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

Theorem plycjlem 19673
Description: Lemma for plycj 19674 and coecj 19675. (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 11606 . . . . 5  |-  ( z  e.  CC  ->  (
* `  z )  e.  CC )
32adantl 452 . . . 4  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
* `  z )  e.  CC )
4 cjf 11605 . . . . . 6  |-  * : CC --> CC
54a1i 10 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  * : CC
--> CC )
65feqmptd 5591 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  *  =  ( z  e.  CC  |->  ( * `  z
) ) )
7 fzfid 11051 . . . . . 6  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  (
0 ... N )  e. 
Fin )
8 plycjlem.3 . . . . . . . . . 10  |-  A  =  (coeff `  F )
98coef3 19630 . . . . . . . . 9  |-  ( F  e.  (Poly `  S
)  ->  A : NN0
--> CC )
109adantr 451 . . . . . . . 8  |-  ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  ->  A : NN0 --> CC )
11 elfznn0 10838 . . . . . . . 8  |-  ( k  e.  ( 0 ... N )  ->  k  e.  NN0 )
12 ffvelrn 5679 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( A `  k
)  e.  CC )
1310, 11, 12syl2an 463 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
14 expcl 11137 . . . . . . . . 9  |-  ( ( x  e.  CC  /\  k  e.  NN0 )  -> 
( x ^ k
)  e.  CC )
1511, 14sylan2 460 . . . . . . . 8  |-  ( ( x  e.  CC  /\  k  e.  ( 0 ... N ) )  ->  ( x ^
k )  e.  CC )
1615adantll 694 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
x ^ k )  e.  CC )
1713, 16mulcld 8871 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  x  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( x ^ k ) )  e.  CC )
187, 17fsumcl 12222 . . . . 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 19636 . . . . 5  |-  ( F  e.  (Poly `  S
)  ->  F  =  ( x  e.  CC  |->  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( x ^ k ) ) ) )
21 fveq2 5541 . . . . 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 5707 . . . 4  |-  ( F  e.  (Poly `  S
)  ->  ( *  o.  F )  =  ( x  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) ) ) ) )
23 oveq1 5881 . . . . . . 7  |-  ( x  =  ( * `  z )  ->  (
x ^ k )  =  ( ( * `
 z ) ^
k ) )
2423oveq2d 5890 . . . . . 6  |-  ( x  =  ( * `  z )  ->  (
( A `  k
)  x.  ( x ^ k ) )  =  ( ( A `
 k )  x.  ( ( * `  z ) ^ k
) ) )
2524sumeq2sdv 12193 . . . . 5  |-  ( x  =  ( * `  z )  ->  sum_ k  e.  ( 0 ... N
) ( ( A `
 k )  x.  ( x ^ k
) )  =  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) )
2625fveq2d 5545 . . . 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 5707 . . 3  |-  ( F  e.  (Poly `  S
)  ->  ( (
*  o.  F )  o.  * )  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
281, 27syl5eq 2340 . 2  |-  ( F  e.  (Poly `  S
)  ->  G  =  ( z  e.  CC  |->  ( * `  sum_ k  e.  ( 0 ... N ) ( ( A `  k
)  x.  ( ( * `  z ) ^ k ) ) ) ) )
29 fzfid 11051 . . . . 5  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  (
0 ... N )  e. 
Fin )
309adantr 451 . . . . . . 7  |-  ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  ->  A : NN0 --> CC )
3130, 11, 12syl2an 463 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  ( A `  k )  e.  CC )
32 expcl 11137 . . . . . . 7  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( ( * `  z ) ^ k
)  e.  CC )
333, 11, 32syl2an 463 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  z
) ^ k )  e.  CC )
3431, 33mulcld 8871 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( A `  k
)  x.  ( ( * `  z ) ^ k ) )  e.  CC )
3529, 34fsumcj 12284 . . . 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 11722 . . . . . 6  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( * `  ( A `  k ) )  x.  ( * `
 ( ( * `
 z ) ^
k ) ) ) )
37 fvco3 5612 . . . . . . . 8  |-  ( ( A : NN0 --> CC  /\  k  e.  NN0 )  -> 
( ( *  o.  A ) `  k
)  =  ( * `
 ( A `  k ) ) )
3830, 11, 37syl2an 463 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( *  o.  A
) `  k )  =  ( * `  ( A `  k ) ) )
39 cjexp 11651 . . . . . . . . 9  |-  ( ( ( * `  z
)  e.  CC  /\  k  e.  NN0 )  -> 
( * `  (
( * `  z
) ^ k ) )  =  ( ( * `  ( * `
 z ) ) ^ k ) )
403, 11, 39syl2an 463 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( (
* `  z ) ^ k ) )  =  ( ( * `
 ( * `  z ) ) ^
k ) )
41 cjcj 11641 . . . . . . . . . 10  |-  ( z  e.  CC  ->  (
* `  ( * `  z ) )  =  z )
4241ad2antlr 707 . . . . . . . . 9  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( * `  z ) )  =  z )
4342oveq1d 5889 . . . . . . . 8  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
( * `  (
* `  z )
) ^ k )  =  ( z ^
k ) )
4440, 43eqtr2d 2329 . . . . . . 7  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
z ^ k )  =  ( * `  ( ( * `  z ) ^ k
) ) )
4538, 44oveq12d 5892 . . . . . 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 2331 . . . . 5  |-  ( ( ( F  e.  (Poly `  S )  /\  z  e.  CC )  /\  k  e.  ( 0 ... N
) )  ->  (
* `  ( ( A `  k )  x.  ( ( * `  z ) ^ k
) ) )  =  ( ( ( *  o.  A ) `  k )  x.  (
z ^ k ) ) )
4746sumeq2dv 12192 . . . 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 2328 . . 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 4122 . 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 2328 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 358    = wceq 1632    e. wcel 1696    e. cmpt 4093    o. ccom 4709   -->wf 5267   ` cfv 5271  (class class class)co 5874   CCcc 8751   0cc0 8753    x. cmul 8758   NN0cn0 9981   ...cfz 10798   ^cexp 11120   *ccj 11597   sum_csu 12174  Polycply 19582  coeffccoe 19584  degcdgr 19585
This theorem is referenced by:  plycj  19674  coecj  19675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-hash 11354  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-clim 11978  df-rlim 11979  df-sum 12175  df-0p 19041  df-ply 19586  df-coe 19588  df-dgr 19589
  Copyright terms: Public domain W3C validator