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Theorem plycjlem 22842
Description: Lemma for plycj 22843 and coecj 22844. (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 13023 . . . . 5  |-  ( z  e.  CC  ->  (
* `  z )  e.  CC )
32adantl 464 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  z )  e.  CC )
4 cjf 13022 . . . . . 6  |-  * : CC --> CC
54a1i 11 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  * : CC
--> CC )
65feqmptd 5901 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  *  =  ( z  e.  CC  |->  ( * `  z
) ) )
7 fzfid 12068 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
0 ... N )  e. 
Fin )
8 plycjlem.3 . . . . . . . . . 10  |-  A  =  (coeff `  F )
98coef3 22798 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
109adantr 463 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  A : NN0 --> CC )
11 elfznn0 11775 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
12 ffvelrn 6005 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1310, 11, 12syl2an 475 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
14 expcl 12169 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ k
)  e.  CC )
1511, 14sylan2 472 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  ( 0 ... N ) )  ->  ( x ^
k )  e.  CC )
1615adantll 711 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
x ^ k )  e.  CC )
1713, 16mulcld 9605 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( x ^ k ) )  e.  CC )
187, 17fsumcl 13640 . . . . 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 22804 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( x ^ k ) ) ) )
21 fveq2 5848 . . . . 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 6040 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  F )  =  ( x  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) ) ) )
23 oveq1 6277 . . . . . . 7  |-  ( x  =  ( * `  z )  ->  (
x ^ k )  =  ( ( * `
 z ) ^
k ) )
2423oveq2d 6286 . . . . . 6  |-  ( x  =  ( * `  z )  ->  (
( A `  k
)  x.  ( x ^ k ) )  =  ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )
2524sumeq2sdv 13611 . . . . 5  |-  ( x  =  ( * `  z )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) )
2625fveq2d 5852 . . . 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 6040 . . 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 12068 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
0 ... N )  e. 
Fin )
309adantr 463 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  A : NN0 --> CC )
3130, 11, 12syl2an 475 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
32 expcl 12169 . . . . . . 7  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  z ) ^ k
)  e.  CC )
333, 11, 32syl2an 475 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  z
) ^ k )  e.  CC )
3431, 33mulcld 9605 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( ( * `  z ) ^ k ) )  e.  CC )
3529, 34fsumcj 13709 . . . 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 13139 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( * `  ( A `  k ) )  x.  ( * `
 ( ( * `
 z ) ^
k ) ) ) )
37 fvco3 5925 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
3830, 11, 37syl2an 475 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
39 cjexp 13068 . . . . . . . . 9  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( * `  z
) ^ k ) )  =  ( ( * `  ( * `
 z ) ) ^ k ) )
403, 11, 39syl2an 475 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (
* `  z ) ^ k ) )  =  ( ( * `
 ( * `  z ) ) ^
k ) )
41 cjcj 13058 . . . . . . . . . 10  |-  ( z  e.  CC  ->  (
* `  ( * `  z ) )  =  z )
4241ad2antlr 724 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( * `  z ) )  =  z )
4342oveq1d 6285 . . . . . . . 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 6288 . . . . . 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 13610 . . . 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 4525 . 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 367    = wceq 1398    e. wcel 1823    |-> cmpt 4497    o. ccom 4992   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   0cc0 9481    x. cmul 9486   NN0cn0 10791   ...cfz 11675   ^cexp 12151   *ccj 13014   sum_csu 13593  Polycply 22750  coeffccoe 22752  degcdgr 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  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 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-sup 7893  df-oi 7927  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-n0 10792  df-z 10861  df-uz 11083  df-rp 11222  df-fz 11676  df-fzo 11800  df-fl 11910  df-seq 12093  df-exp 12152  df-hash 12391  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-clim 13396  df-rlim 13397  df-sum 13594  df-0p 22246  df-ply 22754  df-coe 22756  df-dgr 22757
This theorem is referenced by:  plycj  22843  coecj  22844
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