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Theorem coeid3 22365
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 22364 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( X ^ k ) ) )
433adant2 1010 . 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 11712 . . . 4  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
653ad2ant2 1013 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... N )  C_  (
0 ... M ) )
7 elfznn0 11759 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
81coef3 22357 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
983ad2ant1 1012 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  A : NN0 --> CC )
109ffvelrnda 6012 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
11 expcl 12140 . . . . . 6  |-  ( ( X  e.  CC  /\  k  e.  NN0 )  -> 
( X ^ k
)  e.  CC )
12113ad2antl3 1155 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( X ^
k )  e.  CC )
1310, 12mulcld 9605 . . . 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 3620 . . . . . . 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 994 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  F  e.  (Poly `  S ) )
18 eldifi 3619 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
19 elfzuz 11673 . . . . . . . . . . . 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 11105 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2321, 22syl6eleqr 2559 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  NN0 )
241, 2dgrub 22359 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
25243expia 1193 . . . . . . . . 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 995 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  M  e.  ( ZZ>= `  N )
)
28 eluzel2 11076 . . . . . . . . . 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 11669 . . . . . . . . 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 2678 . . . . . 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 6290 . . . 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 11759 . . . . . . 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 9766 . . . 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 2501 . . 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 12039 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... M )  e.  Fin )
426, 14, 40, 41fsumss 13496 . 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 2501 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 968    = wceq 1374    e. wcel 1762    =/= wne 2655    \ cdif 3466    C_ wss 3469   class class class wbr 4440   -->wf 5575   ` cfv 5579  (class class class)co 6275   CCcc 9479   0cc0 9481    x. cmul 9486    <_ cle 9618   NN0cn0 10784   ZZcz 10853   ZZ>=cuz 11071   ...cfz 11661   ^cexp 12122   sum_csu 13457  Polycply 22309  coeffccoe 22311  degcdgr 22312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-fal 1380  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-sup 7890  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-n0 10785  df-z 10854  df-uz 11072  df-rp 11210  df-fz 11662  df-fzo 11782  df-fl 11886  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-clim 13260  df-rlim 13261  df-sum 13458  df-0p 21805  df-ply 22313  df-coe 22315  df-dgr 22316
This theorem is referenced by:  dvply2g  22408  aannenlem1  22451
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