dvntaylp0 - Metamath Proof Explorer

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Theorem dvntaylp0 24967

Description: The first 𝑁 derivatives of the Taylor polynomial at 𝐵 match the derivatives of the function from which it is derived. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Hypotheses**
Ref	Expression
dvntaylp0.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvntaylp0.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
dvntaylp0.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
dvntaylp0.m	⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
dvntaylp0.b	⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
dvntaylp0.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)

**Assertion**
Ref	Expression
dvntaylp0	⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵))

Proof of Theorem dvntaylp0 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvntaylp0.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
2		elfz3nn0 12996	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	nn0cnd 11945	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		elfznn0 12995	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
6	1, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
7	6	nn0cnd 11945	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8	4, 7	npcand 10990	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
9	8	oveq1d 7150	. . . . . . 7 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
10		dvntaylp0.t	. . . . . . 7 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
11	9, 10	eqtr4di 2851	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
12	11	oveq2d 7151	. . . . 5 ⊢ (𝜑 → (ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
13	12	fveq1d 6647	. . . 4 ⊢ (𝜑 → ((ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((ℂ D^𝑛 𝑇)‘𝑀))
14		dvntaylp0.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
15		dvntaylp0.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
16		dvntaylp0.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
17		fznn0sub 12934	. . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
18	1, 17	syl 17	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
19		dvntaylp0.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
20	8	fveq2d 6649	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
21	20	dmeqd 5738	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
22	19, 21	eleqtrrd 2893	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)))
23	14, 15, 16, 6, 18, 22	dvntaylp 24966	. . . 4 ⊢ (𝜑 → ((ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵))‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
24	13, 23	eqtr3d 2835	. . 3 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
25	24	fveq1d 6647	. 2 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵))
26		cnex 10607	. . . . . . 7 ⊢ ℂ ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
28		elpm2r 8407	. . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
29	27, 14, 15, 16, 28	syl22anc 837	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
30		dvnf 24530	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
31	14, 29, 6, 30	syl3anc 1368	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
32		dvnbss 24531	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
33	14, 29, 6, 32	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
34	15, 33	fssdmd 6503	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝐴)
35	34, 16	sstrd 3925	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
36	18	orcd 870	. . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ₀ ∨ (𝑁 − 𝑀) = +∞))
37		dvnadd 24532	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) ∧ (𝑀 ∈ ℕ₀ ∧ (𝑁 − 𝑀) ∈ ℕ₀)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
38	14, 29, 6, 18, 37	syl22anc 837	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
39	7, 4	pncan3d 10989	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
40	39	fveq2d 6649	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
41	38, 40	eqtrd 2833	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
42	41	dmeqd 5738	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
43	19, 42	eleqtrrd 2893	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)))
44	14, 31, 35, 18, 43	taylplem1 24958	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
45		eqid 2798	. . . 4 ⊢ ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)
46	14, 31, 35, 36, 44, 45	tayl0 24957	. . 3 ⊢ (𝜑 → (𝐵 ∈ dom ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵) ∧ (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵)))
47	46	simprd 499	. 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵))
48	25, 47	eqtrd 2833	1 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⊆ wss 3881 {cpr 4527 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ↑_pm cpm 8390 ℂcc 10524 ℝcr 10525 0cc0 10526 + caddc 10529 +∞cpnf 10661 − cmin 10859 ℕ₀cn0 11885 ...cfz 12885 D^𝑛 cdvn 24467 Tayl ctayl 24948

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-pm 8392 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-xnn0 11956 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-icc 12733 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-fac 13630 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-clim 14837 df-sum 15035 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-perf 21742 df-cn 21832 df-cnp 21833 df-haus 21920 df-tx 22167 df-hmeo 22360 df-fil 22451 df-fm 22543 df-flim 22544 df-flf 22545 df-tsms 22732 df-xms 22927 df-ms 22928 df-tms 22929 df-cncf 23483 df-limc 24469 df-dv 24470 df-dvn 24471 df-tayl 24950

This theorem is referenced by: taylthlem1 24968 taylthlem2 24969

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