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Mirrors > Home > MPE Home > Th. List > taylpfval | Structured version Visualization version GIF version |
Description: Define the Taylor polynomial of a function. The constant Tayl is a function of five arguments: 𝑆 is the base set with respect to evaluate the derivatives (generally ℝ or ℂ), 𝐹 is the function we are approximating, at point 𝐵, to order 𝑁. The result is a polynomial function of 𝑥. (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 |
---|---|
taylpfval | ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | taylpfval.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | taylpfval.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) | |
3 | taylpfval.a | . . . 4 ⊢ (𝜑 → 𝐴 ⊆ 𝑆) | |
4 | taylpfval.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
5 | 4 | orcd 870 | . . . 4 ⊢ (𝜑 → (𝑁 ∈ ℕ0 ∨ 𝑁 = +∞)) |
6 | taylpfval.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑁)) | |
7 | 1, 2, 3, 4, 6 | taylplem1 24958 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D𝑛 𝐹)‘𝑘)) |
8 | taylpfval.t | . . . 4 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵) | |
9 | 1, 2, 3, 5, 7, 8 | taylfval 24954 | . . 3 ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))))) |
10 | cnfldbas 20095 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
11 | cnfld0 20115 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
12 | cnring 20113 | . . . . . . . 8 ⊢ ℂfld ∈ Ring | |
13 | ringcmn 19327 | . . . . . . . 8 ⊢ (ℂfld ∈ Ring → ℂfld ∈ CMnd) | |
14 | 12, 13 | mp1i 13 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂfld ∈ CMnd) |
15 | cnfldtps 23383 | . . . . . . . 8 ⊢ ℂfld ∈ TopSp | |
16 | 15 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ℂfld ∈ TopSp) |
17 | ovex 7168 | . . . . . . . . 9 ⊢ (0[,]𝑁) ∈ V | |
18 | 17 | inex1 5185 | . . . . . . . 8 ⊢ ((0[,]𝑁) ∩ ℤ) ∈ V |
19 | 18 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ V) |
20 | 1, 2, 3, 5, 7 | taylfvallem1 24952 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ ℂ) |
21 | 20 | fmpttd 6856 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))):((0[,]𝑁) ∩ ℤ)⟶ℂ) |
22 | eqid 2798 | . . . . . . . 8 ⊢ (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) = (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) | |
23 | 0z 11980 | . . . . . . . . . . 11 ⊢ 0 ∈ ℤ | |
24 | 4 | nn0zd 12073 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℤ) |
25 | fzval2 12888 | . . . . . . . . . . 11 ⊢ ((0 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (0...𝑁) = ((0[,]𝑁) ∩ ℤ)) | |
26 | 23, 24, 25 | sylancr 590 | . . . . . . . . . 10 ⊢ (𝜑 → (0...𝑁) = ((0[,]𝑁) ∩ ℤ)) |
27 | 26 | adantr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0...𝑁) = ((0[,]𝑁) ∩ ℤ)) |
28 | fzfid 13336 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (0...𝑁) ∈ Fin) | |
29 | 27, 28 | eqeltrrd 2891 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((0[,]𝑁) ∩ ℤ) ∈ Fin) |
30 | ovexd 7170 | . . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℂ) ∧ 𝑘 ∈ ((0[,]𝑁) ∩ ℤ)) → (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) ∈ V) | |
31 | c0ex 10624 | . . . . . . . . 9 ⊢ 0 ∈ V | |
32 | 31 | a1i 11 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 0 ∈ V) |
33 | 22, 29, 30, 32 | fsuppmptdm 8828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) finSupp 0) |
34 | eqid 2798 | . . . . . . 7 ⊢ (TopOpen‘ℂfld) = (TopOpen‘ℂfld) | |
35 | 34 | cnfldhaus 23390 | . . . . . . . 8 ⊢ (TopOpen‘ℂfld) ∈ Haus |
36 | 35 | a1i 11 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (TopOpen‘ℂfld) ∈ Haus) |
37 | 10, 11, 14, 16, 19, 21, 33, 34, 36 | haustsmsid 22746 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = {(ℂfld Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))}) |
38 | 29, 20 | gsumfsum 20158 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = Σ𝑘 ∈ ((0[,]𝑁) ∩ ℤ)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) |
39 | 27 | sumeq1d 15050 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)) = Σ𝑘 ∈ ((0[,]𝑁) ∩ ℤ)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) |
40 | 38, 39 | eqtr4d 2836 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂfld Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) |
41 | 40 | sneqd 4537 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → {(ℂfld Σg (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))} = {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))}) |
42 | 37, 41 | eqtrd 2833 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) = {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))}) |
43 | 42 | xpeq2d 5549 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) = ({𝑥} × {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))})) |
44 | 43 | iuneq2dv 4905 | . . 3 ⊢ (𝜑 → ∪ 𝑥 ∈ ℂ ({𝑥} × (ℂfld tsums (𝑘 ∈ ((0[,]𝑁) ∩ ℤ) ↦ (((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))) = ∪ 𝑥 ∈ ℂ ({𝑥} × {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))})) |
45 | 9, 44 | eqtrd 2833 | . 2 ⊢ (𝜑 → 𝑇 = ∪ 𝑥 ∈ ℂ ({𝑥} × {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))})) |
46 | dfmpt3 6454 | . 2 ⊢ (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))) = ∪ 𝑥 ∈ ℂ ({𝑥} × {Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))}) | |
47 | 45, 46 | eqtr4di 2851 | 1 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∩ cin 3880 ⊆ wss 3881 {csn 4525 {cpr 4527 ∪ ciun 4881 ↦ cmpt 5110 × cxp 5517 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 Fincfn 8492 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 +∞cpnf 10661 − cmin 10859 / cdiv 11286 ℕ0cn0 11885 ℤcz 11969 [,]cicc 12729 ...cfz 12885 ↑cexp 13425 !cfa 13629 Σcsu 15034 TopOpenctopn 16687 Σg cgsu 16706 CMndccmn 18898 Ringcrg 19290 ℂfldccnfld 20091 TopSpctps 21537 Hauscha 21913 tsums ctsu 22731 D𝑛 cdvn 24467 Tayl ctayl 24948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cnp 21833 df-haus 21920 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tsms 22732 df-xms 22927 df-ms 22928 df-limc 24469 df-dv 24470 df-dvn 24471 df-tayl 24950 |
This theorem is referenced by: taylpf 24961 taylpval 24962 taylply2 24963 dvtaylp 24965 |
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