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Theorem dvnfre 26287

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 6896	. . . . . 6 ⊢ (𝑥 = 0 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘0))
2	1	dmeqd 5892	. . . . . 6 ⊢ (𝑥 = 0 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘0))
3	1, 2	feq12d 6708	. . . . 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 6896	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑛))
6	5	dmeqd 5892	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑛))
7	5, 6	feq12d 6708	. . . . 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 6896	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
10	9	dmeqd 5892	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
11	9, 10	feq12d 6708	. . . . 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 6896	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑁))
14	13	dmeqd 5892	. . . . . 6 ⊢ (𝑥 = 𝑁 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑁))
15	13, 14	feq12d 6708	. . . . 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 488	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
18		ax-resscn 11270	. . . . . . 7 ⊢ ℝ ⊆ ℂ
19		fss 6737	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
20	18, 19	mpan2 704	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ)
21		cnex 11294	. . . . . . . . 9 ⊢ ℂ ∈ V
22		reex 11304	. . . . . . . . 9 ⊢ ℝ ∈ V
23		elpm2r 8879	. . . . . . . . 9 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	21, 22, 23	mpanl12 715	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
25	20, 24	sylan 592	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
26		dvn0 26259	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
27	18, 25, 26	sylancr 599	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
28	27	dmeqd 5892	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = dom 𝐹)
29		fdm 6730	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	29	adantr 486	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom 𝐹 = 𝐴)
31	28, 30	eqtrd 2798	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = 𝐴)
32	27, 31	feq12d 6708	. . . . 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 260	. . . 4 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0):dom ((ℝ D^𝑛 𝐹)‘0)⟶ℝ)
34		simprr 785	. . . . . . . . 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 4723	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
36		simprl 783	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝑛 ∈ ℕ₀)
37		dvnbss 26263	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
38	35, 25, 36, 37	mp3an2ani 1497	. . . . . . . . . . 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 486	. . . . . . . . . . 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 3949	. . . . . . . . . 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 781	. . . . . . . . . 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 3923	. . . . . . . . 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 26286	. . . . . . . . 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 596	. . . . . . . 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 26260	. . . . . . . . . 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 1497	. . . . . . . . 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 5892	. . . . . . . . 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 6708	. . . . . . . 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 260	. . . . . . 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 462	. . . . . 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 419	. . . . 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 30	. . . 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 12807	. . 3 ⊢ (𝑁 ∈ ℕ₀ → ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ))
54	53	com12 33	. 2 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → (𝑁 ∈ ℕ₀ → ((ℝ D^𝑛 𝐹)‘𝑁):dom ((ℝ D^𝑛 𝐹)‘𝑁)⟶ℝ))
55	54	3impia 1135	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

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1569 ∈ wcel 2146 Vcvv 3454 ⊆ wss 3900 {cpr 4590 dom cdm 5655 ⟶wf 6543 ‘cfv 6547 (class class class)co 7430 ↑_pm cpm 8862 ℂcc 11211 ℝcr 11212 0cc0 11213 1c1 11214 + caddc 11216 ℕ₀cn0 12619 D cdv 26198 D^𝑛 cdvn 26199

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1944 ax-6 2001 ax-7 2042 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2196 ax-12 2215 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7761 ax-inf2 9655 ax-cnex 11269 ax-resscn 11270 ax-1cn 11271 ax-icn 11272 ax-addcl 11273 ax-addrcl 11274 ax-mulcl 11275 ax-mulrcl 11276 ax-mulcom 11277 ax-addass 11278 ax-mulass 11279 ax-distr 11280 ax-i2m1 11281 ax-1ne0 11282 ax-1rid 11283 ax-rnegex 11284 ax-rrecex 11285 ax-cnre 11286 ax-pre-lttri 11287 ax-pre-lttrn 11288 ax-pre-ltadd 11289 ax-pre-mulgt0 11290 ax-pre-sup 11291

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2101 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6311 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6501 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7888 df-1st 8011 df-2nd 8012 df-frecs 8313 df-wrecs 8344 df-recs 8393 df-rdg 8432 df-1o 8490 df-er 8731 df-map 8863 df-pm 8864 df-en 8988 df-dom 8989 df-sdom 8990 df-fin 8991 df-fi 9416 df-sup 9447 df-inf 9448 df-pnf 11358 df-mnf 11359 df-xr 11360 df-ltxr 11361 df-le 11362 df-sub 11556 df-neg 11557 df-div 11987 df-nn 12349 df-2 12418 df-3 12419 df-4 12420 df-5 12421 df-6 12422 df-7 12423 df-8 12424 df-9 12425 df-n0 12620 df-z 12707 df-dec 12828 df-uz 12979 df-q 13089 df-rp 13134 df-xneg 13254 df-xadd 13255 df-xmul 13256 df-ioo 13493 df-icc 13496 df-fz 13653 df-seq 14158 df-exp 14218 df-cj 15281 df-re 15282 df-im 15283 df-sqrt 15417 df-abs 15418 df-struct 17340 df-slot 17375 df-ndx 17387 df-base 17403 df-plusg 17456 df-mulr 17457 df-starv 17458 df-tset 17462 df-ple 17463 df-ds 17465 df-unif 17466 df-rest 17608 df-topn 17609 df-topgen 17629 df-psmet 21685 df-xmet 21686 df-met 21687 df-bl 21688 df-mopn 21689 df-fbas 21690 df-fg 21691 df-cnfld 21694 df-top 23227 df-topon 23244 df-topsp 23266 df-bases 23279 df-cld 23352 df-ntr 23353 df-cls 23354 df-nei 23431 df-lp 23469 df-perf 23470 df-cn 23560 df-cnp 23561 df-haus 23648 df-fil 24180 df-fm 24272 df-flim 24273 df-flf 24274 df-xms 24654 df-ms 24655 df-cncf 25214 df-limc 26201 df-dv 26202 df-dvn 26203

This theorem is used by: taylthlem2 26716

W3C validator