718, 36syl 17 . . . . . 6  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
3837, 12sylan2 472 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( X ^ k )  e.  CC )
3938mul02d 9812 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( 0  x.  ( X ^
k ) )  =  0 )
4035, 39eqtrd 2443 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  x.  ( X ^ k
) )  =  0 )
41 fzfid 12124 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... M )  e.  Fin )
426, 14, 40, 41fsumss 13696 . 2  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( X ^ k ) )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k )  x.  ( X ^ k ) ) )
434, 42eqtrd 2443 1  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... M ) ( ( A `  k
)  x.  ( X ^ k ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842    =/= wne 2598    \ cdif 3411    C_ wss 3414   class class class wbr 4395   -->wf 5565   ` cfv 5569  (class class class)co 6278   CCcc 9520   0cc0 9522    x. cmul 9527    <_ cle 9659   NN0cn0 10836   ZZcz 10905   ZZ>=cuz 11127   ...cfz 11726   ^cexp 12210   sum_csu 13657  Polycply 22873  coeffccoe 22875  degcdgr 22876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-fal 1411  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-oadd 7171  df-er 7348  df-map 7459  df-pm 7460  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-sup 7935  df-oi 7969  df-card 8352  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-n0 10837  df-z 10906  df-uz 11128  df-rp 11266  df-fz 11727  df-fzo 11855  df-fl 11966  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-clim 13460  df-rlim 13461  df-sum 13658  df-0p 22369  df-ply 22877  df-coe 22879  df-dgr 22880
This theorem is referenced by:  dvply2g  22973  aannenlem1  23016
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