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Theorem dvnfre 24549

Description: The 𝑁-th derivative of a real function is real. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Assertion**
Ref	Expression
dvnfre	⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑁 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ)

Proof of Theorem dvnfre Dummy variables 𝑥 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 0 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘0))
2	1	dmeqd 5774	. . . . . 6 ⊢ (𝑥 = 0 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘0))
3	1, 2	feq12d 6502	. . . . 5 ⊢ (𝑥 = 0 → (((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ))
4	3	imbi2d 343	. . . 4 ⊢ (𝑥 = 0 → (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ) ↔ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ)))
5		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑛))
6	5	dmeqd 5774	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑛))
7	5, 6	feq12d 6502	. . . . 5 ⊢ (𝑥 = 𝑛 → (((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ))
8	7	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑛 → (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ) ↔ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)))
9		fveq2 6670	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
10	9	dmeqd 5774	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
11	9, 10	feq12d 6502	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ))
12	11	imbi2d 343	. . . 4 ⊢ (𝑥 = (𝑛 + 1) → (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ) ↔ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ)))
13		fveq2 6670	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑁))
14	13	dmeqd 5774	. . . . . 6 ⊢ (𝑥 = 𝑁 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑁))
15	13, 14	feq12d 6502	. . . . 5 ⊢ (𝑥 = 𝑁 → (((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ))
16	15	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝑁 → (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑥):dom ((ℝ D^𝑛 𝐹)‘𝑥)⟶ℝ) ↔ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ)))
17		simpl 485	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
18		ax-resscn 10594	. . . . . . 7 ⊢ ℝ ⊆ ℂ
19		fss 6527	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
20	18, 19	mpan2 689	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ)
21		cnex 10618	. . . . . . . . 9 ⊢ ℂ ∈ V
22		reex 10628	. . . . . . . . 9 ⊢ ℝ ∈ V
23		elpm2r 8424	. . . . . . . . 9 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	21, 22, 23	mpanl12 700	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
25	20, 24	sylan 582	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
26		dvn0 24521	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
27	18, 25, 26	sylancr 589	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
28	27	dmeqd 5774	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = dom 𝐹)
29		fdm 6522	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	29	adantr 483	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom 𝐹 = 𝐴)
31	28, 30	eqtrd 2856	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = 𝐴)
32	27, 31	feq12d 6502	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ ↔ 𝐹:𝐴⟶ℝ))
33	17, 32	mpbird 259	. . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ)
34		simprr 771	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)
35	22	prid1 4698	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
36		simprl 769	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝑛 ∈ ℕ₀)
37		dvnbss 24525	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
38	35, 25, 36, 37	mp3an2ani 1464	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
39	30	adantr 483	. . . . . . . . . . 11 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → dom 𝐹 = 𝐴)
40	38, 39	sseqtrd 4007	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ 𝐴)
41		simplr 767	. . . . . . . . . 10 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝐴 ⊆ ℝ)
42	40, 41	sstrd 3977	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ ℝ)
43		dvfre 24548	. . . . . . . . 9 ⊢ ((((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ ∧ dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ ℝ) → (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)):dom (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛))⟶ℝ)
44	34, 42, 43	syl2anc 586	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)):dom (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛))⟶ℝ)
45		dvnp1 24522	. . . . . . . . . 10 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)))
46	18, 25, 36, 45	mp3an2ani 1464	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)))
47	46	dmeqd 5774	. . . . . . . . 9 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)) = dom (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)))
48	46, 47	feq12d 6502	. . . . . . . 8 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → (((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ ↔ (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛)):dom (ℝ D ((ℝ D^𝑛 𝐹)‘𝑛))⟶ℝ))
49	44, 48	mpbird 259	. . . . . . 7 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ)
50	49	expr 459	. . . . . 6 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ 𝑛 ∈ ℕ₀) → (((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ))
51	50	expcom 416	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ)))
52	51	a2d 29	. . . 4 ⊢ (𝑛 ∈ ℕ₀ → (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ) → ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)):dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1))⟶ℝ)))
53	4, 8, 12, 16, 33, 52	nn0ind 12078	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ))
54	53	com12 32	. 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (𝑁 ∈ ℕ₀ → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ))
55	54	3impia 1113	1 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑁 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⊆ wss 3936 {cpr 4569 dom cdm 5555 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_pm cpm 8407 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 ℕ₀cn0 11898 D cdv 24461 D^𝑛 cdvn 24462

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fi 8875 df-sup 8906 df-inf 8907 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-plusg 16578 df-mulr 16579 df-starv 16580 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-rest 16696 df-topn 16697 df-topgen 16717 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-cncf 23486 df-limc 24464 df-dv 24465 df-dvn 24466

This theorem is referenced by: taylthlem2 24962

W3C validator