MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  plycjlem Unicode version

Theorem plycjlem 20155
Description: Lemma for plycj 20156 and coecj 20157. (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 11873 . . . . 5  |-  ( z  e.  CC  ->  (
* `  z )  e.  CC )
32adantl 453 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  z )  e.  CC )
4 cjf 11872 . . . . . 6  |-  * : CC --> CC
54a1i 11 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  * : CC
--> CC )
65feqmptd 5746 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  *  =  ( z  e.  CC  |->  ( * `  z
) ) )
7 fzfid 11275 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
0 ... N )  e. 
Fin )
8 plycjlem.3 . . . . . . . . . 10  |-  A  =  (coeff `  F )
98coef3 20112 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
109adantr 452 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  A : NN0 --> CC )
11 elfznn0 11047 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
12 ffvelrn 5835 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1310, 11, 12syl2an 464 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
14 expcl 11362 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ k
)  e.  CC )
1511, 14sylan2 461 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  ( 0 ... N ) )  ->  ( x ^
k )  e.  CC )
1615adantll 695 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
x ^ k )  e.  CC )
1713, 16mulcld 9072 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( x ^ k ) )  e.  CC )
187, 17fsumcl 12490 . . . . 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 20118 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( x ^ k ) ) ) )
21 fveq2 5695 . . . . 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 5868 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  F )  =  ( x  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) ) ) )
23 oveq1 6055 . . . . . . 7  |-  ( x  =  ( * `  z )  ->  (
x ^ k )  =  ( ( * `
 z ) ^
k ) )
2423oveq2d 6064 . . . . . 6  |-  ( x  =  ( * `  z )  ->  (
( A `  k
)  x.  ( x ^ k ) )  =  ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )
2524sumeq2sdv 12461 . . . . 5  |-  ( x  =  ( * `  z )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) )
2625fveq2d 5699 . . . 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 5868 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( (
*  o.  F )  o.  * )  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
281, 27syl5eq 2456 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
29 fzfid 11275 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
0 ... N )  e. 
Fin )
309adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  A : NN0 --> CC )
3130, 11, 12syl2an 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
32 expcl 11362 . . . . . . 7  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  z ) ^ k
)  e.  CC )
333, 11, 32syl2an 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  z
) ^ k )  e.  CC )
3431, 33mulcld 9072 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( ( * `  z ) ^ k ) )  e.  CC )
3529, 34fsumcj 12552 . . . 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 11989 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( * `  ( A `  k ) )  x.  ( * `
 ( ( * `
 z ) ^
k ) ) ) )
37 fvco3 5767 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
3830, 11, 37syl2an 464 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
39 cjexp 11918 . . . . . . . . 9  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( * `  z
) ^ k ) )  =  ( ( * `  ( * `
 z ) ) ^ k ) )
403, 11, 39syl2an 464 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (
* `  z ) ^ k ) )  =  ( ( * `
 ( * `  z ) ) ^
k ) )
41 cjcj 11908 . . . . . . . . . 10  |-  ( z  e.  CC  ->  (
* `  ( * `  z ) )  =  z )
4241ad2antlr 708 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( * `  z ) )  =  z )
4342oveq1d 6063 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  (
* `  z )
) ^ k )  =  ( z ^
k ) )
4440, 43eqtr2d 2445 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
z ^ k )  =  ( * `  ( ( * `  z ) ^ k
) ) )
4538, 44oveq12d 6066 . . . . . 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 2447 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( ( *  o.  A ) `  k )  x.  (
z ^ k ) ) )
4746sumeq2dv 12460 . . . 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 2444 . . 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 4263 . 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 2444 1  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( ( *  o.  A ) `  k
)  x.  ( z ^ k ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721    e. cmpt 4234    o. ccom 4849   -->wf 5417   ` cfv 5421  (class class class)co 6048   CCcc 8952   0cc0 8954    x. cmul 8959   NN0cn0 10185   ...cfz 11007   ^cexp 11345   *ccj 11864   sum_csu 12442  Polycply 20064  coeffccoe 20066  degcdgr 20067
This theorem is referenced by:  plycj  20156  coecj  20157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-inf2 7560  ax-cnex 9010  ax-resscn 9011  ax-1cn 9012  ax-icn 9013  ax-addcl 9014  ax-addrcl 9015  ax-mulcl 9016  ax-mulrcl 9017  ax-mulcom 9018  ax-addass 9019  ax-mulass 9020  ax-distr 9021  ax-i2m1 9022  ax-1ne0 9023  ax-1rid 9024  ax-rnegex 9025  ax-rrecex 9026  ax-cnre 9027  ax-pre-lttri 9028  ax-pre-lttrn 9029  ax-pre-ltadd 9030  ax-pre-mulgt0 9031  ax-pre-sup 9032  ax-addf 9033
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rmo 2682  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-se 4510  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-isom 5430  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-of 6272  df-1st 6316  df-2nd 6317  df-riota 6516  df-recs 6600  df-rdg 6635  df-1o 6691  df-oadd 6695  df-er 6872  df-map 6987  df-pm 6988  df-en 7077  df-dom 7078  df-sdom 7079  df-fin 7080  df-sup 7412  df-oi 7443  df-card 7790  df-pnf 9086  df-mnf 9087  df-xr 9088  df-ltxr 9089  df-le 9090  df-sub 9257  df-neg 9258  df-div 9642  df-nn 9965  df-2 10022  df-3 10023  df-n0 10186  df-z 10247  df-uz 10453  df-rp 10577  df-fz 11008  df-fzo 11099  df-fl 11165  df-seq 11287  df-exp 11346  df-hash 11582  df-cj 11867  df-re 11868  df-im 11869  df-sqr 12003  df-abs 12004  df-clim 12245  df-rlim 12246  df-sum 12443  df-0p 19523  df-ply 20068  df-coe 20070  df-dgr 20071
  Copyright terms: Public domain W3C validator