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Theorem coeid3 19622
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 19621 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  X  e.  CC )  ->  ( F `  X )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  ( X ^ k ) ) )
433adant2 974 . 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 10831 . . . 4  |-  ( M  e.  ( ZZ>= `  N
)  ->  ( 0 ... N )  C_  ( 0 ... M
) )
653ad2ant2 977 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... N )  C_  (
0 ... M ) )
7 elfznn0 10822 . . . 4  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
81coef3 19614 . . . . . . 7  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
983ad2ant1 976 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  A : NN0 --> CC )
10 ffvelrn 5663 . . . . . 6  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
119, 10sylan 457 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( A `  k )  e.  CC )
12 expcl 11121 . . . . . 6  |-  ( ( X  e.  CC  /\  k  e.  NN0 )  -> 
( X ^ k
)  e.  CC )
13123ad2antl3 1119 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( X ^
k )  e.  CC )
1411, 13mulcld 8855 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  NN0 )  ->  ( ( A `
 k )  x.  ( X ^ k
) )  e.  CC )
157, 14sylan2 460 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( 0 ... N ) )  ->  ( ( A `  k )  x.  ( X ^ k
) )  e.  CC )
16 eldifn 3299 . . . . . . 7  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  -.  k  e.  ( 0 ... N ) )
1716adantl 452 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  -.  k  e.  ( 0 ... N
) )
18 simpl1 958 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  F  e.  (Poly `  S ) )
19 eldifi 3298 . . . . . . . . . . . 12  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( 0 ... M
) )
20 elfzuz 10794 . . . . . . . . . . . 12  |-  ( k  e.  ( 0 ... M )  ->  k  e.  ( ZZ>= `  0 )
)
2119, 20syl 15 . . . . . . . . . . 11  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  ( ZZ>= `  0 )
)
2221adantl 452 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  ( ZZ>= `  0 )
)
23 nn0uz 10262 . . . . . . . . . 10  |-  NN0  =  ( ZZ>= `  0 )
2422, 23syl6eleqr 2374 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  k  e.  NN0 )
251, 2dgrub 19616 . . . . . . . . . 10  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0  /\  ( A `
 k )  =/=  0 )  ->  k  <_  N )
26253expia 1153 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  S )  /\  k  e.  NN0 )  ->  (
( A `  k
)  =/=  0  -> 
k  <_  N )
)
2718, 24, 26syl2anc 642 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  =/=  0  ->  k  <_  N ) )
28 simpl2 959 . . . . . . . . . 10  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  M  e.  ( ZZ>= `  N )
)
29 eluzel2 10235 . . . . . . . . . 10  |-  ( M  e.  ( ZZ>= `  N
)  ->  N  e.  ZZ )
3028, 29syl 15 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  N  e.  ZZ )
31 elfz5 10790 . . . . . . . . 9  |-  ( ( k  e.  ( ZZ>= ` 
0 )  /\  N  e.  ZZ )  ->  (
k  e.  ( 0 ... N )  <->  k  <_  N ) )
3222, 30, 31syl2anc 642 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( k  e.  ( 0 ... N
)  <->  k  <_  N
) )
3327, 32sylibrd 225 . . . . . . 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
) ) )
3433necon1bd 2514 . . . . . 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 ) )
3517, 34mpd 14 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( A `  k )  =  0 )
3635oveq1d 5873 . . . 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 ) ) )
37 elfznn0 10822 . . . . . . 7  |-  ( k  e.  ( 0 ... M )  ->  k  e.  NN0 )
3819, 37syl 15 . . . . . 6  |-  ( k  e.  ( ( 0 ... M )  \ 
( 0 ... N
) )  ->  k  e.  NN0 )
3938, 13sylan2 460 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( X ^ k )  e.  CC )
4039mul02d 9010 . . . 4  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( 0  x.  ( X ^
k ) )  =  0 )
4136, 40eqtrd 2315 . . 3  |-  ( ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  /\  k  e.  ( ( 0 ... M
)  \  ( 0 ... N ) ) )  ->  ( ( A `  k )  x.  ( X ^ k
) )  =  0 )
42 fzfid 11035 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  M  e.  ( ZZ>= `  N )  /\  X  e.  CC )  ->  ( 0 ... M )  e.  Fin )
436, 15, 41, 42fsumss 12198 . 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 ) ) )
444, 43eqtrd 2315 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 set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149    C_ wss 3152   class class class wbr 4023   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737    x. cmul 8742    <_ cle 8868   NN0cn0 9965   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782   ^cexp 11104   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  dvply2g  19665  aannenlem1  19708
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
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