dvnmptconst - Mathbox for Glauco Siliprandi

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Theorem dvnmptconst 45862

Description: The 𝑁-th derivative of a constant function. (Contributed by Glauco Siliprandi, 5-Apr-2020.)

**Hypotheses**
Ref	Expression
dvnmptconst.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
dvnmptconst.x	⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
dvnmptconst.a	⊢ (𝜑 → 𝐴 ∈ ℂ)
dvnmptconst.n	⊢ (𝜑 → 𝑁 ∈ ℕ)

**Assertion**
Ref	Expression
dvnmptconst	⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑆 𝑥,𝑋 𝜑,𝑥 Allowed substitution hint: 𝑁(𝑥)

Proof of Theorem dvnmptconst Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvnmptconst.n	. 2 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		id 22	. 2 ⊢ (𝜑 → 𝜑)
3		fveq2 6920	. . . . 5 ⊢ (𝑛 = 1 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1))
4	3	eqeq1d 2742	. . . 4 ⊢ (𝑛 = 1 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑥 ∈ 𝑋 ↦ 0)))
5	4	imbi2d 340	. . 3 ⊢ (𝑛 = 1 → ((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0)) ↔ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑥 ∈ 𝑋 ↦ 0))))
6		fveq2 6920	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚))
7	6	eqeq1d 2742	. . . 4 ⊢ (𝑛 = 𝑚 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)))
8	7	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑚 → ((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0)) ↔ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0))))
9		fveq2 6920	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)))
10	9	eqeq1d 2742	. . . 4 ⊢ (𝑛 = (𝑚 + 1) → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑥 ∈ 𝑋 ↦ 0)))
11	10	imbi2d 340	. . 3 ⊢ (𝑛 = (𝑚 + 1) → ((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0)) ↔ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑥 ∈ 𝑋 ↦ 0))))
12		fveq2 6920	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁))
13	12	eqeq1d 2742	. . . 4 ⊢ (𝑛 = 𝑁 → (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0) ↔ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0)))
14	13	imbi2d 340	. . 3 ⊢ (𝑛 = 𝑁 → ((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = (𝑥 ∈ 𝑋 ↦ 0)) ↔ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))))
15		dvnmptconst.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
16		recnprss 25959	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	15, 16	syl 17	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
18		dvnmptconst.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
19	18	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
20		restsspw 17491	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
21		dvnmptconst.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
22	20, 21	sselid 4006	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
23		elpwi 4629	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
25		cnex 11265	. . . . . . 7 ⊢ ℂ ∈ V
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
27	19, 24, 26, 15	mptelpm 45083	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
28		dvn1 25982	. . . . 5 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
29	17, 27, 28	syl2anc 583	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
30	15, 21, 18	dvmptconst 45836	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
31	29, 30	eqtrd 2780	. . 3 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑥 ∈ 𝑋 ↦ 0))
32		simp3 1138	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
33		simp1 1136	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝑚 ∈ ℕ)
34		simpr 484	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
35		simpl 482	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)))
36		pm3.35 802	. . . . . . 7 ⊢ ((𝜑 ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0))) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0))
37	34, 35, 36	syl2anc 583	. . . . . 6 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0))
38	37	3adant1 1130	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0))
39	17	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑆 ⊆ ℂ)
40	27	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
41		nnnn0 12560	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
42	41	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑚 ∈ ℕ₀)
43		dvnp1 25981	. . . . . . 7 ⊢ ((𝑆 ⊆ ℂ ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆) ∧ 𝑚 ∈ ℕ₀) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑆 D ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚)))
44	39, 40, 42, 43	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑆 D ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚)))
45		oveq2 7456	. . . . . . 7 ⊢ (((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0) → (𝑆 D ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)))
46	45	3ad2ant3 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝑆 D ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚)) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)))
47		0cnd 11283	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
48	15, 21, 47	dvmptconst 45836	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
49	48	3ad2ant1 1133	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
50	44, 46, 49	3eqtrd 2784	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑥 ∈ 𝑋 ↦ 0))
51	32, 33, 38, 50	syl3anc 1371	. . . 4 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑥 ∈ 𝑋 ↦ 0))
52	51	3exp 1119	. . 3 ⊢ (𝑚 ∈ ℕ → ((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)) = (𝑥 ∈ 𝑋 ↦ 0))))
53	5, 8, 11, 14, 31, 52	nnind 12311	. 2 ⊢ (𝑁 ∈ ℕ → (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0)))
54	1, 2, 53	sylc 65	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 Vcvv 3488 ⊆ wss 3976 𝒫 cpw 4622 {cpr 4650 ↦ cmpt 5249 ‘cfv 6573 (class class class)co 7448 ↑_pm cpm 8885 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 + caddc 11187 ℕcn 12293 ℕ₀cn0 12553 ↾_t crest 17480 TopOpenctopn 17481 ℂ_fldccnfld 21387 D cdv 25918 D^𝑛 cdvn 25919

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

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-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-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-er 8763 df-map 8886 df-pm 8887 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fi 9480 df-sup 9511 df-inf 9512 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-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-seq 14053 df-exp 14113 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-mulr 17325 df-starv 17326 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-rest 17482 df-topn 17483 df-topgen 17503 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-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-cncf 24923 df-limc 25921 df-dv 25922 df-dvn 25923

This theorem is referenced by: dvnprodlem3 45869

W3C validator