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Mirrors > Home > MPE Home > Th. List > taylpf | Structured version Visualization version GIF version |
Description: The Taylor polynomial is a function on the complex numbers (even if the base set of the original function is the reals). (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 |
---|---|
taylpf | ⊢ (𝜑 → 𝑇:ℂ⟶ℂ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fzfid 13095 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0...𝑁) ∈ Fin) | |
2 | taylpfval.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
3 | taylpfval.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
4 | taylpfval.a | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
5 | taylpfval.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
6 | taylpfval.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
7 | 2, 3, 4, 5, 6 | taylplem2 24559 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ (0...𝑁)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ) |
8 | 1, 7 | fsumcl 14875 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ) |
9 | 8 | fmpttd 6651 | . 2 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):ℂ⟶ℂ) |
10 | taylpfval.t | . . . 4 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | |
11 | 2, 3, 4, 5, 6, 10 | taylpfval 24560 | . . 3 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) |
12 | 11 | feq1d 6278 | . 2 ⊢ (𝜑 → (𝑇:ℂ⟶ℂ ↔ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):ℂ⟶ℂ)) |
13 | 9, 12 | mpbird 249 | 1 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ⊆ wss 3792 {cpr 4400 ↦ cmpt 4967 dom cdm 5357 ⟶wf 6133 ‘cfv 6137 (class class class)co 6924 ℂcc 10272 ℝcr 10273 0cc0 10274 · cmul 10279 − cmin 10608 / cdiv 11034 ℕ0cn0 11646 ...cfz 12647 ↑cexp 13182 !cfa 13382 Σcsu 14828 D𝑛 cdvn 24069 Tayl ctayl 24548 |
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 ax-addf 10353 ax-mulf 10354 |
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-iin 4758 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-om 7346 df-1st 7447 df-2nd 7448 df-supp 7579 df-wrecs 7691 df-recs 7753 df-rdg 7791 df-1o 7845 df-oadd 7849 df-er 8028 df-map 8144 df-pm 8145 df-en 8244 df-dom 8245 df-sdom 8246 df-fin 8247 df-fsupp 8566 df-fi 8607 df-sup 8638 df-inf 8639 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 11035 df-nn 11379 df-2 11442 df-3 11443 df-4 11444 df-5 11445 df-6 11446 df-7 11447 df-8 11448 df-9 11449 df-n0 11647 df-z 11733 df-dec 11850 df-uz 11997 df-q 12100 df-rp 12142 df-xneg 12261 df-xadd 12262 df-xmul 12263 df-icc 12498 df-fz 12648 df-fzo 12789 df-seq 13124 df-exp 13183 df-fac 13383 df-hash 13440 df-cj 14250 df-re 14251 df-im 14252 df-sqrt 14386 df-abs 14387 df-clim 14631 df-sum 14829 df-struct 16261 df-ndx 16262 df-slot 16263 df-base 16265 df-sets 16266 df-plusg 16355 df-mulr 16356 df-starv 16357 df-tset 16361 df-ple 16362 df-ds 16364 df-unif 16365 df-rest 16473 df-topn 16474 df-0g 16492 df-gsum 16493 df-topgen 16494 df-mgm 17632 df-sgrp 17674 df-mnd 17685 df-grp 17816 df-minusg 17817 df-cntz 18137 df-cmn 18585 df-abl 18586 df-mgp 18881 df-ur 18893 df-ring 18940 df-cring 18941 df-psmet 20138 df-xmet 20139 df-met 20140 df-bl 20141 df-mopn 20142 df-fbas 20143 df-fg 20144 df-cnfld 20147 df-top 21110 df-topon 21127 df-topsp 21149 df-bases 21162 df-cld 21235 df-ntr 21236 df-cls 21237 df-nei 21314 df-lp 21352 df-perf 21353 df-cnp 21444 df-haus 21531 df-fil 22062 df-fm 22154 df-flim 22155 df-flf 22156 df-tsms 22342 df-xms 22537 df-ms 22538 df-limc 24071 df-dv 24072 df-dvn 24073 df-tayl 24550 |
This theorem is referenced by: dvntaylp 24566 taylthlem1 24568 |
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