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| Mirrors > Home > MPE Home > Th. List > coecjOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of coecj 26308 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 15140 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
| 4 | 3 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) |
| 5 | plyssc 26229 | . . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
| 6 | 5 | sseli 3978 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) |
| 7 | 1, 2, 4, 6 | plycjOLD 26309 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ)) |
| 8 | dgrcl 26262 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
| 9 | 1, 8 | eqeltrid 2844 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑁 ∈ ℕ0) |
| 10 | cjf 15139 | . . 3 ⊢ ∗:ℂ⟶ℂ | |
| 11 | coecjOLD.3 | . . . 4 ⊢ 𝐴 = (coeff‘𝐹) | |
| 12 | 11 | coef3 26261 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) |
| 13 | fco 6758 | . . 3 ⊢ ((∗:ℂ⟶ℂ ∧ 𝐴:ℕ0⟶ℂ) → (∗ ∘ 𝐴):ℕ0⟶ℂ) | |
| 14 | 10, 12, 13 | sylancr 587 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∗ ∘ 𝐴):ℕ0⟶ℂ) |
| 15 | fvco3 7006 | . . . . . . . . 9 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) | |
| 16 | 12, 15 | sylan 580 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) |
| 17 | cj0 15193 | . . . . . . . . . 10 ⊢ (∗‘0) = 0 | |
| 18 | 17 | eqcomi 2745 | . . . . . . . . 9 ⊢ 0 = (∗‘0) |
| 19 | 18 | a1i 11 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 = (∗‘0)) |
| 20 | 16, 19 | eqeq12d 2752 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (∗‘(𝐴‘𝑘)) = (∗‘0))) |
| 21 | 12 | ffvelcdmda 7102 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) |
| 22 | 0cnd 11250 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℂ) | |
| 23 | cj11 15197 | . . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) | |
| 24 | 21, 22, 23 | syl2anc 584 | . . . . . . 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 2984 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 ↔ (𝐴‘𝑘) ≠ 0)) |
| 27 | 11, 1 | dgrub2 26264 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0}) |
| 28 | plyco0 26221 | . . . . . . . 8 ⊢ ((𝑁 ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
| 29 | 9, 12, 28 | syl2anc 584 | . . . . . . 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 3250 | . . . . 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 3145 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁)) |
| 34 | plyco0 26221 | . . . 4 ⊢ ((𝑁 ∈ ℕ0 ∧ (∗ ∘ 𝐴):ℕ0⟶ℂ) → (((∗ ∘ 𝐴) “ (ℤ≥‘(𝑁 + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ 𝑁))) | |
| 35 | 9, 14, 34 | syl2anc 584 | . . 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 26306 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ 𝐴)‘𝑘) · (𝑧↑𝑘)))) |
| 38 | 7, 9, 14, 36, 37 | coeeq 26256 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2939 ∀wral 3060 {csn 4624 class class class wbr 5141 “ cima 5686 ∘ ccom 5687 ⟶wf 6555 ‘cfv 6559 (class class class)co 7429 ℂcc 11149 0cc0 11151 1c1 11152 + caddc 11154 ≤ cle 11292 ℕ0cn0 12522 ℤ≥cuz 12874 ∗ccj 15131 Polycply 26213 coeffccoe 26215 degcdgr 26216 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-inf2 9677 ax-cnex 11207 ax-resscn 11208 ax-1cn 11209 ax-icn 11210 ax-addcl 11211 ax-addrcl 11212 ax-mulcl 11213 ax-mulrcl 11214 ax-mulcom 11215 ax-addass 11216 ax-mulass 11217 ax-distr 11218 ax-i2m1 11219 ax-1ne0 11220 ax-1rid 11221 ax-rnegex 11222 ax-rrecex 11223 ax-cnre 11224 ax-pre-lttri 11225 ax-pre-lttrn 11226 ax-pre-ltadd 11227 ax-pre-mulgt0 11228 ax-pre-sup 11229 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-int 4945 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-tr 5258 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 6319 df-ord 6385 df-on 6386 df-lim 6387 df-suc 6388 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-isom 6568 df-riota 7386 df-ov 7432 df-oprab 7433 df-mpo 7434 df-of 7694 df-om 7884 df-1st 8010 df-2nd 8011 df-frecs 8302 df-wrecs 8333 df-recs 8407 df-rdg 8446 df-1o 8502 df-er 8741 df-map 8864 df-pm 8865 df-en 8982 df-dom 8983 df-sdom 8984 df-fin 8985 df-sup 9478 df-inf 9479 df-oi 9546 df-card 9975 df-pnf 11293 df-mnf 11294 df-xr 11295 df-ltxr 11296 df-le 11297 df-sub 11490 df-neg 11491 df-div 11917 df-nn 12263 df-2 12325 df-3 12326 df-n0 12523 df-z 12610 df-uz 12875 df-rp 13031 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-rlim 15521 df-sum 15719 df-0p 25695 df-ply 26217 df-coe 26219 df-dgr 26220 |
| This theorem is referenced by: (None) |
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