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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 10606 | . . 3 ⊢ ℂ ∈ V | |
2 | plyadd.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
3 | plyf 24715 | . . . 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 24715 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | |
7 | 5, 6 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ) |
8 | ofnegsub 11624 | . . 3 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺)) | |
9 | 1, 4, 7, 8 | mp3an2i 1457 | . 2 ⊢ (𝜑 → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) = (𝐹 ∘f − 𝐺)) |
10 | plybss 24711 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
11 | 2, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
12 | plysub.5 | . . . . 5 ⊢ (𝜑 → -1 ∈ 𝑆) | |
13 | plyconst 24723 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ -1 ∈ 𝑆) → (ℂ × {-1}) ∈ (Poly‘𝑆)) | |
14 | 11, 12, 13 | syl2anc 584 | . . . 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 24735 | . . 3 ⊢ (𝜑 → ((ℂ × {-1}) ∘f · 𝐺) ∈ (Poly‘𝑆)) |
18 | 2, 17, 15 | plyadd 24734 | . 2 ⊢ (𝜑 → (𝐹 ∘f + ((ℂ × {-1}) ∘f · 𝐺)) ∈ (Poly‘𝑆)) |
19 | 9, 18 | eqeltrrd 2911 | 1 ⊢ (𝜑 → (𝐹 ∘f − 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ⊆ wss 3933 {csn 4557 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ∘f cof 7396 ℂcc 10523 1c1 10526 + caddc 10528 · cmul 10530 − cmin 10858 -cneg 10859 Polycply 24701 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-inf2 9092 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-fal 1541 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-oi 8962 df-card 9356 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-n0 11886 df-z 11970 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-clim 14833 df-sum 15031 df-ply 24705 |
This theorem is referenced by: plysubcl 24739 plydivlem2 24810 plydivlem4 24812 plydiveu 24814 qaa 24839 taylply2 24883 mpaaeu 39628 |
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