taylth - Metamath Proof Explorer

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Theorem taylth 26759

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 11331	. . 3 ⊢ ℝ ∈ {ℝ, ℂ}
2	1	a1i 11	. 2 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
3		taylth.f	. . 3 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
4		ax-resscn 11296	. . 3 ⊢ ℝ ⊆ ℂ
5		fss 6756	. . 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 6916	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥))
21		fveq2 6916	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥))
22	20, 21	oveq12d 7469	. . . . . . 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 7458	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
24	23	oveq1d 7466	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑚))
25	22, 24	oveq12d 7469	. . . . . 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 5215	. . . . 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 7461	. . . 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 2874	. . 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 26758	. 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 26757	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 2147 ⊆ wss 3900 ∖ cdif 4048 {csn 4588 {cpr 4590 ↦ cmpt 5192 dom cdm 5658 ⟶wf 6562 ‘cfv 6566 (class class class)co 7451 ℂcc 11237 ℝcr 11238 0cc0 11239 1c1 11240 − cmin 11580 / cdiv 12013 ℕcn 12375 ..^cfzo 13828 ↑cexp 14244 lim_ℂ climc 26237 D^𝑛 cdvn 26239 Tayl ctayl 26737

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 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5238 ax-sep 5255 ax-nul 5266 ax-pow 5334 ax-pr 5398 ax-un 7783 ax-inf2 9679 ax-cnex 11295 ax-resscn 11296 ax-1cn 11297 ax-icn 11298 ax-addcl 11299 ax-addrcl 11300 ax-mulcl 11301 ax-mulrcl 11302 ax-mulcom 11303 ax-addass 11304 ax-mulass 11305 ax-distr 11306 ax-i2m1 11307 ax-1ne0 11308 ax-1rid 11309 ax-rnegex 11310 ax-rrecex 11311 ax-cnre 11312 ax-pre-lttri 11313 ax-pre-lttrn 11314 ax-pre-ltadd 11315 ax-pre-mulgt0 11316 ax-pre-sup 11317 ax-addf 11318

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 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4841 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5109 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5553 df-eprel 5558 df-po 5566 df-so 5567 df-fr 5611 df-se 5612 df-we 5613 df-xp 5664 df-rel 5665 df-cnv 5666 df-co 5667 df-dm 5668 df-rn 5669 df-res 5670 df-ima 5671 df-pred 6330 df-ord 6391 df-on 6392 df-lim 6393 df-suc 6394 df-iota 6520 df-fun 6568 df-fn 6569 df-f 6570 df-f1 6571 df-fo 6572 df-f1o 6573 df-fv 6574 df-isom 6575 df-riota 7408 df-ov 7454 df-oprab 7455 df-mpo 7456 df-of 7725 df-om 7912 df-1st 8035 df-2nd 8036 df-supp 8215 df-tpos 8281 df-frecs 8337 df-wrecs 8368 df-recs 8417 df-rdg 8456 df-1o 8514 df-2o 8515 df-er 8755 df-map 8887 df-pm 8888 df-ixp 8964 df-en 9012 df-dom 9013 df-sdom 9014 df-fin 9015 df-fsupp 9392 df-fi 9440 df-sup 9471 df-inf 9472 df-oi 9541 df-card 10057 df-pnf 11384 df-mnf 11385 df-xr 11386 df-ltxr 11387 df-le 11388 df-sub 11582 df-neg 11583 df-div 12014 df-nn 12376 df-2 12445 df-3 12446 df-4 12447 df-5 12448 df-6 12449 df-7 12450 df-8 12451 df-9 12452 df-n0 12647 df-xnn0 12720 df-z 12734 df-dec 12855 df-uz 13006 df-q 13116 df-rp 13161 df-xneg 13281 df-xadd 13282 df-xmul 13283 df-ioo 13520 df-ioc 13521 df-ico 13522 df-icc 13523 df-fz 13680 df-fzo 13829 df-fl 13972 df-seq 14185 df-exp 14245 df-fac 14459 df-hash 14516 df-cj 15309 df-re 15310 df-im 15311 df-sqrt 15445 df-abs 15446 df-clim 15699 df-rlim 15700 df-sum 15898 df-struct 17369 df-sets 17386 df-slot 17404 df-ndx 17416 df-base 17432 df-ress 17453 df-plusg 17486 df-mulr 17487 df-starv 17488 df-sca 17489 df-vsca 17490 df-ip 17491 df-tset 17492 df-ple 17493 df-ds 17495 df-unif 17496 df-hom 17497 df-cco 17498 df-euf 17499 df-rest 17643 df-topn 17644 df-0g 17662 df-gsum 17663 df-topgen 17664 df-pt 17665 df-prds 17668 df-xrs 17724 df-qtop 17729 df-imas 17730 df-xps 17732 df-mre 17806 df-mrc 17807 df-acs 17809 df-mgm 18866 df-sgrp 18958 df-mnd 18974 df-submnd 19029 df-grp 19197 df-minusg 19198 df-mulg 19328 df-subg 19383 df-cntz 19582 df-cmn 20047 df-abl 20048 df-mgp 20412 df-rng 20426 df-ur 20459 df-ring 20512 df-cring 20513 df-oppr 20618 df-dvdsr 20638 df-unit 20639 df-invr 20669 df-dvr 20682 df-subrng 20849 df-subrg 20873 df-drng 21033 df-psmet 21721 df-xmet 21722 df-met 21723 df-bl 21724 df-mopn 21725 df-fbas 21726 df-fg 21727 df-cnfld 21730 df-refld 21966 df-top 23267 df-topon 23284 df-topsp 23306 df-bases 23319 df-cld 23392 df-ntr 23393 df-cls 23394 df-nei 23471 df-lp 23509 df-perf 23510 df-cn 23600 df-cnp 23601 df-haus 23688 df-cmp 23760 df-tx 23936 df-hmeo 24129 df-fil 24220 df-fm 24312 df-flim 24313 df-flf 24314 df-tsms 24501 df-xms 24694 df-ms 24695 df-tms 24696 df-cncf 25254 df-0p 26046 df-limc 26241 df-dv 26242 df-dvn 26243 df-ply 26561 df-idp 26562 df-coe 26563 df-dgr 26564 df-tayl 26739

This theorem is used by: (None)

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