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Mirrors > Home > MPE Home > Th. List > coecj | Structured version Visualization version GIF version |
Description: Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.) |
Ref | Expression |
---|---|
plycj.1 | ⊢ 𝑁 = (deg‘𝐹) |
plycj.2 | ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) |
coecj.3 | ⊢ 𝐴 = (coeff‘𝐹) |
Ref | Expression |
---|---|
coecj | ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plycj.1 | . . 3 ⊢ 𝑁 = (deg‘𝐹) | |
2 | plycj.2 | . . 3 ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) | |
3 | cjcl 14456 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
4 | 3 | adantl 485 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) |
5 | plyssc 24797 | . . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
6 | 5 | sseli 3911 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
7 | 1, 2, 4, 6 | plycj 24874 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ)) |
8 | dgrcl 24830 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
9 | 1, 8 | eqeltrid 2894 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0) |
10 | cjf 14455 | . . 3 ⊢ ∗:ℂ⟶ℂ | |
11 | coecj.3 | . . . 4 ⊢ 𝐴 = (coeff‘𝐹) | |
12 | 11 | coef3 24829 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
13 | fco 6505 | . . 3 ⊢ ((∗:ℂ⟶ℂ ∧ 𝐴:ℕ0⟶ℂ) → (∗ ∘ 𝐴):ℕ0⟶ℂ) | |
14 | 10, 12, 13 | sylancr 590 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∗ ∘ 𝐴):ℕ0⟶ℂ) |
15 | fvco3 6737 | . . . . . . . . 9 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) | |
16 | 12, 15 | sylan 583 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) |
17 | cj0 14509 | . . . . . . . . . 10 ⊢ (∗‘0) = 0 | |
18 | 17 | eqcomi 2807 | . . . . . . . . 9 ⊢ 0 = (∗‘0) |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 = (∗‘0)) |
20 | 16, 19 | eqeq12d 2814 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (∗‘(𝐴‘𝑘)) = (∗‘0))) |
21 | 12 | ffvelrnda 6828 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
22 | 0cnd 10623 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℂ) | |
23 | cj11 14513 | . . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) | |
24 | 21, 22, 23 | syl2anc 587 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) |
25 | 20, 24 | bitrd 282 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (𝐴‘𝑘) = 0)) |
26 | 25 | necon3bid 3031 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 ↔ (𝐴‘𝑘) ≠ 0)) |
27 | 11, 1 | dgrub2 24832 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
28 | plyco0 24789 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
29 | 9, 12, 28 | syl2anc 587 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
30 | 27, 29 | mpbid 235 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
31 | 30 | r19.21bi 3173 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
32 | 26, 31 | sylbid 243 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
33 | 32 | ralrimiva 3149 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
34 | plyco0 24789 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (∗ ∘ 𝐴):ℕ0⟶ℂ) → (((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
35 | 9, 14, 34 | syl2anc 587 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
36 | 33, 35 | mpbird 260 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0}) |
37 | 1, 2, 11 | plycjlem 24873 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘)))) |
38 | 7, 9, 14, 36, 37 | coeeq 24824 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 {csn 4525 class class class wbr 5030 “ cima 5522 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 0cc0 10526 1c1 10527 + caddc 10529 ≤ cle 10665 ℕ0cn0 11885 ℤ≥cuz 12231 ∗ccj 14447 Polycply 24781 coeffccoe 24783 degcdgr 24784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12886 df-fzo 13029 df-fl 13157 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-rlim 14838 df-sum 15035 df-0p 24274 df-ply 24785 df-coe 24787 df-dgr 24788 |
This theorem is referenced by: (None) |
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