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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 10355 | . . . 4 ⊢ ℂ ∈ V | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → ℂ ∈ V) |
3 | plyadd.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (Poly‘𝑆)) | |
4 | plyf 24402 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:ℂ⟶ℂ) |
6 | plyadd.2 | . . . 4 ⊢ (𝜑 → 𝐺 ∈ (Poly‘𝑆)) | |
7 | plyf 24402 | . . . 4 ⊢ (𝐺 ∈ (Poly‘𝑆) → 𝐺:ℂ⟶ℂ) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺:ℂ⟶ℂ) |
9 | ofnegsub 11377 | . . 3 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ 𝐺:ℂ⟶ℂ) → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) | |
10 | 2, 5, 8, 9 | syl3anc 1439 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) = (𝐹 ∘𝑓 − 𝐺)) |
11 | plybss 24398 | . . . . . 6 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝑆 ⊆ ℂ) | |
12 | 3, 11 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
13 | plysub.5 | . . . . 5 ⊢ (𝜑 → -1 ∈ 𝑆) | |
14 | plyconst 24410 | . . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ -1 ∈ 𝑆) → (ℂ × {-1}) ∈ (Poly‘𝑆)) | |
15 | 12, 13, 14 | syl2anc 579 | . . . 4 ⊢ (𝜑 → (ℂ × {-1}) ∈ (Poly‘𝑆)) |
16 | plyadd.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆) | |
17 | plymul.4 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 · 𝑦) ∈ 𝑆) | |
18 | 15, 6, 16, 17 | plymul 24422 | . . 3 ⊢ (𝜑 → ((ℂ × {-1}) ∘𝑓 · 𝐺) ∈ (Poly‘𝑆)) |
19 | 3, 18, 16 | plyadd 24421 | . 2 ⊢ (𝜑 → (𝐹 ∘𝑓 + ((ℂ × {-1}) ∘𝑓 · 𝐺)) ∈ (Poly‘𝑆)) |
20 | 10, 19 | eqeltrrd 2860 | 1 ⊢ (𝜑 → (𝐹 ∘𝑓 − 𝐺) ∈ (Poly‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 {csn 4398 × cxp 5355 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ∘𝑓 cof 7174 ℂcc 10272 1c1 10275 + caddc 10277 · cmul 10279 − cmin 10608 -cneg 10609 Polycply 24388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5008 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 ax-un 7228 ax-inf2 8837 ax-cnex 10330 ax-resscn 10331 ax-1cn 10332 ax-icn 10333 ax-addcl 10334 ax-addrcl 10335 ax-mulcl 10336 ax-mulrcl 10337 ax-mulcom 10338 ax-addass 10339 ax-mulass 10340 ax-distr 10341 ax-i2m1 10342 ax-1ne0 10343 ax-1rid 10344 ax-rnegex 10345 ax-rrecex 10346 ax-cnre 10347 ax-pre-lttri 10348 ax-pre-lttrn 10349 ax-pre-ltadd 10350 ax-pre-mulgt0 10351 ax-pre-sup 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4674 df-int 4713 df-iun 4757 df-br 4889 df-opab 4951 df-mpt 4968 df-tr 4990 df-id 5263 df-eprel 5268 df-po 5276 df-so 5277 df-fr 5316 df-se 5317 df-we 5318 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-pred 5935 df-ord 5981 df-on 5982 df-lim 5983 df-suc 5984 df-iota 6101 df-fun 6139 df-fn 6140 df-f 6141 df-f1 6142 df-fo 6143 df-f1o 6144 df-fv 6145 df-isom 6146 df-riota 6885 df-ov 6927 df-oprab 6928 df-mpt2 6929 df-of 7176 df-om 7346 df-1st 7447 df-2nd 7448 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-sup 8638 df-oi 8706 df-card 9100 df-pnf 10415 df-mnf 10416 df-xr 10417 df-ltxr 10418 df-le 10419 df-sub 10610 df-neg 10611 df-div 11036 df-nn 11380 df-2 11443 df-3 11444 df-n0 11648 df-z 11734 df-uz 11998 df-rp 12143 df-fz 12649 df-fzo 12790 df-seq 13125 df-exp 13184 df-hash 13442 df-cj 14252 df-re 14253 df-im 14254 df-sqrt 14388 df-abs 14389 df-clim 14636 df-sum 14834 df-ply 24392 |
This theorem is referenced by: plysubcl 24426 plydivlem2 24497 plydivlem4 24499 plydiveu 24501 qaa 24526 taylply2 24570 mpaaeu 38693 |
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