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Theorem dvnfre 25469

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 6892	. . . . . 6 ⊢ (𝑥 = 0 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘0))
2	1	dmeqd 5906	. . . . . 6 ⊢ (𝑥 = 0 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘0))
3	1, 2	feq12d 6706	. . . . 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 341	. . . 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 6892	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑛))
6	5	dmeqd 5906	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑛))
7	5, 6	feq12d 6706	. . . . 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 341	. . . 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 6892	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
10	9	dmeqd 5906	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
11	9, 10	feq12d 6706	. . . . 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 341	. . . 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 6892	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑁))
14	13	dmeqd 5906	. . . . . 6 ⊢ (𝑥 = 𝑁 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑁))
15	13, 14	feq12d 6706	. . . . 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 341	. . . 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 484	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
18		ax-resscn 11167	. . . . . . 7 ⊢ ℝ ⊆ ℂ
19		fss 6735	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
20	18, 19	mpan2 690	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ)
21		cnex 11191	. . . . . . . . 9 ⊢ ℂ ∈ V
22		reex 11201	. . . . . . . . 9 ⊢ ℝ ∈ V
23		elpm2r 8839	. . . . . . . . 9 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	21, 22, 23	mpanl12 701	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
25	20, 24	sylan 581	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
26		dvn0 25441	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
27	18, 25, 26	sylancr 588	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
28	27	dmeqd 5906	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = dom 𝐹)
29		fdm 6727	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	29	adantr 482	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom 𝐹 = 𝐴)
31	28, 30	eqtrd 2773	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = 𝐴)
32	27, 31	feq12d 6706	. . . . 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 257	. . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ)
34		simprr 772	. . . . . . . . 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 4767	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
36		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝑛 ∈ ℕ₀)
37		dvnbss 25445	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
38	35, 25, 36, 37	mp3an2ani 1469	. . . . . . . . . . 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 482	. . . . . . . . . . 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 4023	. . . . . . . . . 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 768	. . . . . . . . . 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 3993	. . . . . . . . 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 25468	. . . . . . . . 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 585	. . . . . . . 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 25442	. . . . . . . . . 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 1469	. . . . . . . . 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 5906	. . . . . . . . 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 6706	. . . . . . . 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 257	. . . . . . 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 458	. . . . . 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 415	. . . . 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 12657	. . 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 1118	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 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ⊆ wss 3949 {cpr 4631 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ↑_pm cpm 8821 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 ℕ₀cn0 12472 D cdv 25380 D^𝑛 cdvn 25381

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fi 9406 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-icc 13331 df-fz 13485 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-struct 17080 df-slot 17115 df-ndx 17127 df-base 17145 df-plusg 17210 df-mulr 17211 df-starv 17212 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-rest 17368 df-topn 17369 df-topgen 17389 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-cncf 24394 df-limc 25383 df-dv 25384 df-dvn 25385

This theorem is referenced by: taylthlem2 25886

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