dvn2bss - Metamath Proof Explorer

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Theorem dvn2bss 26068

Description: An N-times differentiable point is an M-times differentiable point, if 𝑀 ≤ 𝑁. (Contributed by Mario Carneiro, 30-Dec-2016.)

**Assertion**
Ref	Expression
dvn2bss	⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑀))

**Proof of Theorem dvn2bss**
Step	Hyp	Ref	Expression
1		simp1 1152	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
2		simp2 1153	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
3		elfznn0 13647	. . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
4	3	3ad2ant3 1151	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℕ₀)
5		elfzuz3 13548	. . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
6	5	3ad2ant3 1151	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑀))
7		uznn0sub 12896	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
8	6, 7	syl 18	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑁 − 𝑀) ∈ ℕ₀)
9		dvnadd 26067	. . . . 5 ⊢ (((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))))
10	1, 2, 4, 8, 9	syl22anc 851	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))))
11	4	nn0cnd 12566	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℂ)
12		elfzuz2 13556	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘0))
13	12	3ad2ant3 1151	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		nn0uz 12899	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
15	13, 14	eleqtrrdi 2872	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
16	15	nn0cnd 12566	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
17	11, 16	pncan3d 11571	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
18	17	fveq2d 6885	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘(𝑀 + (𝑁 − 𝑀))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
19	10, 18	eqtrd 2796	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
20	19	dmeqd 5895	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
21		cnex 11180	. . . . 5 ⊢ ℂ ∈ V
22	21	a1i 11	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ℂ ∈ V)
23		dvnf 26065	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
24	3, 23	syl3an3 1181	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ)
25		dvnbss 26066	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
26	3, 25	syl3an3 1181	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
27		elpmi 8842	. . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑_pm 𝑆) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
28	27	3ad2ant2 1150	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
29	28	simprd 500	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom 𝐹 ⊆ 𝑆)
30	26, 29	sstrd 3946	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
31		elpm2r 8841	. . . 4 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (((𝑆 D^𝑛 𝐹)‘𝑀):dom ((𝑆 D^𝑛 𝐹)‘𝑀)⟶ℂ ∧ dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
32	22, 1, 24, 30, 31	syl22anc 851	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
33		dvnbss 26066	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑀))
34	1, 32, 8, 33	syl3anc 1396	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑀))‘(𝑁 − 𝑀)) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑀))
35	20, 34	eqsstrrd 3971	1 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑀))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 ∧ w3a 1101 = wceq 1568 ∈ wcel 2141 Vcvv 3453 ⊆ wss 3904 {cpr 4590 dom cdm 5661 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ↑_pm cpm 8824 ℂcc 11097 ℝcr 11098 0cc0 11099 + caddc 11102 − cmin 11440 ℕ₀cn0 12503 ℤ_≥cuz 12861 ...cfz 13534 D^𝑛 cdvn 26002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 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

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-er 8693 df-map 8825 df-pm 8826 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fi 9370 df-sup 9401 df-inf 9402 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-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-seq 14037 df-exp 14097 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-struct 17206 df-slot 17241 df-ndx 17253 df-base 17269 df-plusg 17322 df-mulr 17323 df-starv 17324 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-rest 17474 df-topn 17475 df-topgen 17495 df-psmet 21493 df-xmet 21494 df-met 21495 df-bl 21496 df-mopn 21497 df-fbas 21498 df-fg 21499 df-cnfld 21502 df-top 23030 df-topon 23047 df-topsp 23069 df-bases 23082 df-cld 23155 df-ntr 23156 df-cls 23157 df-nei 23234 df-lp 23272 df-perf 23273 df-cnp 23364 df-haus 23451 df-fil 23982 df-fm 24074 df-flim 24075 df-flf 24076 df-xms 24456 df-ms 24457 df-limc 26004 df-dv 26005 df-dvn 26006

This theorem is referenced by: taylplem1 26502 taylply2 26507 taylply 26508 taylthlem1 26512 taylthlem2 26513

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