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Theorem plyrecj 22543
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 12063 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
0 ... (deg `  F
) )  e.  Fin )
2 0re 9608 . . . . . . . . 9  |-  0  e.  RR
3 eqid 2467 . . . . . . . . . 10  |-  (coeff `  F )  =  (coeff `  F )
43coef2 22496 . . . . . . . . 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 11782 . . . . . . 7  |-  ( x  e.  ( 0 ... (deg `  F )
)  ->  x  e.  NN0 )
8 ffvelrn 6030 . . . . . . 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 9634 . . . . 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 12164 . . . . . 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 9628 . . . 4  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) )  e.  CC )
151, 14fsumcj 13604 . . 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 13034 . . . . 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 13039 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( (coeff `  F ) `  x ) )  =  ( (coeff `  F
) `  x )
)
18 cjexp 12963 . . . . . . 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 6313 . . . . 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 2508 . . . 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 13505 . . 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 2508 . 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 2467 . . . 4  |-  (deg `  F )  =  (deg
`  F )
253, 24coeid2 22504 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( F `  A )  =  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )
2625fveq2d 5876 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( * `  sum_ x  e.  ( 0 ... (deg `  F )
) ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) ) )
27 cjcl 12918 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
283, 24coeid2 22504 . . 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 2518 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 1379    e. wcel 1767   -->wf 5590   ` cfv 5594  (class class class)co 6295   CCcc 9502   RRcr 9503   0cc0 9504    x. cmul 9509   NN0cn0 10807   ...cfz 11684   ^cexp 12146   *ccj 12909   sum_csu 13488  Polycply 22449  coeffccoe 22451  degcdgr 22452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6535  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-map 7434  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-hash 12386  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-clim 13291  df-rlim 13292  df-sum 13489  df-0p 21945  df-ply 22453  df-coe 22455  df-dgr 22456
This theorem is referenced by:  plyreres  22546  aacjcl  22590
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