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Theorem dvnfre 25921

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 6842	. . . . . 6 ⊢ (𝑥 = 0 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘0))
2	1	dmeqd 5862	. . . . . 6 ⊢ (𝑥 = 0 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘0))
3	1, 2	feq12d 6658	. . . . 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 340	. . . 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 6842	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑛))
6	5	dmeqd 5862	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑛))
7	5, 6	feq12d 6658	. . . . 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 340	. . . 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 6842	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
10	9	dmeqd 5862	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
11	9, 10	feq12d 6658	. . . . 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 340	. . . 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 6842	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑁))
14	13	dmeqd 5862	. . . . . 6 ⊢ (𝑥 = 𝑁 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑁))
15	13, 14	feq12d 6658	. . . . 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 340	. . . 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 482	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
18		ax-resscn 11097	. . . . . . 7 ⊢ ℝ ⊆ ℂ
19		fss 6686	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
20	18, 19	mpan2 692	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ)
21		cnex 11121	. . . . . . . . 9 ⊢ ℂ ∈ V
22		reex 11131	. . . . . . . . 9 ⊢ ℝ ∈ V
23		elpm2r 8794	. . . . . . . . 9 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	21, 22, 23	mpanl12 703	. . . . . . . 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 25893	. . . . . . 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 5862	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = dom 𝐹)
29		fdm 6679	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom 𝐹 = 𝐴)
31	28, 30	eqtrd 2772	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = 𝐴)
32	27, 31	feq12d 6658	. . . . 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 773	. . . . . . . . 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 4707	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
36		simprl 771	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝑛 ∈ ℕ₀)
37		dvnbss 25897	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
38	35, 25, 36, 37	mp3an2ani 1471	. . . . . . . . . . 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 480	. . . . . . . . . . 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 3959	. . . . . . . . . 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 769	. . . . . . . . . 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 3933	. . . . . . . . 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 25920	. . . . . . . . 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 25894	. . . . . . . . . 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 1471	. . . . . . . . 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 5862	. . . . . . . . 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 6658	. . . . . . . 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 456	. . . . . 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 413	. . . . 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 12626	. . 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 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ⊆ wss 3890 {cpr 4570 dom cdm 5632 ⟶wf 6496 ‘cfv 6500 (class class class)co 7369 ↑_pm cpm 8776 ℂcc 11038 ℝcr 11039 0cc0 11040 1c1 11041 + caddc 11043 ℕ₀cn0 12439 D cdv 25832 D^𝑛 cdvn 25833

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5308 ax-pr 5376 ax-un 7691 ax-inf2 9564 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7820 df-1st 7944 df-2nd 7945 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-pm 8778 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fi 9326 df-sup 9357 df-inf 9358 df-pnf 11183 df-mnf 11184 df-xr 11185 df-ltxr 11186 df-le 11187 df-sub 11381 df-neg 11382 df-div 11810 df-nn 12177 df-2 12246 df-3 12247 df-4 12248 df-5 12249 df-6 12250 df-7 12251 df-8 12252 df-9 12253 df-n0 12440 df-z 12527 df-dec 12647 df-uz 12791 df-q 12901 df-rp 12945 df-xneg 13065 df-xadd 13066 df-xmul 13067 df-ioo 13304 df-icc 13307 df-fz 13464 df-seq 13966 df-exp 14026 df-cj 15063 df-re 15064 df-im 15065 df-sqrt 15199 df-abs 15200 df-struct 17119 df-slot 17154 df-ndx 17166 df-base 17182 df-plusg 17235 df-mulr 17236 df-starv 17237 df-tset 17241 df-ple 17242 df-ds 17244 df-unif 17245 df-rest 17387 df-topn 17388 df-topgen 17408 df-psmet 21346 df-xmet 21347 df-met 21348 df-bl 21349 df-mopn 21350 df-fbas 21351 df-fg 21352 df-cnfld 21355 df-top 22861 df-topon 22878 df-topsp 22900 df-bases 22913 df-cld 22986 df-ntr 22987 df-cls 22988 df-nei 23065 df-lp 23103 df-perf 23104 df-cn 23194 df-cnp 23195 df-haus 23282 df-fil 23813 df-fm 23905 df-flim 23906 df-flf 23907 df-xms 24287 df-ms 24288 df-cncf 24847 df-limc 25835 df-dv 25836 df-dvn 25837

This theorem is referenced by: taylthlem2 26341

W3C validator