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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 13956 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (0...(deg‘𝐹)) ∈ Fin) | |
2 | 0re 11232 | . . . . . . . . 9 ⊢ 0 ∈ ℝ | |
3 | eqid 2727 | . . . . . . . . . 10 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
4 | 3 | coef2 26139 | . . . . . . . . 9 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 0 ∈ ℝ) → (coeff‘𝐹):ℕ0⟶ℝ) |
5 | 2, 4 | mpan2 690 | . . . . . . . 8 ⊢ (𝐹 ∈ (Poly‘ℝ) → (coeff‘𝐹):ℕ0⟶ℝ) |
6 | 5 | adantr 480 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (coeff‘𝐹):ℕ0⟶ℝ) |
7 | elfznn0 13612 | . . . . . . 7 ⊢ (𝑥 ∈ (0...(deg‘𝐹)) → 𝑥 ∈ ℕ0) | |
8 | ffvelcdm 7085 | . . . . . . 7 ⊢ (((coeff‘𝐹):ℕ0⟶ℝ ∧ 𝑥 ∈ ℕ0) → ((coeff‘𝐹)‘𝑥) ∈ ℝ) | |
9 | 6, 7, 8 | syl2an 595 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑥) ∈ ℝ) |
10 | 9 | recnd 11258 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((coeff‘𝐹)‘𝑥) ∈ ℂ) |
11 | simpr 484 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → 𝐴 ∈ ℂ) | |
12 | expcl 14062 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ0) → (𝐴↑𝑥) ∈ ℂ) | |
13 | 11, 7, 12 | syl2an 595 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (𝐴↑𝑥) ∈ ℂ) |
14 | 10, 13 | mulcld 11250 | . . . 4 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥)) ∈ ℂ) |
15 | 1, 14 | fsumcj 15774 | . . 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 15186 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = ((∗‘((coeff‘𝐹)‘𝑥)) · (∗‘(𝐴↑𝑥)))) |
17 | 9 | cjred 15191 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘((coeff‘𝐹)‘𝑥)) = ((coeff‘𝐹)‘𝑥)) |
18 | cjexp 15115 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝑥 ∈ ℕ0) → (∗‘(𝐴↑𝑥)) = ((∗‘𝐴)↑𝑥)) | |
19 | 11, 7, 18 | syl2an 595 | . . . . . 6 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(𝐴↑𝑥)) = ((∗‘𝐴)↑𝑥)) |
20 | 17, 19 | oveq12d 7432 | . . . . 5 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → ((∗‘((coeff‘𝐹)‘𝑥)) · (∗‘(𝐴↑𝑥))) = (((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
21 | 16, 20 | eqtrd 2767 | . . . 4 ⊢ (((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) ∧ 𝑥 ∈ (0...(deg‘𝐹))) → (∗‘(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) = (((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
22 | 21 | sumeq2dv 15667 | . . 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 2767 | . 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 2727 | . . . 4 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
25 | 3, 24 | coeid2 26147 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (𝐹‘𝐴) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥))) |
26 | 25 | fveq2d 6895 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (∗‘Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · (𝐴↑𝑥)))) |
27 | cjcl 15070 | . . 3 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
28 | 3, 24 | coeid2 26147 | . . 3 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ (∗‘𝐴) ∈ ℂ) → (𝐹‘(∗‘𝐴)) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
29 | 27, 28 | sylan2 592 | . 2 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (𝐹‘(∗‘𝐴)) = Σ𝑥 ∈ (0...(deg‘𝐹))(((coeff‘𝐹)‘𝑥) · ((∗‘𝐴)↑𝑥))) |
30 | 23, 26, 29 | 3eqtr4d 2777 | 1 ⊢ ((𝐹 ∈ (Poly‘ℝ) ∧ 𝐴 ∈ ℂ) → (∗‘(𝐹‘𝐴)) = (𝐹‘(∗‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11122 ℝcr 11123 0cc0 11124 · cmul 11129 ℕ0cn0 12488 ...cfz 13502 ↑cexp 14044 ∗ccj 15061 Σcsu 15650 Polycply 26092 coeffccoe 26094 degcdgr 26095 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-map 8836 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-fz 13503 df-fzo 13646 df-fl 13775 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-clim 15450 df-rlim 15451 df-sum 15651 df-0p 25573 df-ply 26096 df-coe 26098 df-dgr 26099 |
This theorem is referenced by: plyreres 26191 aacjcl 26236 |
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