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Theorem plyrecj 21889
Description: A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.)
Assertion
Ref Expression
plyrecj  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( F `  (
* `  A )
) )

Proof of Theorem plyrecj
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fzfid 11916 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
0 ... (deg `  F
) )  e.  Fin )
2 0re 9501 . . . . . . . . 9  |-  0  e.  RR
3 eqid 2454 . . . . . . . . . 10  |-  (coeff `  F )  =  (coeff `  F )
43coef2 21842 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
52, 4mpan2 671 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  (coeff `  F
) : NN0 --> RR )
65adantr 465 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (coeff `  F ) : NN0 --> RR )
7 elfznn0 11602 . . . . . . 7  |-  ( x  e.  ( 0 ... (deg `  F )
)  ->  x  e.  NN0 )
8 ffvelrn 5953 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> RR  /\  x  e.  NN0 )  ->  (
(coeff `  F ) `  x )  e.  RR )
96, 7, 8syl2an 477 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  RR )
109recnd 9527 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  CC )
11 simpr 461 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  A  e.  CC )
12 expcl 12004 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( A ^ x
)  e.  CC )
1311, 7, 12syl2an 477 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( A ^
x )  e.  CC )
1410, 13mulcld 9521 . . . 4  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) )  e.  CC )
151, 14fsumcj 13395 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( * `
 ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) ) )
1610, 13cjmuld 12832 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) )  =  ( ( * `  ( (coeff `  F ) `  x
) )  x.  (
* `  ( A ^ x ) ) ) )
179cjred 12837 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( (coeff `  F ) `  x ) )  =  ( (coeff `  F
) `  x )
)
18 cjexp 12761 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( * `  ( A ^ x ) )  =  ( ( * `
 A ) ^
x ) )
1911, 7, 18syl2an 477 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( A ^ x ) )  =  ( ( * `  A ) ^ x ) )
2017, 19oveq12d 6221 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( * `
 ( (coeff `  F ) `  x
) )  x.  (
* `  ( A ^ x ) ) )  =  ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
2116, 20eqtrd 2495 . . . 4  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) )  =  ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
2221sumeq2dv 13302 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  sum_ x  e.  ( 0 ... (deg `  F ) ) ( * `  ( ( (coeff `  F ) `  x )  x.  ( A ^ x ) ) )  =  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( ( * `  A ) ^ x
) ) )
2315, 22eqtrd 2495 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
24 eqid 2454 . . . 4  |-  (deg `  F )  =  (deg
`  F )
253, 24coeid2 21850 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( F `  A )  =  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )
2625fveq2d 5806 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( * `  sum_ x  e.  ( 0 ... (deg `  F )
) ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) ) )
27 cjcl 12716 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
283, 24coeid2 21850 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  (
* `  A )  e.  CC )  ->  ( F `  ( * `  A ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
2927, 28sylan2 474 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( F `  ( * `  A ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
3023, 26, 293eqtr4d 2505 1  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( F `  (
* `  A )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   -->wf 5525   ` cfv 5529  (class class class)co 6203   CCcc 9395   RRcr 9396   0cc0 9397    x. cmul 9402   NN0cn0 10694   ...cfz 11558   ^cexp 11986   *ccj 12707   sum_csu 13285  Polycply 21795  coeffccoe 21797  degcdgr 21798
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 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7962  ax-cnex 9453  ax-resscn 9454  ax-1cn 9455  ax-icn 9456  ax-addcl 9457  ax-addrcl 9458  ax-mulcl 9459  ax-mulrcl 9460  ax-mulcom 9461  ax-addass 9462  ax-mulass 9463  ax-distr 9464  ax-i2m1 9465  ax-1ne0 9466  ax-1rid 9467  ax-rnegex 9468  ax-rrecex 9469  ax-cnre 9470  ax-pre-lttri 9471  ax-pre-lttrn 9472  ax-pre-ltadd 9473  ax-pre-mulgt0 9474  ax-pre-sup 9475  ax-addf 9476
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 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-of 6433  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-map 7329  df-pm 7330  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7806  df-oi 7839  df-card 8224  df-pnf 9535  df-mnf 9536  df-xr 9537  df-ltxr 9538  df-le 9539  df-sub 9712  df-neg 9713  df-div 10109  df-nn 10438  df-2 10495  df-3 10496  df-n0 10695  df-z 10762  df-uz 10977  df-rp 11107  df-fz 11559  df-fzo 11670  df-fl 11763  df-seq 11928  df-exp 11987  df-hash 12225  df-cj 12710  df-re 12711  df-im 12712  df-sqr 12846  df-abs 12847  df-clim 13088  df-rlim 13089  df-sum 13286  df-0p 21291  df-ply 21799  df-coe 21801  df-dgr 21802
This theorem is referenced by:  plyreres  21892  aacjcl  21936
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