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Theorem coeid3 21688
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 21687 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( X ^ k ) ) )
433adant2 1007 . 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 11490 . . . 4  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
653ad2ant2 1010 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... N )  C_  (
0 ... M ) )
7 elfznn0 11473 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
81coef3 21680 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
983ad2ant1 1009 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  A : NN0 --> CC )
109ffvelrnda 5838 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
11 expcl 11875 . . . . . 6  |-  ( ( X  e.  CC  /\  k  e.  NN0 )  -> 
( X ^ k
)  e.  CC )
12113ad2antl3 1152 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( X ^
k )  e.  CC )
1310, 12mulcld 9398 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  x.  ( X ^ k
) )  e.  CC )
147, 13sylan2 474 . . 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 3474 . . . . . . 7  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
1615adantl 466 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
17 simpl1 991 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  F  e.  (Poly `  S ) )
18 eldifi 3473 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
19 elfzuz 11441 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
2018, 19syl 16 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
2120adantl 466 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
22 nn0uz 10887 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2321, 22syl6eleqr 2529 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  NN0 )
241, 2dgrub 21682 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
25243expia 1189 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
2617, 23, 25syl2anc 661 . . . . . . . 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 992 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  M  e.  ( ZZ>= `  N )
)
28 eluzel2 10858 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
2927, 28syl 16 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  N  e.  ZZ )
30 elfz5 11437 . . . . . . . . 9  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
3121, 29, 30syl2anc 661 . . . . . . . 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 2674 . . . . . 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 6101 . . . 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 11473 . . . . . . 7  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
3718, 36syl 16 . . . . . 6  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
3837, 12sylan2 474 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( X ^ k )  e.  CC )
3938mul02d 9559 . . . 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 2470 . . 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 11787 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... M )  e.  Fin )
426, 14, 40, 41fsumss 13194 . 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 2470 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 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601    \ cdif 3320    C_ wss 3323   class class class wbr 4287   -->wf 5409   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274    x. cmul 9279    <_ cle 9411   NN0cn0 10571   ZZcz 10638   ZZ>=cuz 10853   ...cfz 11429   ^cexp 11857   sum_csu 13155  Polycply 21632  coeffccoe 21634  degcdgr 21635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-inf2 7839  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352  ax-addf 9353
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-se 4675  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-of 6315  df-om 6472  df-1st 6572  df-2nd 6573  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-pm 7209  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-sup 7683  df-oi 7716  df-card 8101  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-n0 10572  df-z 10639  df-uz 10854  df-rp 10984  df-fz 11430  df-fzo 11541  df-fl 11634  df-seq 11799  df-exp 11858  df-hash 12096  df-cj 12580  df-re 12581  df-im 12582  df-sqr 12716  df-abs 12717  df-clim 12958  df-rlim 12959  df-sum 13156  df-0p 21128  df-ply 21636  df-coe 21638  df-dgr 21639
This theorem is referenced by:  dvply2g  21731  aannenlem1  21774
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