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Mirrors > Home > MPE Home > Th. List > taylply | Structured version Visualization version GIF version |
Description: The Taylor polynomial is a polynomial of degree (at most) 𝑁. (Contributed by Mario Carneiro, 31-Dec-2016.) |
Ref | Expression |
---|---|
taylpfval.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
taylpfval.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
taylpfval.a | ⊢ (𝜑 → 𝐴 ⊆ 𝑆) |
taylpfval.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
taylpfval.b | ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
taylpfval.t | ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) |
Ref | Expression |
---|---|
taylply | ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | taylpfval.s | . 2 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | taylpfval.f | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | taylpfval.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | taylpfval.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | taylpfval.b | . 2 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
6 | taylpfval.t | . 2 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | |
7 | cnring 20164 | . . 3 ⊢ ℂfld ∈ Ring | |
8 | cnfldbas 20146 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
9 | 8 | subrgid 19174 | . . 3 ⊢ (ℂfld ∈ Ring → ℂ ∈ (SubRing‘ℂfld)) |
10 | 7, 9 | mp1i 13 | . 2 ⊢ (𝜑 → ℂ ∈ (SubRing‘ℂfld)) |
11 | cnex 10353 | . . . . . . . 8 ⊢ ℂ ∈ V | |
12 | 11 | a1i 11 | . . . . . . 7 ⊢ (𝜑 → ℂ ∈ V) |
13 | elpm2r 8158 | . . . . . . 7 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) | |
14 | 12, 1, 2, 3, 13 | syl22anc 829 | . . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
15 | dvnbss 24128 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑁 ∈ ℕ0) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) | |
16 | 1, 14, 4, 15 | syl3anc 1439 | . . . . 5 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom 𝐹) |
17 | 2, 16 | fssdmd 6306 | . . . 4 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ 𝐴) |
18 | recnprss 24105 | . . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ) | |
19 | 1, 18 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ) |
20 | 3, 19 | sstrd 3831 | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ) |
21 | 17, 20 | sstrd 3831 | . . 3 ⊢ (𝜑 → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ ℂ) |
22 | 21, 5 | sseldd 3822 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℂ) |
23 | 1 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ}) |
24 | 14 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑pm 𝑆)) |
25 | elfznn0 12751 | . . . . . 6 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ0) | |
26 | 25 | adantl 475 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ0) |
27 | dvnf 24127 | . . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ ℕ0) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) | |
28 | 23, 24, 26, 27 | syl3anc 1439 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D𝑛 𝐹)‘𝑘):dom ((𝑆 D𝑛 𝐹)‘𝑘)⟶ℂ) |
29 | simpr 479 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁)) | |
30 | dvn2bss 24130 | . . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm 𝑆) ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) | |
31 | 23, 24, 29, 30 | syl3anc 1439 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
32 | 5 | adantr 474 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) |
33 | 31, 32 | sseldd 3822 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
34 | 28, 33 | ffvelrnd 6624 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ) |
35 | faccl 13388 | . . . . 5 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
36 | 26, 35 | syl 17 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ) |
37 | 36 | nncnd 11392 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ) |
38 | 36 | nnne0d 11425 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0) |
39 | 34, 37, 38 | divcld 11151 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ) |
40 | 1, 2, 3, 4, 5, 6, 10, 22, 39 | taylply2 24559 | 1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℂ) ∧ (deg‘𝑇) ≤ 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 Vcvv 3398 ⊆ wss 3792 {cpr 4400 class class class wbr 4886 dom cdm 5355 ⟶wf 6131 ‘cfv 6135 (class class class)co 6922 ↑pm cpm 8141 ℂcc 10270 ℝcr 10271 0cc0 10272 ≤ cle 10412 ℕcn 11374 ℕ0cn0 11642 ...cfz 12643 !cfa 13378 Ringcrg 18934 SubRingcsubrg 19168 ℂfldccnfld 20142 D𝑛 cdvn 24065 Polycply 24377 degcdgr 24380 Tayl ctayl 24544 |
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 ax-mulf 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 4672 df-int 4711 df-iun 4755 df-iin 4756 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-supp 7577 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-fsupp 8564 df-fi 8605 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-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-q 12096 df-rp 12138 df-xneg 12257 df-xadd 12258 df-xmul 12259 df-icc 12494 df-fz 12644 df-fzo 12785 df-fl 12912 df-seq 13120 df-exp 13179 df-fac 13379 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-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-rest 16469 df-topn 16470 df-0g 16488 df-gsum 16489 df-topgen 16490 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-subg 17975 df-cntz 18133 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-subrg 19170 df-psmet 20134 df-xmet 20135 df-met 20136 df-bl 20137 df-mopn 20138 df-fbas 20139 df-fg 20140 df-cnfld 20143 df-top 21106 df-topon 21123 df-topsp 21145 df-bases 21158 df-cld 21231 df-ntr 21232 df-cls 21233 df-nei 21310 df-lp 21348 df-perf 21349 df-cnp 21440 df-haus 21527 df-fil 22058 df-fm 22150 df-flim 22151 df-flf 22152 df-tsms 22338 df-xms 22533 df-ms 22534 df-0p 23874 df-limc 24067 df-dv 24068 df-dvn 24069 df-ply 24381 df-idp 24382 df-coe 24383 df-dgr 24384 df-tayl 24546 |
This theorem is referenced by: (None) |
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