dvnmptconst - Mathbox for Glauco Siliprandi

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Theorem dvnmptconst 42949

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 6658	. . . . 5 ⊢ (𝑛 = 1 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1))
4	3	eqeq1d 2760	. . . 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 344	. . 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 6658	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚))
7	6	eqeq1d 2760	. . . 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 344	. . 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 6658	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)))
10	9	eqeq1d 2760	. . . 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 344	. . 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 6658	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁))
13	12	eqeq1d 2760	. . . 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 344	. . 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 24603	. . . . . 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 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
20		restsspw 16763	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
21		dvnmptconst.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
22	20, 21	sseldi 3890	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
23		elpwi 4503	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
24	22, 23	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
25		cnex 10656	. . . . . . 7 ⊢ ℂ ∈ V
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
27	19, 24, 26, 15	mptelpm 42171	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
28		dvn1 24625	. . . . 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 587	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
30	15, 21, 18	dvmptconst 42923	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
31	29, 30	eqtrd 2793	. . 3 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑥 ∈ 𝑋 ↦ 0))
32		simp3 1135	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
33		simp1 1133	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝑚 ∈ ℕ)
34		simpr 488	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
35		simpl 486	. . . . . . 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 587	. . . . . 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 1127	. . . . 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 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑆 ⊆ ℂ)
40	27	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
41		nnnn0 11941	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
42	41	3ad2ant2 1131	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑚 ∈ ℕ₀)
43		dvnp1 24624	. . . . . . 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 1368	. . . . . 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 7158	. . . . . . 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 1132	. . . . . 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 10672	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
48	15, 21, 47	dvmptconst 42923	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
49	48	3ad2ant1 1130	. . . . . 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 2797	. . . . 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 1368	. . . 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 1116	. . 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 11692	. 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3409 ⊆ wss 3858 𝒫 cpw 4494 {cpr 4524 ↦ cmpt 5112 ‘cfv 6335 (class class class)co 7150 ↑_pm cpm 8417 ℂcc 10573 ℝcr 10574 0cc0 10575 1c1 10576 + caddc 10578 ℕcn 11674 ℕ₀cn0 11934 ↾_t crest 16752 TopOpenctopn 16753 ℂ_fldccnfld 20166 D cdv 24562 D^𝑛 cdvn 24563

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-rep 5156 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-inf2 9137 ax-cnex 10631 ax-resscn 10632 ax-1cn 10633 ax-icn 10634 ax-addcl 10635 ax-addrcl 10636 ax-mulcl 10637 ax-mulrcl 10638 ax-mulcom 10639 ax-addass 10640 ax-mulass 10641 ax-distr 10642 ax-i2m1 10643 ax-1ne0 10644 ax-1rid 10645 ax-rnegex 10646 ax-rrecex 10647 ax-cnre 10648 ax-pre-lttri 10649 ax-pre-lttrn 10650 ax-pre-ltadd 10651 ax-pre-mulgt0 10652 ax-pre-sup 10653

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rmo 3078 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-pss 3877 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-tp 4527 df-op 4529 df-uni 4799 df-int 4839 df-iun 4885 df-iin 4886 df-br 5033 df-opab 5095 df-mpt 5113 df-tr 5139 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5483 df-we 5485 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-pred 6126 df-ord 6172 df-on 6173 df-lim 6174 df-suc 6175 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7580 df-1st 7693 df-2nd 7694 df-wrecs 7957 df-recs 8018 df-rdg 8056 df-1o 8112 df-er 8299 df-map 8418 df-pm 8419 df-en 8528 df-dom 8529 df-sdom 8530 df-fin 8531 df-fi 8908 df-sup 8939 df-inf 8940 df-pnf 10715 df-mnf 10716 df-xr 10717 df-ltxr 10718 df-le 10719 df-sub 10910 df-neg 10911 df-div 11336 df-nn 11675 df-2 11737 df-3 11738 df-4 11739 df-5 11740 df-6 11741 df-7 11742 df-8 11743 df-9 11744 df-n0 11935 df-z 12021 df-dec 12138 df-uz 12283 df-q 12389 df-rp 12431 df-xneg 12548 df-xadd 12549 df-xmul 12550 df-icc 12786 df-fz 12940 df-seq 13419 df-exp 13480 df-cj 14506 df-re 14507 df-im 14508 df-sqrt 14642 df-abs 14643 df-struct 16543 df-ndx 16544 df-slot 16545 df-base 16547 df-plusg 16636 df-mulr 16637 df-starv 16638 df-tset 16642 df-ple 16643 df-ds 16645 df-unif 16646 df-rest 16754 df-topn 16755 df-topgen 16775 df-psmet 20158 df-xmet 20159 df-met 20160 df-bl 20161 df-mopn 20162 df-fbas 20163 df-fg 20164 df-cnfld 20167 df-top 21594 df-topon 21611 df-topsp 21633 df-bases 21646 df-cld 21719 df-ntr 21720 df-cls 21721 df-nei 21798 df-lp 21836 df-perf 21837 df-cn 21927 df-cnp 21928 df-haus 22015 df-fil 22546 df-fm 22638 df-flim 22639 df-flf 22640 df-xms 23022 df-ms 23023 df-cncf 23579 df-limc 24565 df-dv 24566 df-dvn 24567

This theorem is referenced by: dvnprodlem3 42956

W3C validator