|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| 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.2 | ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) | 
| coecj.3 | ⊢ 𝐴 = (coeff‘𝐹) | 
| Ref | Expression | 
|---|---|
| coecj | ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | plycj.2 | . . 3 ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) | |
| 2 | cjcl 15145 | . . . 4 ⊢ (𝑥 ∈ ℂ → (∗‘𝑥) ∈ ℂ) | |
| 3 | 2 | adantl 481 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑥 ∈ ℂ) → (∗‘𝑥) ∈ ℂ) | 
| 4 | plyssc 26240 | . . . 4 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
| 5 | 4 | sseli 3978 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹 ∈ (Poly‘ℂ)) | 
| 6 | 1, 3, 5 | plycj 26318 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 ∈ (Poly‘ℂ)) | 
| 7 | dgrcl 26273 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
| 8 | cjf 15144 | . . 3 ⊢ ∗:ℂ⟶ℂ | |
| 9 | coecj.3 | . . . 4 ⊢ 𝐴 = (coeff‘𝐹) | |
| 10 | 9 | coef3 26272 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐴:ℕ0⟶ℂ) | 
| 11 | fco 6759 | . . 3 ⊢ ((∗:ℂ⟶ℂ ∧ 𝐴:ℕ0⟶ℂ) → (∗ ∘ 𝐴):ℕ0⟶ℂ) | |
| 12 | 8, 10, 11 | sylancr 587 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → (∗ ∘ 𝐴):ℕ0⟶ℂ) | 
| 13 | fvco3 7007 | . . . . . . . . 9 ⊢ ((𝐴:ℕ0⟶ℂ ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) | |
| 14 | 10, 13 | sylan 580 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ 𝐴)‘𝑘) = (∗‘(𝐴‘𝑘))) | 
| 15 | cj0 15198 | . . . . . . . . . 10 ⊢ (∗‘0) = 0 | |
| 16 | 15 | eqcomi 2745 | . . . . . . . . 9 ⊢ 0 = (∗‘0) | 
| 17 | 16 | a1i 11 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 = (∗‘0)) | 
| 18 | 14, 17 | eqeq12d 2752 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (∗‘(𝐴‘𝑘)) = (∗‘0))) | 
| 19 | 10 | ffvelcdmda 7103 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (𝐴‘𝑘) ∈ ℂ) | 
| 20 | 0cnd 11255 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → 0 ∈ ℂ) | |
| 21 | cj11 15202 | . . . . . . . 8 ⊢ (((𝐴‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) | |
| 22 | 19, 20, 21 | syl2anc 584 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((∗‘(𝐴‘𝑘)) = (∗‘0) ↔ (𝐴‘𝑘) = 0)) | 
| 23 | 18, 22 | bitrd 279 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) = 0 ↔ (𝐴‘𝑘) = 0)) | 
| 24 | 23 | necon3bid 2984 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 ↔ (𝐴‘𝑘) ≠ 0)) | 
| 25 | eqid 2736 | . . . . . . . 8 ⊢ (deg‘𝐹) = (deg‘𝐹) | |
| 26 | 9, 25 | dgrub2 26275 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (𝐴 “ (ℤ≥‘((deg‘𝐹) + 1))) = {0}) | 
| 27 | plyco0 26232 | . . . . . . . 8 ⊢ (((deg‘𝐹) ∈ ℕ0 ∧ 𝐴:ℕ0⟶ℂ) → ((𝐴 “ (ℤ≥‘((deg‘𝐹) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹)))) | |
| 28 | 7, 10, 27 | syl2anc 584 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((𝐴 “ (ℤ≥‘((deg‘𝐹) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹)))) | 
| 29 | 26, 28 | mpbid 232 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹))) | 
| 30 | 29 | r19.21bi 3250 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝐴‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹))) | 
| 31 | 24, 30 | sylbid 240 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝑘 ∈ ℕ0) → (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹))) | 
| 32 | 31 | ralrimiva 3145 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹))) | 
| 33 | plyco0 26232 | . . . 4 ⊢ (((deg‘𝐹) ∈ ℕ0 ∧ (∗ ∘ 𝐴):ℕ0⟶ℂ) → (((∗ ∘ 𝐴) “ (ℤ≥‘((deg‘𝐹) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹)))) | |
| 34 | 7, 12, 33 | syl2anc 584 | . . 3 ⊢ (𝐹 ∈ (Poly‘𝑆) → (((∗ ∘ 𝐴) “ (ℤ≥‘((deg‘𝐹) + 1))) = {0} ↔ ∀𝑘 ∈ ℕ0 (((∗ ∘ 𝐴)‘𝑘) ≠ 0 → 𝑘 ≤ (deg‘𝐹)))) | 
| 35 | 32, 34 | mpbird 257 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → ((∗ ∘ 𝐴) “ (ℤ≥‘((deg‘𝐹) + 1))) = {0}) | 
| 36 | 25, 1, 9 | plycjlem 26317 | . 2 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑦 ∈ ℂ ↦ Σ𝑧 ∈ (0...(deg‘𝐹))(((∗ ∘ 𝐴)‘𝑧) · (𝑦↑𝑧)))) | 
| 37 | 6, 7, 12, 35, 36 | coeeq 26267 | 1 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐺) = (∗ ∘ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 {csn 4625 class class class wbr 5142 “ cima 5687 ∘ ccom 5688 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 ℂcc 11154 0cc0 11156 1c1 11157 + caddc 11159 ≤ cle 11297 ℕ0cn0 12528 ℤ≥cuz 12879 ∗ccj 15136 Polycply 26224 coeffccoe 26226 degcdgr 26227 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 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 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-map 8869 df-pm 8870 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-n0 12529 df-z 12616 df-uz 12880 df-rp 13036 df-fz 13549 df-fzo 13696 df-fl 13833 df-seq 14044 df-exp 14104 df-hash 14371 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-clim 15525 df-rlim 15526 df-sum 15724 df-0p 25706 df-ply 26228 df-coe 26230 df-dgr 26231 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |