dvntaylp0 - Metamath Proof Explorer

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Theorem dvntaylp0 26256

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 13558	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	nn0cnd 12481	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		elfznn0 13557	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
6	1, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
7	6	nn0cnd 12481	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8	4, 7	npcand 11513	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
9	8	oveq1d 7384	. . . . . . 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 2782	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
12	11	oveq2d 7385	. . . . 5 ⊢ (𝜑 → (ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
13	12	fveq1d 6842	. . . 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 13493	. . . . . 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 6844	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
21	20	dmeqd 5859	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
22	19, 21	eleqtrrd 2831	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)))
23	14, 15, 16, 6, 18, 22	dvntaylp 26255	. . . 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 2766	. . 3 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
25	24	fveq1d 6842	. 2 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵))
26		cnex 11125	. . . . . . 7 ⊢ ℂ ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
28		elpm2r 8795	. . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
29	27, 14, 15, 16, 28	syl22anc 838	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
30		dvnf 25805	. . . . 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 1373	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
32		dvnbss 25806	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
33	14, 29, 6, 32	syl3anc 1373	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
34	15, 33	fssdmd 6688	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝐴)
35	34, 16	sstrd 3954	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
36	18	orcd 873	. . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ₀ ∨ (𝑁 − 𝑀) = +∞))
37		dvnadd 25807	. . . . . . . . 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 838	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
39	7, 4	pncan3d 11512	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
40	39	fveq2d 6844	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
41	38, 40	eqtrd 2764	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
42	41	dmeqd 5859	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
43	19, 42	eleqtrrd 2831	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)))
44	14, 31, 35, 18, 43	taylplem1 26246	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
45		eqid 2729	. . . 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 26245	. . 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 495	. 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵))
48	25, 47	eqtrd 2764	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 1540 ∈ wcel 2109 Vcvv 3444 ⊆ wss 3911 {cpr 4587 dom cdm 5631 ⟶wf 6495 ‘cfv 6499 (class class class)co 7369 ↑_pm cpm 8777 ℂcc 11042 ℝcr 11043 0cc0 11044 + caddc 11047 +∞cpnf 11181 − cmin 11381 ℕ₀cn0 12418 ...cfz 13444 D^𝑛 cdvn 25741 Tayl ctayl 26236

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-inf2 9570 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121 ax-pre-sup 11122 ax-addf 11123

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4868 df-int 4907 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-se 5585 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-isom 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-of 7633 df-om 7823 df-1st 7947 df-2nd 7948 df-supp 8117 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-1o 8411 df-2o 8412 df-er 8648 df-map 8778 df-pm 8779 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9289 df-fi 9338 df-sup 9369 df-inf 9370 df-oi 9439 df-card 9868 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-2 12225 df-3 12226 df-4 12227 df-5 12228 df-6 12229 df-7 12230 df-8 12231 df-9 12232 df-n0 12419 df-xnn0 12492 df-z 12506 df-dec 12626 df-uz 12770 df-q 12884 df-rp 12928 df-xneg 13048 df-xadd 13049 df-xmul 13050 df-icc 13289 df-fz 13445 df-fzo 13592 df-seq 13943 df-exp 14003 df-fac 14215 df-hash 14272 df-cj 15041 df-re 15042 df-im 15043 df-sqrt 15177 df-abs 15178 df-clim 15430 df-sum 15629 df-struct 17093 df-sets 17110 df-slot 17128 df-ndx 17140 df-base 17156 df-ress 17177 df-plusg 17209 df-mulr 17210 df-starv 17211 df-sca 17212 df-vsca 17213 df-ip 17214 df-tset 17215 df-ple 17216 df-ds 17218 df-unif 17219 df-hom 17220 df-cco 17221 df-rest 17361 df-topn 17362 df-0g 17380 df-gsum 17381 df-topgen 17382 df-pt 17383 df-prds 17386 df-xrs 17441 df-qtop 17446 df-imas 17447 df-xps 17449 df-mre 17523 df-mrc 17524 df-acs 17526 df-mgm 18543 df-sgrp 18622 df-mnd 18638 df-submnd 18687 df-grp 18844 df-minusg 18845 df-mulg 18976 df-cntz 19225 df-cmn 19688 df-abl 19689 df-mgp 20026 df-ur 20067 df-ring 20120 df-cring 20121 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22757 df-topon 22774 df-topsp 22796 df-bases 22809 df-cld 22882 df-ntr 22883 df-cls 22884 df-nei 22961 df-lp 22999 df-perf 23000 df-cn 23090 df-cnp 23091 df-haus 23178 df-tx 23425 df-hmeo 23618 df-fil 23709 df-fm 23801 df-flim 23802 df-flf 23803 df-tsms 23990 df-xms 24184 df-ms 24185 df-tms 24186 df-cncf 24747 df-limc 25743 df-dv 25744 df-dvn 25745 df-tayl 26238

This theorem is referenced by: taylthlem1 26257 taylthlem2 26258 taylthlem2OLD 26259

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