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Theorem plycjlem 21877
Description: Lemma for plycj 21878 and coecj 21879. (Contributed by Mario Carneiro, 24-Jul-2014.)
Hypotheses
Ref Expression
plycj.1  |-  N  =  (deg `  F )
plycj.2  |-  G  =  ( ( *  o.  F )  o.  *
)
plycjlem.3  |-  A  =  (coeff `  F )
Assertion
Ref Expression
plycjlem  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
Distinct variable groups:    z, k, A    k, F, z    k, N, z    S, k, z
Allowed substitution hints:    G( z, k)

Proof of Theorem plycjlem
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 plycj.2 . . 3  |-  G  =  ( ( *  o.  F )  o.  *
)
2 cjcl 12713 . . . . 5  |-  ( z  e.  CC  ->  (
* `  z )  e.  CC )
32adantl 466 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  z )  e.  CC )
4 cjf 12712 . . . . . 6  |-  * : CC --> CC
54a1i 11 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  * : CC
--> CC )
65feqmptd 5854 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  *  =  ( z  e.  CC  |->  ( * `  z
) ) )
7 fzfid 11913 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
0 ... N )  e. 
Fin )
8 plycjlem.3 . . . . . . . . . 10  |-  A  =  (coeff `  F )
98coef3 21834 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
109adantr 465 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  A : NN0 --> CC )
11 elfznn0 11599 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
12 ffvelrn 5951 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1310, 11, 12syl2an 477 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
14 expcl 12001 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ k
)  e.  CC )
1511, 14sylan2 474 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  ( 0 ... N ) )  ->  ( x ^
k )  e.  CC )
1615adantll 713 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
x ^ k )  e.  CC )
1713, 16mulcld 9518 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( x ^ k ) )  e.  CC )
187, 17fsumcl 13329 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) )  e.  CC )
19 plycj.1 . . . . . 6  |-  N  =  (deg `  F )
208, 19coeid 21840 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( x ^ k ) ) ) )
21 fveq2 5800 . . . . 5  |-  ( z  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k )  x.  (
x ^ k ) )  ->  ( * `  z )  =  ( * `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) ) )
2218, 20, 6, 21fmptco 5986 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  F )  =  ( x  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) ) ) )
23 oveq1 6208 . . . . . . 7  |-  ( x  =  ( * `  z )  ->  (
x ^ k )  =  ( ( * `
 z ) ^
k ) )
2423oveq2d 6217 . . . . . 6  |-  ( x  =  ( * `  z )  ->  (
( A `  k
)  x.  ( x ^ k ) )  =  ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )
2524sumeq2sdv 13300 . . . . 5  |-  ( x  =  ( * `  z )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) )
2625fveq2d 5804 . . . 4  |-  ( x  =  ( * `  z )  ->  (
* `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) )  =  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) )
273, 6, 22, 26fmptco 5986 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( (
*  o.  F )  o.  * )  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
281, 27syl5eq 2507 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
29 fzfid 11913 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
0 ... N )  e. 
Fin )
309adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  A : NN0 --> CC )
3130, 11, 12syl2an 477 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
32 expcl 12001 . . . . . . 7  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  z ) ^ k
)  e.  CC )
333, 11, 32syl2an 477 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  z
) ^ k )  e.  CC )
3431, 33mulcld 9518 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( ( * `  z ) ^ k ) )  e.  CC )
3529, 34fsumcj 13392 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )  = 
sum_ k  e.  ( 0 ... N ) ( * `  (
( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) )
3631, 33cjmuld 12829 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( * `  ( A `  k ) )  x.  ( * `
 ( ( * `
 z ) ^
k ) ) ) )
37 fvco3 5878 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
3830, 11, 37syl2an 477 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
39 cjexp 12758 . . . . . . . . 9  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( * `  z
) ^ k ) )  =  ( ( * `  ( * `
 z ) ) ^ k ) )
403, 11, 39syl2an 477 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (
* `  z ) ^ k ) )  =  ( ( * `
 ( * `  z ) ) ^
k ) )
41 cjcj 12748 . . . . . . . . . 10  |-  ( z  e.  CC  ->  (
* `  ( * `  z ) )  =  z )
4241ad2antlr 726 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( * `  z ) )  =  z )
4342oveq1d 6216 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  (
* `  z )
) ^ k )  =  ( z ^
k ) )
4440, 43eqtr2d 2496 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
z ^ k )  =  ( * `  ( ( * `  z ) ^ k
) ) )
4538, 44oveq12d 6219 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( ( *  o.  A ) `  k
)  x.  ( z ^ k ) )  =  ( ( * `
 ( A `  k ) )  x.  ( * `  (
( * `  z
) ^ k ) ) ) )
4636, 45eqtr4d 2498 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( ( *  o.  A ) `  k )  x.  (
z ^ k ) ) )
4746sumeq2dv 13299 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  sum_ k  e.  ( 0 ... N
) ( * `  ( ( A `  k )  x.  (
( * `  z
) ^ k ) ) )  =  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) )
4835, 47eqtrd 2495 . . 3  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )  = 
sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k )  x.  (
z ^ k ) ) )
4948mpteq2dva 4487 . 2  |-  ( F  e.  (Poly `  S
)  ->  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) )  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
5028, 49eqtrd 2495 1  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    |-> cmpt 4459    o. ccom 4953   -->wf 5523   ` cfv 5527  (class class class)co 6201   CCcc 9392   0cc0 9394    x. cmul 9399   NN0cn0 10691   ...cfz 11555   ^cexp 11983   *ccj 12704   sum_csu 13282  Polycply 21786  coeffccoe 21788  degcdgr 21789
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 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-inf2 7959  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471  ax-pre-sup 9472  ax-addf 9473
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-se 4789  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-isom 5536  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-of 6431  df-om 6588  df-1st 6688  df-2nd 6689  df-recs 6943  df-rdg 6977  df-1o 7031  df-oadd 7035  df-er 7212  df-map 7327  df-pm 7328  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-sup 7803  df-oi 7836  df-card 8221  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710  df-div 10106  df-nn 10435  df-2 10492  df-3 10493  df-n0 10692  df-z 10759  df-uz 10974  df-rp 11104  df-fz 11556  df-fzo 11667  df-fl 11760  df-seq 11925  df-exp 11984  df-hash 12222  df-cj 12707  df-re 12708  df-im 12709  df-sqr 12843  df-abs 12844  df-clim 13085  df-rlim 13086  df-sum 13283  df-0p 21282  df-ply 21790  df-coe 21792  df-dgr 21793
This theorem is referenced by:  plycj  21878  coecj  21879
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