dvntaylp0 - Metamath Proof Explorer

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Theorem dvntaylp0 26500

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 13648	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
3	1, 2	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	nn0cnd 12566	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		elfznn0 13647	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
6	1, 5	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
7	6	nn0cnd 12566	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8	4, 7	npcand 11572	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
9	8	oveq1d 7426	. . . . . . 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 2822	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
12	11	oveq2d 7427	. . . . 5 ⊢ (𝜑 → (ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
13	12	fveq1d 6884	. . . 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 13583	. . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ ℕ₀)
18	1, 17	syl 18	. . . . 5 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
19		dvntaylp0.b	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
20	8	fveq2d 6886	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
21	20	dmeqd 5896	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
22	19, 21	eleqtrrd 2872	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)))
23	14, 15, 16, 6, 18, 22	dvntaylp 26499	. . . 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 2806	. . 3 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
25	24	fveq1d 6884	. 2 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵))
26		cnex 11180	. . . . . . 7 ⊢ ℂ ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
28		elpm2r 8841	. . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
29	27, 14, 15, 16, 28	syl22anc 851	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
30		dvnf 26054	. . . . 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 1396	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
32		dvnbss 26055	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
33	14, 29, 6, 32	syl3anc 1396	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
34	15, 33	fssdmd 6725	. . . . 5 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝐴)
35	34, 16	sstrd 3955	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
36	18	orcd 886	. . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ₀ ∨ (𝑁 − 𝑀) = +∞))
37		dvnadd 26056	. . . . . . . . 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 851	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
39	7, 4	pncan3d 11571	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
40	39	fveq2d 6886	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
41	38, 40	eqtrd 2804	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
42	41	dmeqd 5896	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
43	19, 42	eleqtrrd 2872	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)))
44	14, 31, 35, 18, 43	taylplem1 26491	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
45		eqid 2769	. . . 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 26490	. . 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 500	. 2 ⊢ (𝜑 → (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑀)‘𝐵))
48	25, 47	eqtrd 2804	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 1567 ∈ wcel 2149 Vcvv 3463 ⊆ wss 3913 {cpr 4596 dom cdm 5662 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_pm cpm 8824 ℂcc 11097 ℝcr 11098 0cc0 11099 + caddc 11102 +∞cpnf 11239 − cmin 11440 ℕ₀cn0 12503 ...cfz 13534 D^𝑛 cdvn 25991 Tayl ctayl 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-icc 13378 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-fac 14309 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-ur 20263 df-ring 20316 df-cring 20317 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-tsms 24252 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-limc 25993 df-dv 25994 df-dvn 25995 df-tayl 26483

This theorem is referenced by: taylthlem1 26501 taylthlem2 26502

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