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Mirrors > Home > MPE Home > Th. List > plysub | Structured version Visualization version GIF version |
Description: The difference of two polynomials is a polynomial. (Contributed by Mario Carneiro, 21-Jul-2014.) |
Ref | Expression |
---|---|
plyadd.1 | ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) |
plyadd.2 | ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) |
plyadd.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) |
plymul.4 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) |
plysub.5 | ⊢ (𝜑 → -1 ∈ 𝑆) |
Ref | Expression |
---|---|
plysub | ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (Poly‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 10698 | . . 3 ⊢ ℂ ∈ V | |
2 | plyadd.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
3 | plyf 24949 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
5 | plyadd.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | |
6 | plyf 24949 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ) |
8 | ofnegsub 11716 | . . 3 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺)) | |
9 | 1, 4, 7, 8 | mp3an2i 1467 | . 2 ⊢ (𝜑 → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺)) |
10 | plybss 24945 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
11 | 2, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | plysub.5 | . . . . 5 ⊢ (𝜑 → -1 ∈ 𝑆) | |
13 | plyconst 24957 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ -1 ∈ 𝑆) → (ℂ × {-1}) ∈ (Poly‘𝑆)) | |
14 | 11, 12, 13 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (ℂ × {-1}) ∈ (Poly‘𝑆)) |
15 | plyadd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
16 | plymul.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
17 | 14, 5, 15, 16 | plymul 24969 | . . 3 ⊢ (𝜑 → ((ℂ × {-1}) ∘f · 𝐺) ∈ (Poly‘𝑆)) |
18 | 2, 17, 15 | plyadd 24968 | . 2 ⊢ (𝜑 → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) ∈ (Poly‘𝑆)) |
19 | 9, 18 | eqeltrrd 2834 | 1 ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ⊆ wss 3843 {csn 4516 × cxp 5523 ⟶wf 6335 ‘cfv 6339 (class class class)co 7172 ∘f cof 7425 ℂcc 10615 1c1 10618 + caddc 10620 · cmul 10622 − cmin 10950 -cneg 10951 Polycply 24935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-inf2 9179 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-pre-sup 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-se 5484 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-isom 6348 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-sup 8981 df-oi 9049 df-card 9443 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-div 11378 df-nn 11719 df-2 11781 df-3 11782 df-n0 11979 df-z 12065 df-uz 12327 df-rp 12475 df-fz 12984 df-fzo 13127 df-seq 13463 df-exp 13524 df-hash 13785 df-cj 14550 df-re 14551 df-im 14552 df-sqrt 14686 df-abs 14687 df-clim 14937 df-sum 15138 df-ply 24939 |
This theorem is referenced by: plysubcl 24973 plydivlem2 25044 plydivlem4 25046 plydiveu 25048 qaa 25073 taylply2 25117 mpaaeu 40569 |
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