dvn2bss - Metamath Proof Explorer

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Theorem dvn2bss 23912

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 1129	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
2		simp2 1130	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
3		elfznn0 12639	. . . . . 6 ⊢ (𝑀 ∈ (0...𝑁) → 𝑀 ∈ ℕ₀)
4	3	3ad2ant3 1128	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℕ₀)
5		elfzuz3 12545	. . . . . . 7 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘𝑀))
6	5	3ad2ant3 1128	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘𝑀))
7		uznn0sub 11920	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝑁 − 𝑀) ∈ ℕ₀)
8	6, 7	syl 17	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑁 − 𝑀) ∈ ℕ₀)
9		dvnadd 23911	. . . . 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 1476	. . . 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 11554	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑀 ∈ ℂ)
12		elfzuz2 12552	. . . . . . . . 9 ⊢ (𝑀 ∈ (0...𝑁) → 𝑁 ∈ (ℤ_≥‘0))
13	12	3ad2ant3 1128	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ (ℤ_≥‘0))
14		nn0uz 11923	. . . . . . . 8 ⊢ ℕ₀ = (ℤ_≥‘0)
15	13, 14	syl6eleqr 2860	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℕ₀)
16	15	nn0cnd 11554	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → 𝑁 ∈ ℂ)
17	11, 16	pncan3d 10596	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝑀 + (𝑁 − 𝑀)) = 𝑁)
18	17	fveq2d 6336	. . . 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 2804	. . 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 5464	. 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 10218	. . . . 5 ⊢ ℂ ∈ V
22	21	a1i 11	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ℂ ∈ V)
23		dvnf 23909	. . . . 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 1168	. . . 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 23910	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
26	3, 25	syl3an3 1168	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ dom 𝐹)
27		elpmi 8027	. . . . . . 7 ⊢ (𝐹 ∈ (ℂ ↑_pm 𝑆) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
28	27	3ad2ant2 1127	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ 𝑆))
29	28	simprd 477	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom 𝐹 ⊆ 𝑆)
30	26, 29	sstrd 3760	. . . 4 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑀) ⊆ 𝑆)
31		elpm2r 8026	. . . 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 1476	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑀 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑀) ∈ (ℂ ↑_pm 𝑆))
33		dvnbss 23910	. . 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 1475	. 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	eqsstr3d 3787	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 382 ∧ w3a 1070 = wceq 1630 ∈ wcel 2144 Vcvv 3349 ⊆ wss 3721 {cpr 4316 dom cdm 5249 ⟶wf 6027 ‘cfv 6031 (class class class)co 6792 ↑_pm cpm 8009 ℂcc 10135 ℝcr 10136 0cc0 10137 + caddc 10140 − cmin 10467 ℕ₀cn0 11493 ℤ_≥cuz 11887 ...cfz 12532 D^𝑛 cdvn 23847

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 ax-inf2 8701 ax-cnex 10193 ax-resscn 10194 ax-1cn 10195 ax-icn 10196 ax-addcl 10197 ax-addrcl 10198 ax-mulcl 10199 ax-mulrcl 10200 ax-mulcom 10201 ax-addass 10202 ax-mulass 10203 ax-distr 10204 ax-i2m1 10205 ax-1ne0 10206 ax-1rid 10207 ax-rnegex 10208 ax-rrecex 10209 ax-cnre 10210 ax-pre-lttri 10211 ax-pre-lttrn 10212 ax-pre-ltadd 10213 ax-pre-mulgt0 10214 ax-pre-sup 10215

This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-nel 3046 df-ral 3065 df-rex 3066 df-reu 3067 df-rmo 3068 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-int 4610 df-iun 4654 df-iin 4655 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7212 df-1st 7314 df-2nd 7315 df-wrecs 7558 df-recs 7620 df-rdg 7658 df-1o 7712 df-oadd 7716 df-er 7895 df-map 8010 df-pm 8011 df-en 8109 df-dom 8110 df-sdom 8111 df-fin 8112 df-fi 8472 df-sup 8503 df-inf 8504 df-pnf 10277 df-mnf 10278 df-xr 10279 df-ltxr 10280 df-le 10281 df-sub 10469 df-neg 10470 df-div 10886 df-nn 11222 df-2 11280 df-3 11281 df-4 11282 df-5 11283 df-6 11284 df-7 11285 df-8 11286 df-9 11287 df-n0 11494 df-z 11579 df-dec 11695 df-uz 11888 df-q 11991 df-rp 12035 df-xneg 12150 df-xadd 12151 df-xmul 12152 df-icc 12386 df-fz 12533 df-seq 13008 df-exp 13067 df-cj 14046 df-re 14047 df-im 14048 df-sqrt 14182 df-abs 14183 df-struct 16065 df-ndx 16066 df-slot 16067 df-base 16069 df-plusg 16161 df-mulr 16162 df-starv 16163 df-tset 16167 df-ple 16168 df-ds 16171 df-unif 16172 df-rest 16290 df-topn 16291 df-topgen 16311 df-psmet 19952 df-xmet 19953 df-met 19954 df-bl 19955 df-mopn 19956 df-fbas 19957 df-fg 19958 df-cnfld 19961 df-top 20918 df-topon 20935 df-topsp 20957 df-bases 20970 df-cld 21043 df-ntr 21044 df-cls 21045 df-nei 21122 df-lp 21160 df-perf 21161 df-cnp 21252 df-haus 21339 df-fil 21869 df-fm 21961 df-flim 21962 df-flf 21963 df-xms 22344 df-ms 22345 df-limc 23849 df-dv 23850 df-dvn 23851

This theorem is referenced by: taylplem1 24336 taylply2 24341 taylply 24342 taylthlem1 24346 taylthlem2 24347

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