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

Theorem plyrecj 19660
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 11035 . . . 4  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
0 ... (deg `  F
) )  e.  Fin )
2 0re 8838 . . . . . . . . 9  |-  0  e.  RR
3 eqid 2283 . . . . . . . . . 10  |-  (coeff `  F )  =  (coeff `  F )
43coef2 19613 . . . . . . . . 9  |-  ( ( F  e.  (Poly `  RR )  /\  0  e.  RR )  ->  (coeff `  F ) : NN0 --> RR )
52, 4mpan2 652 . . . . . . . 8  |-  ( F  e.  (Poly `  RR )  ->  (coeff `  F
) : NN0 --> RR )
65adantr 451 . . . . . . 7  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (coeff `  F ) : NN0 --> RR )
7 elfznn0 10822 . . . . . . 7  |-  ( x  e.  ( 0 ... (deg `  F )
)  ->  x  e.  NN0 )
8 ffvelrn 5663 . . . . . . 7  |-  ( ( (coeff `  F ) : NN0 --> RR  /\  x  e.  NN0 )  ->  (
(coeff `  F ) `  x )  e.  RR )
96, 7, 8syl2an 463 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  RR )
109recnd 8861 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( (coeff `  F ) `  x
)  e.  CC )
11 simpr 447 . . . . . 6  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  A  e.  CC )
12 expcl 11121 . . . . . 6  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( A ^ x
)  e.  CC )
1311, 7, 12syl2an 463 . . . . 5  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( A ^
x )  e.  CC )
1410, 13mulcld 8855 . . . 4  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) )  e.  CC )
151, 14fsumcj 12268 . . 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 11706 . . . . 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 11711 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( (coeff `  F ) `  x ) )  =  ( (coeff `  F
) `  x )
)
18 cjexp 11635 . . . . . . 7  |-  ( ( A  e.  CC  /\  x  e.  NN0 )  -> 
( * `  ( A ^ x ) )  =  ( ( * `
 A ) ^
x ) )
1911, 7, 18syl2an 463 . . . . . 6  |-  ( ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  /\  x  e.  ( 0 ... (deg `  F ) ) )  ->  ( * `  ( A ^ x ) )  =  ( ( * `  A ) ^ x ) )
2017, 19oveq12d 5876 . . . . 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 2315 . . . 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 12176 . . 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 2315 . 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 2283 . . . 4  |-  (deg `  F )  =  (deg
`  F )
253, 24coeid2 19621 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  ( F `  A )  =  sum_ x  e.  ( 0 ... (deg `  F ) ) ( ( (coeff `  F
) `  x )  x.  ( A ^ x
) ) )
2625fveq2d 5529 . 2  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( * `  sum_ x  e.  ( 0 ... (deg `  F )
) ( ( (coeff `  F ) `  x
)  x.  ( A ^ x ) ) ) )
27 cjcl 11590 . . 3  |-  ( A  e.  CC  ->  (
* `  A )  e.  CC )
283, 24coeid2 19621 . . 3  |-  ( ( F  e.  (Poly `  RR )  /\  (
* `  A )  e.  CC )  ->  ( F `  ( * `  A ) )  = 
sum_ x  e.  (
0 ... (deg `  F
) ) ( ( (coeff `  F ) `  x )  x.  (
( * `  A
) ^ x ) ) )
2927, 28sylan2 460 . 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 2325 1  |-  ( ( F  e.  (Poly `  RR )  /\  A  e.  CC )  ->  (
* `  ( F `  A ) )  =  ( F `  (
* `  A )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   -->wf 5251   ` cfv 5255  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737    x. cmul 8742   NN0cn0 9965   ...cfz 10782   ^cexp 11104   *ccj 11581   sum_csu 12158  Polycply 19566  coeffccoe 19568  degcdgr 19569
This theorem is referenced by:  plyreres  19663  aacjcl  19707
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-rlim 11963  df-sum 12159  df-0p 19025  df-ply 19570  df-coe 19572  df-dgr 19573
  Copyright terms: Public domain W3C validator