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Mirrors > Home > MPE Home > Th. List > coecjOLD | Structured version Visualization version GIF version |
Description: Obsolete version of coecj 26314 as of 22-Sep-2025. Double conjugation of a polynomial causes the coefficients to be conjugated. (Contributed by Mario Carneiro, 24-Jul-2014.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
plycjOLD.1 | ⊢ 𝑁 = (deg‘𝐹) |
plycjOLD.2 | ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) |
coecjOLD.3 | ⊢ 𝐴 = (coeff‘𝐹) |
Ref | Expression |
---|---|
coecjOLD | ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plycjOLD.1 | . . 3 ⊢ 𝑁 = (deg‘𝐹) | |
2 | plycjOLD.2 | . . 3 ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) | |
3 | cjcl 15130 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
4 | 3 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) |
5 | plyssc 26235 | . . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
6 | 5 | sseli 3991 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
7 | 1, 2, 4, 6 | plycjOLD 26315 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ)) |
8 | dgrcl 26268 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
9 | 1, 8 | eqeltrid 2841 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0) |
10 | cjf 15129 | . . 3 ⊢ ∗:ℂ⟶ℂ | |
11 | coecjOLD.3 | . . . 4 ⊢ 𝐴 = (coeff‘𝐹) | |
12 | 11 | coef3 26267 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
13 | fco 6755 | . . 3 ⊢ ((∗:ℂ⟶ℂ ∧ 𝐴:ℕ0⟶ℂ) → (∗ ∘ 𝐴):ℕ0⟶ℂ) | |
14 | 10, 12, 13 | sylancr 586 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∗ ∘ 𝐴):ℕ0⟶ℂ) |
15 | fvco3 7002 | . . . . . . . . 9 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) | |
16 | 12, 15 | sylan 579 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) |
17 | cj0 15183 | . . . . . . . . . 10 ⊢ (∗‘0) = 0 | |
18 | 17 | eqcomi 2742 | . . . . . . . . 9 ⊢ 0 = (∗‘0) |
19 | 18 | a1i 11 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 = (∗‘0)) |
20 | 16, 19 | eqeq12d 2749 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (∗‘(𝐴‘𝑘)) = (∗‘0))) |
21 | 12 | ffvelcdmda 7098 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
22 | 0cnd 11245 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℂ) | |
23 | cj11 15187 | . . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) | |
24 | 21, 22, 23 | syl2anc 583 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) |
25 | 20, 24 | bitrd 279 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (𝐴‘𝑘) = 0)) |
26 | 25 | necon3bid 2981 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 ↔ (𝐴‘𝑘) ≠ 0)) |
27 | 11, 1 | dgrub2 26270 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
28 | plyco0 26227 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
29 | 9, 12, 28 | syl2anc 583 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
30 | 27, 29 | mpbid 232 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
31 | 30 | r19.21bi 3247 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
32 | 26, 31 | sylbid 240 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
33 | 32 | ralrimiva 3142 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
34 | plyco0 26227 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (∗ ∘ 𝐴):ℕ0⟶ℂ) → (((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
35 | 9, 14, 34 | syl2anc 583 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) |
36 | 33, 35 | mpbird 257 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0}) |
37 | 1, 2, 11 | plycjlem 26312 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘)))) |
38 | 7, 9, 14, 36, 37 | coeeq 26262 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 ≠ wne 2936 ∀wral 3057 {csn 4630 class class class wbr 5149 “ cima 5686 ∘ ccom 5687 ⟶wf 6554 ‘cfv 6558 (class class class)co 7425 ℂcc 11144 0cc0 11146 1c1 11147 + caddc 11149 ≤ cle 11287 ℕ0cn0 12517 ℤ≥cuz 12869 ∗ccj 15121 Polycply 26219 coeffccoe 26221 degcdgr 26222 |
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 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-int 4954 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5635 df-se 5636 df-we 5637 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-pred 6317 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-isom 6567 df-riota 7381 df-ov 7428 df-oprab 7429 df-mpo 7430 df-of 7691 df-om 7881 df-1st 8007 df-2nd 8008 df-frecs 8299 df-wrecs 8330 df-recs 8404 df-rdg 8443 df-1o 8499 df-er 8738 df-map 8861 df-pm 8862 df-en 8979 df-dom 8980 df-sdom 8981 df-fin 8982 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11485 df-neg 11486 df-div 11912 df-nn 12258 df-2 12320 df-3 12321 df-n0 12518 df-z 12605 df-uz 12870 df-rp 13026 df-fz 13538 df-fzo 13682 df-fl 13818 df-seq 14029 df-exp 14089 df-hash 14356 df-cj 15124 df-re 15125 df-im 15126 df-sqrt 15260 df-abs 15261 df-clim 15510 df-rlim 15511 df-sum 15709 df-0p 25700 df-ply 26223 df-coe 26225 df-dgr 26226 |
This theorem is referenced by: (None) |
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