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Theorem plyrecj 20065
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 11240 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
0 ... (deg `  F
) )  e.  Fin )
2 0re 9025 . . . . . . . . 9  |-  0  e.  RR
3 eqid 2388 . . . . . . . . . 10  |-  (coeff `  F )  =  (coeff `  F )
43coef2 20018 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
52, 4mpan2 653 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  (coeff `  F
) : NN0 --> RR )
65adantr 452 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (coeff `  F ) : NN0 --> RR )
7 elfznn0 11016 . . . . . . 7  |-  ( x  e.  ( 0 ... (deg `  F )
)  ->  x  e.  NN0 )
8 ffvelrn 5808 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> RR  /\  x  e.  NN0 )  ->  (
(coeff `  F ) `  x )  e.  RR )
96, 7, 8syl2an 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  RR )
109recnd 9048 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  CC )
11 simpr 448 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  A  e.  CC )
12 expcl 11327 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( A ^ x
)  e.  CC )
1311, 7, 12syl2an 464 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( A ^
x )  e.  CC )
1410, 13mulcld 9042 . . . 4  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) )  e.  CC )
151, 14fsumcj 12517 . . 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 11954 . . . . 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 11959 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( (coeff `  F ) `  x ) )  =  ( (coeff `  F
) `  x )
)
18 cjexp 11883 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( * `  ( A ^ x ) )  =  ( ( * `
 A ) ^
x ) )
1911, 7, 18syl2an 464 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( A ^ x ) )  =  ( ( * `  A ) ^ x ) )
2017, 19oveq12d 6039 . . . . 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 2420 . . . 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 12425 . . 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 2420 . 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 2388 . . . 4  |-  (deg `  F )  =  (deg
`  F )
253, 24coeid2 20026 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( F `  A )  =  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )
2625fveq2d 5673 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( * `  sum_ x  e.  ( 0 ... (deg `  F )
) ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) ) )
27 cjcl 11838 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
283, 24coeid2 20026 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  (
* `  A )  e.  CC )  ->  ( F `  ( * `  A ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
2927, 28sylan2 461 . 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 2430 1  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( F `  (
* `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1717   -->wf 5391   ` cfv 5395  (class class class)co 6021   CCcc 8922   RRcr 8923   0cc0 8924    x. cmul 8929   NN0cn0 10154   ...cfz 10976   ^cexp 11310   *ccj 11829   sum_csu 12407  Polycply 19971  coeffccoe 19973  degcdgr 19974
This theorem is referenced by:  plyreres  20068  aacjcl  20112
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003
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 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-oadd 6665  df-er 6842  df-map 6957  df-pm 6958  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-sup 7382  df-oi 7413  df-card 7760  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-n0 10155  df-z 10216  df-uz 10422  df-rp 10546  df-fz 10977  df-fzo 11067  df-fl 11130  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-clim 12210  df-rlim 12211  df-sum 12408  df-0p 19430  df-ply 19975  df-coe 19977  df-dgr 19978
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