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Mirrors > Home > MPE Home > Th. List > plycjOLD | Structured version Visualization version GIF version |
Description: Obsolete version of plycj 26307 as of 22-Sep-2025. The double conjugation of a polynomial is a polynomial. (The single conjugation is not because our definition of polynomial includes only holomorphic functions, i.e. no dependence on (∗‘𝑧) independently of 𝑧.) (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) ∘ ∗) |
plycjOLD.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆) |
plycjOLD.4 | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
Ref | Expression |
---|---|
plycjOLD | ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | plycjOLD.4 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
2 | plycjOLD.1 | . . . . 5 ⊢ 𝑁 = (deg‘𝐹) | |
3 | plycjOLD.2 | . . . . 5 ⊢ 𝐺 = ((∗ ∘ 𝐹) ∘ ∗) | |
4 | eqid 2736 | . . . . 5 ⊢ (coeff‘𝐹) = (coeff‘𝐹) | |
5 | 2, 3, 4 | plycjlem 26306 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ (coeff‘𝐹))‘𝑘) · (𝑧↑𝑘)))) |
6 | 1, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 = (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ (coeff‘𝐹))‘𝑘) · (𝑧↑𝑘)))) |
7 | plybss 26223 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
9 | 0cnd 11250 | . . . . . 6 ⊢ (𝜑 → 0 ∈ ℂ) | |
10 | 9 | snssd 4807 | . . . . 5 ⊢ (𝜑 → {0} ⊆ ℂ) |
11 | 8, 10 | unssd 4191 | . . . 4 ⊢ (𝜑 → (𝑆 ∪ {0}) ⊆ ℂ) |
12 | dgrcl 26262 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → (deg‘𝐹) ∈ ℕ0) | |
13 | 1, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → (deg‘𝐹) ∈ ℕ0) |
14 | 2, 13 | eqeltrid 2844 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
15 | 4 | coef 26259 | . . . . . . 7 ⊢ (𝐹 ∈ (Poly‘𝑆) → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
16 | 1, 15 | syl 17 | . . . . . 6 ⊢ (𝜑 → (coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0})) |
17 | elfznn0 13656 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
18 | fvco3 7006 | . . . . . 6 ⊢ (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((∗ ∘ (coeff‘𝐹))‘𝑘) = (∗‘((coeff‘𝐹)‘𝑘))) | |
19 | 16, 17, 18 | syl2an 596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((∗ ∘ (coeff‘𝐹))‘𝑘) = (∗‘((coeff‘𝐹)‘𝑘))) |
20 | ffvelcdm 7099 | . . . . . . 7 ⊢ (((coeff‘𝐹):ℕ0⟶(𝑆 ∪ {0}) ∧ 𝑘 ∈ ℕ0) → ((coeff‘𝐹)‘𝑘) ∈ (𝑆 ∪ {0})) | |
21 | 16, 17, 20 | syl2an 596 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((coeff‘𝐹)‘𝑘) ∈ (𝑆 ∪ {0})) |
22 | plycjOLD.3 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑆) → (∗‘𝑥) ∈ 𝑆) | |
23 | 22 | ralrimiva 3145 | . . . . . . . . . 10 ⊢ (𝜑 → ∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆) |
24 | fveq2 6904 | . . . . . . . . . . . 12 ⊢ (𝑥 = ((coeff‘𝐹)‘𝑘) → (∗‘𝑥) = (∗‘((coeff‘𝐹)‘𝑘))) | |
25 | 24 | eleq1d 2825 | . . . . . . . . . . 11 ⊢ (𝑥 = ((coeff‘𝐹)‘𝑘) → ((∗‘𝑥) ∈ 𝑆 ↔ (∗‘((coeff‘𝐹)‘𝑘)) ∈ 𝑆)) |
26 | 25 | rspccv 3618 | . . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑆 (∗‘𝑥) ∈ 𝑆 → (((coeff‘𝐹)‘𝑘) ∈ 𝑆 → (∗‘((coeff‘𝐹)‘𝑘)) ∈ 𝑆)) |
27 | 23, 26 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → (((coeff‘𝐹)‘𝑘) ∈ 𝑆 → (∗‘((coeff‘𝐹)‘𝑘)) ∈ 𝑆)) |
28 | elsni 4641 | . . . . . . . . . . . . 13 ⊢ (((coeff‘𝐹)‘𝑘) ∈ {0} → ((coeff‘𝐹)‘𝑘) = 0) | |
29 | 28 | fveq2d 6908 | . . . . . . . . . . . 12 ⊢ (((coeff‘𝐹)‘𝑘) ∈ {0} → (∗‘((coeff‘𝐹)‘𝑘)) = (∗‘0)) |
30 | cj0 15193 | . . . . . . . . . . . 12 ⊢ (∗‘0) = 0 | |
31 | 29, 30 | eqtrdi 2792 | . . . . . . . . . . 11 ⊢ (((coeff‘𝐹)‘𝑘) ∈ {0} → (∗‘((coeff‘𝐹)‘𝑘)) = 0) |
32 | fvex 6917 | . . . . . . . . . . . 12 ⊢ (∗‘((coeff‘𝐹)‘𝑘)) ∈ V | |
33 | 32 | elsn 4639 | . . . . . . . . . . 11 ⊢ ((∗‘((coeff‘𝐹)‘𝑘)) ∈ {0} ↔ (∗‘((coeff‘𝐹)‘𝑘)) = 0) |
34 | 31, 33 | sylibr 234 | . . . . . . . . . 10 ⊢ (((coeff‘𝐹)‘𝑘) ∈ {0} → (∗‘((coeff‘𝐹)‘𝑘)) ∈ {0}) |
35 | 34 | a1i 11 | . . . . . . . . 9 ⊢ (𝜑 → (((coeff‘𝐹)‘𝑘) ∈ {0} → (∗‘((coeff‘𝐹)‘𝑘)) ∈ {0})) |
36 | 27, 35 | orim12d 967 | . . . . . . . 8 ⊢ (𝜑 → ((((coeff‘𝐹)‘𝑘) ∈ 𝑆 ∨ ((coeff‘𝐹)‘𝑘) ∈ {0}) → ((∗‘((coeff‘𝐹)‘𝑘)) ∈ 𝑆 ∨ (∗‘((coeff‘𝐹)‘𝑘)) ∈ {0}))) |
37 | elun 4152 | . . . . . . . 8 ⊢ (((coeff‘𝐹)‘𝑘) ∈ (𝑆 ∪ {0}) ↔ (((coeff‘𝐹)‘𝑘) ∈ 𝑆 ∨ ((coeff‘𝐹)‘𝑘) ∈ {0})) | |
38 | elun 4152 | . . . . . . . 8 ⊢ ((∗‘((coeff‘𝐹)‘𝑘)) ∈ (𝑆 ∪ {0}) ↔ ((∗‘((coeff‘𝐹)‘𝑘)) ∈ 𝑆 ∨ (∗‘((coeff‘𝐹)‘𝑘)) ∈ {0})) | |
39 | 36, 37, 38 | 3imtr4g 296 | . . . . . . 7 ⊢ (𝜑 → (((coeff‘𝐹)‘𝑘) ∈ (𝑆 ∪ {0}) → (∗‘((coeff‘𝐹)‘𝑘)) ∈ (𝑆 ∪ {0}))) |
40 | 39 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((coeff‘𝐹)‘𝑘) ∈ (𝑆 ∪ {0}) → (∗‘((coeff‘𝐹)‘𝑘)) ∈ (𝑆 ∪ {0}))) |
41 | 21, 40 | mpd 15 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (∗‘((coeff‘𝐹)‘𝑘)) ∈ (𝑆 ∪ {0})) |
42 | 19, 41 | eqeltrd 2840 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((∗ ∘ (coeff‘𝐹))‘𝑘) ∈ (𝑆 ∪ {0})) |
43 | 11, 14, 42 | elplyd 26231 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((∗ ∘ (coeff‘𝐹))‘𝑘) · (𝑧↑𝑘))) ∈ (Poly‘(𝑆 ∪ {0}))) |
44 | 6, 43 | eqeltrd 2840 | . 2 ⊢ (𝜑 → 𝐺 ∈ (Poly‘(𝑆 ∪ {0}))) |
45 | plyun0 26226 | . 2 ⊢ (Poly‘(𝑆 ∪ {0})) = (Poly‘𝑆) | |
46 | 44, 45 | eleqtrdi 2850 | 1 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3060 ∪ cun 3948 ⊆ wss 3950 {csn 4624 ↦ cmpt 5223 ∘ ccom 5687 ⟶wf 6555 ‘cfv 6559 (class class class)co 7429 ℂcc 11149 0cc0 11151 · cmul 11156 ℕ0cn0 12522 ...cfz 13543 ↑cexp 14098 ∗ccj 15131 Σcsu 15718 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: coecjOLD 26310 |
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