taylth - Metamath Proof Explorer

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Theorem taylth 26702

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 11292	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		taylth.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4		ax-resscn 11257	. . 3 ⊢ ℝ ⊆ ℂ
5		fss 6726	. . 3 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
6	3, 4, 5	sylancl 598	. 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 486	. . 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 486	. . 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 486	. . 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 486	. . 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 486	. . 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 783	. . 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 785	. . . 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 6885	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥))
21		fveq2 6885	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥))
22	20, 21	oveq12d 7438	. . . . . . 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 7427	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
24	23	oveq1d 7435	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑚))
25	22, 24	oveq12d 7438	. . . . . 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 5209	. . . . 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 7430	. . . 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 2871	. . 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 26701	. 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 26700	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ∖ cdif 3896 ⊆ wss 3899 {csn 4584 {cpr 4586 ↦ cmpt 5186 dom cdm 5651 ⟶wf 6534 ‘cfv 6538 (class class class)co 7420 ℂcc 11198 ℝcr 11199 0cc0 11200 1c1 11201 − cmin 11541 / cdiv 11973 ℕcn 12335 ..^cfzo 13788 ↑cexp 14204 lim_ℂ climc 26182 D^𝑛 cdvn 26184 Tayl ctayl 26680

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278 ax-addf 11279

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-of 7693 df-om 7878 df-1st 8001 df-2nd 8002 df-supp 8178 df-tpos 8243 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-2o 8477 df-er 8717 df-map 8849 df-pm 8850 df-ixp 8926 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-fsupp 9354 df-fi 9403 df-sup 9434 df-inf 9435 df-oi 9504 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-4 12407 df-5 12408 df-6 12409 df-7 12410 df-8 12411 df-9 12412 df-n0 12607 df-xnn0 12680 df-z 12694 df-dec 12815 df-uz 12966 df-q 13076 df-rp 13121 df-xneg 13241 df-xadd 13242 df-xmul 13243 df-ioo 13480 df-ioc 13481 df-ico 13482 df-icc 13483 df-fz 13640 df-fzo 13789 df-fl 13932 df-seq 14145 df-exp 14205 df-fac 14418 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-rlim 15656 df-sum 15854 df-struct 17325 df-sets 17342 df-slot 17360 df-ndx 17372 df-base 17388 df-ress 17409 df-plusg 17441 df-mulr 17442 df-starv 17443 df-sca 17444 df-vsca 17445 df-ip 17446 df-tset 17447 df-ple 17448 df-ds 17450 df-unif 17451 df-hom 17452 df-cco 17453 df-rest 17593 df-topn 17594 df-0g 17612 df-gsum 17613 df-topgen 17614 df-pt 17615 df-prds 17618 df-xrs 17674 df-qtop 17679 df-imas 17680 df-xps 17682 df-mre 17756 df-mrc 17757 df-acs 17759 df-mgm 18816 df-sgrp 18908 df-mnd 18924 df-submnd 18979 df-grp 19147 df-minusg 19148 df-mulg 19278 df-subg 19333 df-cntz 19531 df-cmn 19996 df-abl 19997 df-mgp 20361 df-rng 20375 df-ur 20408 df-ring 20461 df-cring 20462 df-oppr 20567 df-dvdsr 20587 df-unit 20588 df-invr 20618 df-dvr 20631 df-subrng 20798 df-subrg 20822 df-drng 20982 df-psmet 21670 df-xmet 21671 df-met 21672 df-bl 21673 df-mopn 21674 df-fbas 21675 df-fg 21676 df-cnfld 21679 df-refld 21911 df-top 23212 df-topon 23229 df-topsp 23251 df-bases 23264 df-cld 23337 df-ntr 23338 df-cls 23339 df-nei 23416 df-lp 23454 df-perf 23455 df-cn 23545 df-cnp 23546 df-haus 23633 df-cmp 23705 df-tx 23881 df-hmeo 24074 df-fil 24165 df-fm 24257 df-flim 24258 df-flf 24259 df-tsms 24446 df-xms 24639 df-ms 24640 df-tms 24641 df-cncf 25199 df-0p 25991 df-limc 26186 df-dv 26187 df-dvn 26188 df-ply 26506 df-idp 26507 df-coe 26508 df-dgr 26509 df-tayl 26682

This theorem is used by: (None)

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