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Mirrors > Home > MPE Home > Th. List > plyrecj | Structured version Visualization version GIF version |
Description: A polynomial with real coefficients distributes under conjugation. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
plyrecj | ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13546 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (0...(deg‘𝐹)) ∈ Fin) | |
2 | 0re 10835 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
3 | eqid 2737 | . . . . . . . . . 10 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
4 | 3 | coef2 25125 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ) |
5 | 2, 4 | mpan2 691 | . . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (coeff‘𝐹):ℕ0⟶ℝ) |
6 | 5 | adantr 484 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℝ) |
7 | elfznn0 13205 | . . . . . . 7 ⊢ (𝑥 ∈ (0...(deg‘𝐹)) → 𝑥 ∈ ℕ0) | |
8 | ffvelrn 6902 | . . . . . . 7 ⊢ (((coeff‘𝐹):ℕ0⟶ℝ ∧ 𝑥 ∈ ℕ0) → ((coeff‘𝐹)‘𝑥) ∈ ℝ) | |
9 | 6, 7, 8 | syl2an 599 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑥) ∈ ℝ) |
10 | 9 | recnd 10861 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑥) ∈ ℂ) |
11 | simpr 488 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ) | |
12 | expcl 13653 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ0) → (𝐴↑𝑥) ∈ ℂ) | |
13 | 11, 7, 12 | syl2an 599 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (𝐴↑𝑥) ∈ ℂ) |
14 | 10, 13 | mulcld 10853 | . . . 4 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥)) ∈ ℂ) |
15 | 1, 14 | fsumcj 15374 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = Σ𝑥 ∈ (0...(deg‘𝐹))(∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥)))) |
16 | 10, 13 | cjmuld 14784 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = ((∗‘((coeff‘𝐹)‘𝑥)) · (∗‘(𝐴↑𝑥)))) |
17 | 9 | cjred 14789 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘((coeff‘𝐹)‘𝑥)) = ((coeff‘𝐹)‘𝑥)) |
18 | cjexp 14713 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ0) → (∗‘(𝐴↑𝑥)) = ((∗‘𝐴)↑𝑥)) | |
19 | 11, 7, 18 | syl2an 599 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(𝐴↑𝑥)) = ((∗‘𝐴)↑𝑥)) |
20 | 17, 19 | oveq12d 7231 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((∗‘((coeff‘𝐹)‘𝑥)) · (∗‘(𝐴↑𝑥))) = (((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
21 | 16, 20 | eqtrd 2777 | . . . 4 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = (((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
22 | 21 | sumeq2dv 15267 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → Σ𝑥 ∈ (0...(deg‘𝐹))(∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
23 | 15, 22 | eqtrd 2777 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
24 | eqid 2737 | . . . 4 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
25 | 3, 24 | coeid2 25133 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) |
26 | 25 | fveq2d 6721 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (∗‘Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥)))) |
27 | cjcl 14668 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
28 | 3, 24 | coeid2 25133 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ (∗‘𝐴) ∈ ℂ) → (𝐹‘(∗‘𝐴)) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
29 | 27, 28 | sylan2 596 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (𝐹‘(∗‘𝐴)) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
30 | 23, 26, 29 | 3eqtr4d 2787 | 1 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ⟶wf 6376 ‘cfv 6380 (class class class)co 7213 ℂcc 10727 ℝcr 10728 0cc0 10729 · cmul 10734 ℕ0cn0 12090 ...cfz 13095 ↑cexp 13635 ∗ccj 14659 Σcsu 15249 Polycply 25078 coeffccoe 25080 degcdgr 25081 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-inf2 9256 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-pre-sup 10807 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-isom 6389 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-pm 8511 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-sup 9058 df-inf 9059 df-oi 9126 df-card 9555 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-div 11490 df-nn 11831 df-2 11893 df-3 11894 df-n0 12091 df-z 12177 df-uz 12439 df-rp 12587 df-fz 13096 df-fzo 13239 df-fl 13367 df-seq 13575 df-exp 13636 df-hash 13897 df-cj 14662 df-re 14663 df-im 14664 df-sqrt 14798 df-abs 14799 df-clim 15049 df-rlim 15050 df-sum 15250 df-0p 24567 df-ply 25082 df-coe 25084 df-dgr 25085 |
This theorem is referenced by: plyreres 25176 aacjcl 25220 |
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