dvntaylp0 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvntaylp0		Structured version Visualization version GIF version

Theorem dvntaylp0 26432

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 13678	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	nn0cnd 12615	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		elfznn0 13677	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
6	1, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
7	6	nn0cnd 12615	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8	4, 7	npcand 11651	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
9	8	oveq1d 7463	. . . . . . 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 2798	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
12	11	oveq2d 7464	. . . . 5 ⊢ (𝜑 → (ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
13	12	fveq1d 6922	. . . 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 13616	. . . . . 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 6924	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
21	20	dmeqd 5930	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
22	19, 21	eleqtrrd 2847	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)))
23	14, 15, 16, 6, 18, 22	dvntaylp 26431	. . . 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 2782	. . 3 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
25	24	fveq1d 6922	. 2 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵))
26		cnex 11265	. . . . . . 7 ⊢ ℂ ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
28		elpm2r 8903	. . . . . 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 25983	. . . . 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 1371	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
32		dvnbss 25984	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
33	14, 29, 6, 32	syl3anc 1371	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
34	15, 33	fssdmd 6765	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝐴)
35	34, 16	sstrd 4019	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
36	18	orcd 872	. . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ₀ ∨ (𝑁 − 𝑀) = +∞))
37		dvnadd 25985	. . . . . . . . 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 11650	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
40	39	fveq2d 6924	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
41	38, 40	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
42	41	dmeqd 5930	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
43	19, 42	eleqtrrd 2847	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)))
44	14, 31, 35, 18, 43	taylplem1 26422	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
45		eqid 2740	. . . 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 26421	. . 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 2780	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 1537 ∈ wcel 2108 Vcvv 3488 ⊆ wss 3976 {cpr 4650 dom cdm 5700 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ↑_pm cpm 8885 ℂcc 11182 ℝcr 11183 0cc0 11184 + caddc 11187 +∞cpnf 11321 − cmin 11520 ℕ₀cn0 12553 ...cfz 13567 D^𝑛 cdvn 25919 Tayl ctayl 26412

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-xnn0 12626 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-fac 14323 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-grp 18976 df-minusg 18977 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-abl 19825 df-mgp 20162 df-ur 20209 df-ring 20262 df-cring 20263 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-tsms 24156 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922 df-dvn 25923 df-tayl 26414

This theorem is referenced by: taylthlem1 26433 taylthlem2 26434 taylthlem2OLD 26435

W3C validator