taylth - Metamath Proof Explorer

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Theorem taylth 24965

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 10631	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		taylth.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4		ax-resscn 10596	. . 3 ⊢ ℝ ⊆ ℂ
5		fss 6529	. . 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 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 483	. . 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 483	. . 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 483	. . 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 483	. . 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 483	. . 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 769	. . 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 771	. . . 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 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥))
21		fveq2 6672	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥))
22	20, 21	oveq12d 7176	. . . . . . 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 7165	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
24	23	oveq1d 7173	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑚))
25	22, 24	oveq12d 7176	. . . . . 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 5171	. . . . 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 7168	. . . 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 2925	. . 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 24964	. 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 24963	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ⊆ wss 3938 {csn 4569 {cpr 4571 ↦ cmpt 5148 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 1c1 10540 − cmin 10872 / cdiv 11299 ℕcn 11640 ..^cfzo 13036 ↑cexp 13432 lim_ℂ climc 24462 D^𝑛 cdvn 24464 Tayl ctayl 24943

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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-supp 7833 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-2o 8105 df-oadd 8108 df-er 8291 df-map 8410 df-pm 8411 df-ixp 8464 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-fsupp 8836 df-fi 8877 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-q 12352 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-ioo 12745 df-ioc 12746 df-ico 12747 df-icc 12748 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-rlim 14848 df-sum 15045 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-hom 16591 df-cco 16592 df-rest 16698 df-topn 16699 df-0g 16717 df-gsum 16718 df-topgen 16719 df-pt 16720 df-prds 16723 df-xrs 16777 df-qtop 16782 df-imas 16783 df-xps 16785 df-mre 16859 df-mrc 16860 df-acs 16862 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-submnd 17959 df-grp 18108 df-minusg 18109 df-mulg 18227 df-subg 18278 df-cntz 18449 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-dvr 19435 df-drng 19506 df-subrg 19535 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 df-mopn 20543 df-fbas 20544 df-fg 20545 df-cnfld 20548 df-refld 20751 df-top 21504 df-topon 21521 df-topsp 21543 df-bases 21556 df-cld 21629 df-ntr 21630 df-cls 21631 df-nei 21708 df-lp 21746 df-perf 21747 df-cn 21837 df-cnp 21838 df-haus 21925 df-cmp 21997 df-tx 22172 df-hmeo 22365 df-fil 22456 df-fm 22548 df-flim 22549 df-flf 22550 df-tsms 22737 df-xms 22932 df-ms 22933 df-tms 22934 df-cncf 23488 df-0p 24273 df-limc 24466 df-dv 24467 df-dvn 24468 df-ply 24780 df-idp 24781 df-coe 24782 df-dgr 24783 df-tayl 24945

This theorem is referenced by: (None)

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