dvnmptconst - Mathbox for Glauco Siliprandi

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Theorem dvnmptconst 46777

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 23	. 2 ⊢ (𝜑 → 𝜑)
3		fveq2 6882	. . . . 5 ⊢ (𝑛 = 1 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1))
4	3	eqeq1d 2764	. . . 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 343	. . 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 6882	. . . . 5 ⊢ (𝑛 = 𝑚 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚))
7	6	eqeq1d 2764	. . . 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 343	. . 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 6882	. . . . 5 ⊢ (𝑛 = (𝑚 + 1) → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘(𝑚 + 1)))
10	9	eqeq1d 2764	. . . 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 343	. . 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 6882	. . . . 5 ⊢ (𝑛 = 𝑁 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑛) = ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁))
13	12	eqeq1d 2764	. . . 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 343	. . 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 26138	. . . . . 6 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	15, 16	syl 18	. . . . 5 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
18		dvnmptconst.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℂ)
19	18	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ℂ)
20		restsspw 17522	. . . . . . . 8 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) ⊆ 𝒫 𝑆
21		dvnmptconst.x	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
22	20, 21	sselid 3932	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝒫 𝑆)
23		elpwi 4567	. . . . . . 7 ⊢ (𝑋 ∈ 𝒫 𝑆 → 𝑋 ⊆ 𝑆)
24	22, 23	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆)
25		cnex 11209	. . . . . . 7 ⊢ ℂ ∈ V
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → ℂ ∈ V)
27	19, 24, 26, 15	mptelpm 46016	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
28		dvn1 26160	. . . . 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 596	. . . 4 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)))
30	15, 21, 18	dvmptconst 46751	. . . 4 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ 0))
31	29, 30	eqtrd 2797	. . 3 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘1) = (𝑥 ∈ 𝑋 ↦ 0))
32		simp3 1156	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
33		simp1 1154	. . . . 5 ⊢ ((𝑚 ∈ ℕ ∧ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝑚 ∈ ℕ)
34		simpr 490	. . . . . . 7 ⊢ (((𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) ∧ 𝜑) → 𝜑)
35		simpl 488	. . . . . . 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		pm3.35 815	. . . . . . 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 596	. . . . . 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 1148	. . . . 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 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑆 ⊆ ℂ)
40	27	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (ℂ ↑_pm 𝑆))
41		nnnn0 12539	. . . . . . . 8 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℕ₀)
42	41	3ad2ant2 1152	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑚) = (𝑥 ∈ 𝑋 ↦ 0)) → 𝑚 ∈ ℕ₀)
43		dvnp1 26159	. . . . . . 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 1398	. . . . . 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 7425	. . . . . . 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 1153	. . . . . 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 11227	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℂ)
48	15, 21, 47	dvmptconst 46751	. . . . . . 7 ⊢ (𝜑 → (𝑆 D (𝑥 ∈ 𝑋 ↦ 0)) = (𝑥 ∈ 𝑋 ↦ 0))
49	48	3ad2ant1 1151	. . . . . 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 2801	. . . . 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 1398	. . . 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 1137	. . 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 12279	. 2 ⊢ (𝑁 ∈ ℕ → (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0)))
54	1, 2, 53	sylc 66	1 ⊢ (𝜑 → ((𝑆 D^𝑛 (𝑥 ∈ 𝑋 ↦ 𝐴))‘𝑁) = (𝑥 ∈ 𝑋 ↦ 0))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 Vcvv 3453 ⊆ wss 3902 𝒫 cpw 4560 {cpr 4589 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7417 ↑_pm cpm 8831 ℂcc 11126 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 ℕcn 12261 ℕ₀cn0 12532 ↾_t crest 17511 TopOpenctopn 17512 ℂ_fldccnfld 21591 D cdv 26097 D^𝑛 cdvn 26098

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-pm 8833 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-fi 9385 df-sup 9416 df-inf 9417 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-4 12333 df-5 12334 df-6 12335 df-7 12336 df-8 12337 df-9 12338 df-n0 12533 df-z 12620 df-dec 12741 df-uz 12892 df-q 13002 df-rp 13047 df-xneg 13167 df-xadd 13168 df-xmul 13169 df-icc 13409 df-fz 13566 df-seq 14070 df-exp 14130 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-struct 17245 df-slot 17280 df-ndx 17292 df-base 17308 df-plusg 17361 df-mulr 17362 df-starv 17363 df-tset 17367 df-ple 17368 df-ds 17370 df-unif 17371 df-rest 17513 df-topn 17514 df-topgen 17534 df-psmet 21583 df-xmet 21584 df-met 21585 df-bl 21586 df-mopn 21587 df-fbas 21588 df-fg 21589 df-cnfld 21592 df-top 23125 df-topon 23142 df-topsp 23164 df-bases 23177 df-cld 23250 df-ntr 23251 df-cls 23252 df-nei 23329 df-lp 23367 df-perf 23368 df-cn 23458 df-cnp 23459 df-haus 23546 df-fil 24078 df-fm 24170 df-flim 24171 df-flf 24172 df-xms 24552 df-ms 24553 df-cncf 25112 df-limc 26100 df-dv 26101 df-dvn 26102

This theorem is used by: dvnprodlem3 46784

W3C validator