dvntaylp0 - Metamath Proof Explorer

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Theorem dvntaylp0 26348

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 13549	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ ℕ₀)
3	1, 2	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
4	3	nn0cnd 12476	. . . . . . . . 9 ⊢ (𝜑 → 𝑁 ∈ ℂ)
5		elfznn0 13548	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
6	1, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
7	6	nn0cnd 12476	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℂ)
8	4, 7	npcand 11508	. . . . . . . 8 ⊢ (𝜑 → ((𝑁 − 𝑀) + 𝑀) = 𝑁)
9	8	oveq1d 7383	. . . . . . 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 2790	. . . . . 6 ⊢ (𝜑 → (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
12	11	oveq2d 7384	. . . . 5 ⊢ (𝜑 → (ℂ D^𝑛 (((𝑁 − 𝑀) + 𝑀)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
13	12	fveq1d 6844	. . . 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 13484	. . . . . 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 6846	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
21	20	dmeqd 5862	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
22	19, 21	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘((𝑁 − 𝑀) + 𝑀)))
23	14, 15, 16, 6, 18, 22	dvntaylp 26347	. . . 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 2774	. . 3 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑀) = ((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵))
25	24	fveq1d 6844	. 2 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑀)‘𝐵) = (((𝑁 − 𝑀)(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑀))𝐵)‘𝐵))
26		cnex 11119	. . . . . . 7 ⊢ ℂ ∈ V
27	26	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
28		elpm2r 8794	. . . . . 6 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
29	27, 14, 15, 16, 28	syl22anc 839	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
30		dvnf 25897	. . . . 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 1374	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
32		dvnbss 25898	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
33	14, 29, 6, 32	syl3anc 1374	. . . . . 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 3946	. . . 4 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
36	18	orcd 874	. . . 4 ⊢ (𝜑 → ((𝑁 − 𝑀) ∈ ℕ₀ ∨ (𝑁 − 𝑀) = +∞))
37		dvnadd 25899	. . . . . . . . 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 839	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
39	7, 4	pncan3d 11507	. . . . . . . . 9 ⊢ (𝜑 → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
40	39	fveq2d 6846	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
41	38, 40	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
42	41	dmeqd 5862	. . . . . 6 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
43	19, 42	eleqtrrd 2840	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)))
44	14, 31, 35, 18, 43	taylplem1 26338	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ((0[,](𝑁 − 𝑀)) ∩ ℤ)) → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘𝑘))
45		eqid 2737	. . . 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 26337	. . 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 2772	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 1542 ∈ wcel 2114 Vcvv 3442 ⊆ wss 3903 {cpr 4584 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 ↑_pm cpm 8776 ℂcc 11036 ℝcr 11037 0cc0 11038 + caddc 11041 +∞cpnf 11175 − cmin 11376 ℕ₀cn0 12413 ...cfz 13435 D^𝑛 cdvn 25833 Tayl ctayl 26328

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-icc 13280 df-fz 13436 df-fzo 13583 df-seq 13937 df-exp 13997 df-fac 14209 df-hash 14266 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-clim 15423 df-sum 15622 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-mulg 19010 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-ur 20129 df-ring 20182 df-cring 20183 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-tsms 24083 df-xms 24276 df-ms 24277 df-tms 24278 df-cncf 24839 df-limc 25835 df-dv 25836 df-dvn 25837 df-tayl 26330

This theorem is referenced by: taylthlem1 26349 taylthlem2 26350 taylthlem2OLD 26351

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