taylth - Metamath Proof Explorer

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Theorem taylth 26340

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.)

**Hypotheses**
Ref	Expression
taylth.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
taylth.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
taylth.d	⊢ (𝜑 → dom ((ℝ D^𝑛 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)))

**Assertion**
Ref	Expression
taylth	⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝑇 𝑥,𝑁 𝜑,𝑥 Allowed substitution hint: 𝑅(𝑥)

Proof of Theorem taylth Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		reelprrecn 11118	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		taylth.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4		ax-resscn 11083	. . 3 ⊢ ℝ ⊆ ℂ
5		fss 6678	. . 3 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
6	3, 4, 5	sylancl 586	. 2 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
7		taylth.a	. 2 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
8		taylth.d	. 2 ⊢ (𝜑 → dom ((ℝ D^𝑛 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 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 𝐹:𝐴⟶ℝ)
14	7	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 𝐴 ⊆ ℝ)
15	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → dom ((ℝ D^𝑛 𝐹)‘𝑁) = 𝐴)
16	9	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 𝑁 ∈ ℕ)
17	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 𝐵 ∈ 𝐴)
18		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 𝑚 ∈ (1..^𝑁))
19		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))
20		fveq2 6834	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥))
21		fveq2 6834	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥))
22	20, 21	oveq12d 7376	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)))
23		oveq1 7365	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
24	23	oveq1d 7373	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑚))
25	22, 24	oveq12d 7376	. . . . . 6 ⊢ (𝑦 = 𝑥 → (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚)) = (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)))
26	25	cbvmptv 5202	. . . . 5 ⊢ (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)))
27	26	oveq1i 7368	. . . 4 ⊢ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵)
28	19, 27	eleqtrdi 2846	. . 3 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵))
29	13, 14, 15, 16, 17, 11, 18, 28	taylthlem2 26338	. 2 ⊢ ((𝜑 ∧ (𝑚 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦)) / ((𝑦 − 𝐵)↑𝑚))) lim_ℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑚 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑚 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑚 + 1)))) lim_ℂ 𝐵))
30	2, 6, 7, 8, 9, 10, 11, 12, 29	taylthlem1 26337	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∖ cdif 3898 ⊆ wss 3901 {csn 4580 {cpr 4582 ↦ cmpt 5179 dom cdm 5624 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11024 ℝcr 11025 0cc0 11026 1c1 11027 − cmin 11364 / cdiv 11794 ℕcn 12145 ..^cfzo 13570 ↑cexp 13984 lim_ℂ climc 25819 D^𝑛 cdvn 25821 Tayl ctayl 26316

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-tpos 8168 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-xnn0 12475 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-seq 13925 df-exp 13985 df-fac 14197 df-hash 14254 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-clim 15411 df-rlim 15412 df-sum 15610 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-submnd 18709 df-grp 18866 df-minusg 18867 df-mulg 18998 df-subg 19053 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-oppr 20273 df-dvdsr 20293 df-unit 20294 df-invr 20324 df-dvr 20337 df-subrng 20479 df-subrg 20503 df-drng 20664 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-refld 21560 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-cmp 23331 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-tsms 24071 df-xms 24264 df-ms 24265 df-tms 24266 df-cncf 24827 df-0p 25627 df-limc 25823 df-dv 25824 df-dvn 25825 df-ply 26149 df-idp 26150 df-coe 26151 df-dgr 26152 df-tayl 26318

This theorem is referenced by: (None)

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