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Mirrors > Home > MPE Home > Th. List > facth | Structured version Visualization version GIF version |
Description: The factor theorem. If a polynomial 𝐹 has a root at 𝐴, then 𝐺 = 𝑥 − 𝐴 is a factor of 𝐹 (and the other factor is 𝐹 quot 𝐺). This is part of Metamath 100 proof #89. (Contributed by Mario Carneiro, 26-Jul-2014.) |
Ref | Expression |
---|---|
facth.1 | ⊢ 𝐺 = (Xp ∘𝑓 − (ℂ × {𝐴})) |
Ref | Expression |
---|---|
facth | ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 = (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | facth.1 | . . . . 5 ⊢ 𝐺 = (Xp ∘𝑓 − (ℂ × {𝐴})) | |
2 | eqid 2778 | . . . . 5 ⊢ (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) | |
3 | 1, 2 | plyrem 24497 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ) → (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (ℂ × {(𝐹‘𝐴)})) |
4 | 3 | 3adant3 1123 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (ℂ × {(𝐹‘𝐴)})) |
5 | simp3 1129 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐹‘𝐴) = 0) | |
6 | 5 | sneqd 4410 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → {(𝐹‘𝐴)} = {0}) |
7 | 6 | xpeq2d 5385 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (ℂ × {(𝐹‘𝐴)}) = (ℂ × {0})) |
8 | 4, 7 | eqtrd 2814 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (ℂ × {0})) |
9 | cnex 10353 | . . . 4 ⊢ ℂ ∈ V | |
10 | 9 | a1i 11 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → ℂ ∈ V) |
11 | simp1 1127 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 ∈ (Poly‘𝑆)) | |
12 | plyf 24391 | . . . 4 ⊢ (𝐹 ∈ (Poly‘𝑆) → 𝐹:ℂ⟶ℂ) | |
13 | 11, 12 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹:ℂ⟶ℂ) |
14 | 1 | plyremlem 24496 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴})) |
15 | 14 | 3ad2ant2 1125 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐺 ∈ (Poly‘ℂ) ∧ (deg‘𝐺) = 1 ∧ (◡𝐺 “ {0}) = {𝐴})) |
16 | 15 | simp1d 1133 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐺 ∈ (Poly‘ℂ)) |
17 | plyssc 24393 | . . . . . . 7 ⊢ (Poly‘𝑆) ⊆ (Poly‘ℂ) | |
18 | 17, 11 | sseldi 3819 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 ∈ (Poly‘ℂ)) |
19 | 15 | simp2d 1134 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (deg‘𝐺) = 1) |
20 | ax-1ne0 10341 | . . . . . . . . 9 ⊢ 1 ≠ 0 | |
21 | 20 | a1i 11 | . . . . . . . 8 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 1 ≠ 0) |
22 | 19, 21 | eqnetrd 3036 | . . . . . . 7 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (deg‘𝐺) ≠ 0) |
23 | fveq2 6446 | . . . . . . . . 9 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = (deg‘0𝑝)) | |
24 | dgr0 24455 | . . . . . . . . 9 ⊢ (deg‘0𝑝) = 0 | |
25 | 23, 24 | syl6eq 2830 | . . . . . . . 8 ⊢ (𝐺 = 0𝑝 → (deg‘𝐺) = 0) |
26 | 25 | necon3i 3001 | . . . . . . 7 ⊢ ((deg‘𝐺) ≠ 0 → 𝐺 ≠ 0𝑝) |
27 | 22, 26 | syl 17 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐺 ≠ 0𝑝) |
28 | quotcl2 24494 | . . . . . 6 ⊢ ((𝐹 ∈ (Poly‘ℂ) ∧ 𝐺 ∈ (Poly‘ℂ) ∧ 𝐺 ≠ 0𝑝) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) | |
29 | 18, 16, 27, 28 | syl3anc 1439 | . . . . 5 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) |
30 | plymulcl 24414 | . . . . 5 ⊢ ((𝐺 ∈ (Poly‘ℂ) ∧ (𝐹 quot 𝐺) ∈ (Poly‘ℂ)) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) | |
31 | 16, 29, 30 | syl2anc 579 | . . . 4 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ)) |
32 | plyf 24391 | . . . 4 ⊢ ((𝐺 ∘𝑓 · (𝐹 quot 𝐺)) ∈ (Poly‘ℂ) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)):ℂ⟶ℂ) | |
33 | 31, 32 | syl 17 | . . 3 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → (𝐺 ∘𝑓 · (𝐹 quot 𝐺)):ℂ⟶ℂ) |
34 | ofsubeq0 11371 | . . 3 ⊢ ((ℂ ∈ V ∧ 𝐹:ℂ⟶ℂ ∧ (𝐺 ∘𝑓 · (𝐹 quot 𝐺)):ℂ⟶ℂ) → ((𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (ℂ × {0}) ↔ 𝐹 = (𝐺 ∘𝑓 · (𝐹 quot 𝐺)))) | |
35 | 10, 13, 33, 34 | syl3anc 1439 | . 2 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → ((𝐹 ∘𝑓 − (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) = (ℂ × {0}) ↔ 𝐹 = (𝐺 ∘𝑓 · (𝐹 quot 𝐺)))) |
36 | 8, 35 | mpbid 224 | 1 ⊢ ((𝐹 ∈ (Poly‘𝑆) ∧ 𝐴 ∈ ℂ ∧ (𝐹‘𝐴) = 0) → 𝐹 = (𝐺 ∘𝑓 · (𝐹 quot 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ≠ wne 2969 Vcvv 3398 {csn 4398 × cxp 5353 ◡ccnv 5354 “ cima 5358 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ∘𝑓 cof 7172 ℂcc 10270 0cc0 10272 1c1 10273 · cmul 10277 − cmin 10606 0𝑝c0p 23873 Polycply 24377 Xpcidp 24378 degcdgr 24380 quot cquot 24482 |
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 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 |
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 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-of 7174 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-pm 8143 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-inf 8637 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-rlim 14628 df-sum 14825 df-0p 23874 df-ply 24381 df-idp 24382 df-coe 24383 df-dgr 24384 df-quot 24483 |
This theorem is referenced by: fta1lem 24499 vieta1lem1 24502 vieta1lem2 24503 |
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