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Mirrors > Home > MPE Home > Th. List > taylth | Structured version Visualization version GIF version |
Description: Taylor's theorem. The Taylor polynomial of a 𝑁-times differentiable function is such that the error term goes to zero faster than (𝑥 − 𝐵)↑𝑁. This is Metamath 100 proof #35. (Contributed by Mario Carneiro, 1-Jan-2017.) |
Ref | Expression |
---|---|
taylth.f | ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) |
taylth.a | ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
taylth.d | ⊢ (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) |
taylth.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
taylth.b | ⊢ (𝜑 → 𝐵 ∈ 𝐴) |
taylth.t | ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵) |
taylth.r | ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) |
Ref | Expression |
---|---|
taylth | ⊢ (𝜑 → 0 ∈ (𝑅 limℂ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reelprrecn 10623 | . . 3 ⊢ ℝ ∈ {ℝ, ℂ} | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ}) |
3 | taylth.f | . . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ) | |
4 | ax-resscn 10588 | . . 3 ⊢ ℝ ⊆ ℂ | |
5 | fss 6521 | . . 3 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ) | |
6 | 3, 4, 5 | sylancl 588 | . 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ) |
7 | taylth.a | . 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ) | |
8 | taylth.d | . 2 ⊢ (𝜑 → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) | |
9 | taylth.n | . 2 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
10 | taylth.b | . 2 ⊢ (𝜑 → 𝐵 ∈ 𝐴) | |
11 | taylth.t | . 2 ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵) | |
12 | taylth.r | . 2 ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) | |
13 | 3 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 𝐹:𝐴⟶ℝ) |
14 | 7 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 𝐴 ⊆ ℝ) |
15 | 8 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → dom ((ℝ D𝑛 𝐹)‘𝑁) = 𝐴) |
16 | 9 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 𝑁 ∈ ℕ) |
17 | 10 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 𝐵 ∈ 𝐴) |
18 | simprl 769 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 𝑚 ∈ (1..^𝑁)) | |
19 | simprr 771 | . . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵)) | |
20 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) = (((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥)) | |
21 | fveq2 6664 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦) = (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) | |
22 | 20, 21 | oveq12d 7168 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) = ((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥))) |
23 | oveq1 7157 | . . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵)) | |
24 | 23 | oveq1d 7165 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑚)) |
25 | 22, 24 | oveq12d 7168 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚)) = (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) |
26 | 25 | cbvmptv 5161 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) |
27 | 26 | oveq1i 7160 | . . . 4 ⊢ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) limℂ 𝐵) |
28 | 19, 27 | eleqtrdi 2923 | . . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) limℂ 𝐵)) |
29 | 13, 14, 15, 16, 17, 11, 18, 28 | taylthlem2 24956 | . 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) limℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D𝑛 𝐹)‘(𝑁 − (𝑚 + 1)))‘𝑥) − (((ℂ D𝑛 𝑇)‘(𝑁 − (𝑚 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑚 + 1)))) limℂ 𝐵)) |
30 | 2, 6, 7, 8, 9, 10, 11, 12, 29 | taylthlem1 24955 | 1 ⊢ (𝜑 → 0 ∈ (𝑅 limℂ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 {cpr 4562 ↦ cmpt 5138 dom cdm 5549 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 − cmin 10864 / cdiv 11291 ℕcn 11632 ..^cfzo 13027 ↑cexp 13423 limℂ climc 24454 D𝑛 cdvn 24456 Tayl ctayl 24935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-tpos 7886 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-xnn0 11962 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-rlim 14840 df-sum 15037 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-submnd 17951 df-grp 18100 df-minusg 18101 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-oppr 19367 df-dvdsr 19385 df-unit 19386 df-invr 19416 df-dvr 19427 df-drng 19498 df-subrg 19527 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-refld 20743 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-cmp 21989 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-tsms 22729 df-xms 22924 df-ms 22925 df-tms 22926 df-cncf 23480 df-0p 24265 df-limc 24458 df-dv 24459 df-dvn 24460 df-ply 24772 df-idp 24773 df-coe 24774 df-dgr 24775 df-tayl 24937 |
This theorem is referenced by: (None) |
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