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Theorem coeid3 21824
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 21823 . . 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 11599 . . . 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 11582 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
81coef3 21816 . . . . . . 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 5942 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
11 expcl 11984 . . . . . 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 9507 . . . 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 3577 . . . . . . 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 3576 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
19 elfzuz 11550 . . . . . . . . . . . 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 10996 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2321, 22syl6eleqr 2550 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  NN0 )
241, 2dgrub 21818 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
25243expia 1190 . . . . . . . . 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 10967 . . . . . . . . . 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 11546 . . . . . . . . 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 2666 . . . . . 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 6205 . . . 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 11582 . . . . . . 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 9668 . . . 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 2492 . . 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 11896 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... M )  e.  Fin )
426, 14, 40, 41fsumss 13304 . 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 2492 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 1370    e. wcel 1758    =/= wne 2644    \ cdif 3423    C_ wss 3426   class class class wbr 4390   -->wf 5512   ` cfv 5516  (class class class)co 6190   CCcc 9381   0cc0 9383    x. cmul 9388    <_ cle 9520   NN0cn0 10680   ZZcz 10747   ZZ>=cuz 10962   ...cfz 11538   ^cexp 11966   sum_csu 13265  Polycply 21768  coeffccoe 21770  degcdgr 21771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-inf2 7948  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460  ax-pre-sup 9461  ax-addf 9462
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-int 4227  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-se 4778  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-isom 5525  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-om 6577  df-1st 6677  df-2nd 6678  df-recs 6932  df-rdg 6966  df-1o 7020  df-oadd 7024  df-er 7201  df-map 7316  df-pm 7317  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-sup 7792  df-oi 7825  df-card 8210  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-div 10095  df-nn 10424  df-2 10481  df-3 10482  df-n0 10681  df-z 10748  df-uz 10963  df-rp 11093  df-fz 11539  df-fzo 11650  df-fl 11743  df-seq 11908  df-exp 11967  df-hash 12205  df-cj 12690  df-re 12691  df-im 12692  df-sqr 12826  df-abs 12827  df-clim 13068  df-rlim 13069  df-sum 13266  df-0p 21264  df-ply 21772  df-coe 21774  df-dgr 21775
This theorem is referenced by:  dvply2g  21867  aannenlem1  21910
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