dvnmptconst - Mathbox for Glauco Siliprandi

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Theorem dvnmptconst 44643

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 6888	. . . . 5 ⊢ (𝑛 = 1 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1))
4	3	eqeq1d 2734	. . . 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 6888	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚))
7	6	eqeq1d 2734	. . . 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 6888	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)))
10	9	eqeq1d 2734	. . . 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 6888	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁))
13	12	eqeq1d 2734	. . . 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 25412	. . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
20		restsspw 17373	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
21		dvnmptconst.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
22	20, 21	sselid 3979	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
23		elpwi 4608	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
25		cnex 11187	. . . . . . 7 ⊢ ℂ ∈ V
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
27	19, 24, 26, 15	mptelpm 43857	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
28		dvn1 25434	. . . . 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 584	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
30	15, 21, 18	dvmptconst 44617	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
31	29, 30	eqtrd 2772	. . 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 485	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
35		simpl 483	. . . . . . 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		pm3.35 801	. . . . . . 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 584	. . . . . 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 12475	. . . . . . . 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 25433	. . . . . . 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 7413	. . . . . . 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 11203	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
48	15, 21, 47	dvmptconst 44617	. . . . . . 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 2776	. . . . 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 12226	. 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3947 𝒫 cpw 4601 {cpr 4629 ↦ cmpt 5230 ‘cfv 6540 (class class class)co 7405 ↑_pm cpm 8817 ℂcc 11104 ℝcr 11105 0cc0 11106 1c1 11107 + caddc 11109 ℕcn 12208 ℕ₀cn0 12468 ↾_t crest 17362 TopOpenctopn 17363 ℂ_fldccnfld 20936 D cdv 25371 D^𝑛 cdvn 25372

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fi 9402 df-sup 9433 df-inf 9434 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-icc 13327 df-fz 13481 df-seq 13963 df-exp 14024 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-struct 17076 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-starv 17208 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-rest 17364 df-topn 17365 df-topgen 17385 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-cncf 24385 df-limc 25374 df-dv 25375 df-dvn 25376

This theorem is referenced by: dvnprodlem3 44650

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