dvnmptconst - Mathbox for Glauco Siliprandi

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Theorem dvnmptconst 44742

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 6891	. . . . 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 6891	. . . . 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 6891	. . . . 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 6891	. . . . 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 25428	. . . . . 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 17379	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
21		dvnmptconst.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
22	20, 21	sselid 3980	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
23		elpwi 4609	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
25		cnex 11193	. . . . . . 7 ⊢ ℂ ∈ V
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
27	19, 24, 26, 15	mptelpm 43960	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
28		dvn1 25450	. . . . 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 44716	. . . 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 12481	. . . . . . . 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 25449	. . . . . . 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 7419	. . . . . . 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 11209	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
48	15, 21, 47	dvmptconst 44716	. . . . . . 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 12232	. 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 3948 𝒫 cpw 4602 {cpr 4630 ↦ cmpt 5231 ‘cfv 6543 (class class class)co 7411 ↑_pm cpm 8823 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 ℕcn 12214 ℕ₀cn0 12474 ↾_t crest 17368 TopOpenctopn 17369 ℂ_fldccnfld 20950 D cdv 25387 D^𝑛 cdvn 25388

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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fi 9408 df-sup 9439 df-inf 9440 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-icc 13333 df-fz 13487 df-seq 13969 df-exp 14030 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17082 df-slot 17117 df-ndx 17129 df-base 17147 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-rest 17370 df-topn 17371 df-topgen 17391 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-xms 23833 df-ms 23834 df-cncf 24401 df-limc 25390 df-dv 25391 df-dvn 25392

This theorem is referenced by: dvnprodlem3 44749

W3C validator