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Theorem dvnfre 26080

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 6882	. . . . . 6 ⊢ (𝑥 = 0 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘0))
2	1	dmeqd 5896	. . . . . 6 ⊢ (𝑥 = 0 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘0))
3	1, 2	feq12d 6694	. . . . 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 6882	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑛))
6	5	dmeqd 5896	. . . . . 6 ⊢ (𝑥 = 𝑛 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑛))
7	5, 6	feq12d 6694	. . . . 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 6882	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
10	9	dmeqd 5896	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘(𝑛 + 1)))
11	9, 10	feq12d 6694	. . . . 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 6882	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((ℝ D^𝑛 𝐹)‘𝑥) = ((ℝ D^𝑛 𝐹)‘𝑁))
14	13	dmeqd 5896	. . . . . 6 ⊢ (𝑥 = 𝑁 → dom ((ℝ D^𝑛 𝐹)‘𝑥) = dom ((ℝ D^𝑛 𝐹)‘𝑁))
15	13, 14	feq12d 6694	. . . . 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 487	. . . . 5 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹:𝐴⟶ℝ)
18		ax-resscn 11157	. . . . . . 7 ⊢ ℝ ⊆ ℂ
19		fss 6723	. . . . . . . . 9 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
20	18, 19	mpan2 703	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → 𝐹:𝐴⟶ℂ)
21		cnex 11181	. . . . . . . . 9 ⊢ ℂ ∈ V
22		reex 11191	. . . . . . . . 9 ⊢ ℝ ∈ V
23		elpm2r 8842	. . . . . . . . 9 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	21, 22, 23	mpanl12 714	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
25	20, 24	sylan 591	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → 𝐹 ∈ (ℂ ↑_pm ℝ))
26		dvn0 26052	. . . . . . 7 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ)) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
27	18, 25, 26	sylancr 598	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → ((ℝ D^𝑛 𝐹)‘0) = 𝐹)
28	27	dmeqd 5896	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = dom 𝐹)
29		fdm 6716	. . . . . . . 8 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	29	adantr 485	. . . . . . 7 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom 𝐹 = 𝐴)
31	28, 30	eqtrd 2804	. . . . . 6 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) → dom ((ℝ D^𝑛 𝐹)‘0) = 𝐴)
32	27, 31	feq12d 6694	. . . . 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 784	. . . . . . . . 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 4733	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
36		simprl 782	. . . . . . . . . . . 12 ⊢ (((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ) ∧ (𝑛 ∈ ℕ₀ ∧ ((ℝ D^𝑛 𝐹)‘𝑛):dom ((ℝ D^𝑛 𝐹)‘𝑛)⟶ℝ)) → 𝑛 ∈ ℕ₀)
37		dvnbss 26056	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
38	35, 25, 36, 37	mp3an2ani 1494	. . . . . . . . . . 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 485	. . . . . . . . . . 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 3981	. . . . . . . . . 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 780	. . . . . . . . . 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 3955	. . . . . . . . 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 26079	. . . . . . . . 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 595	. . . . . . . 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 26053	. . . . . . . . . 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 1494	. . . . . . . . 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 5896	. . . . . . . . 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 6694	. . . . . . . 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 461	. . . . . 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 418	. . . . 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 12691	. . 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 1133	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 400 ∧ w3a 1101 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ⊆ wss 3913 {cpr 4596 dom cdm 5662 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_pm cpm 8825 ℂcc 11098 ℝcr 11099 0cc0 11100 1c1 11101 + caddc 11103 ℕ₀cn0 12504 D cdv 25991 D^𝑛 cdvn 25992

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9610 ax-cnex 11156 ax-resscn 11157 ax-1cn 11158 ax-icn 11159 ax-addcl 11160 ax-addrcl 11161 ax-mulcl 11162 ax-mulrcl 11163 ax-mulcom 11164 ax-addass 11165 ax-mulass 11166 ax-distr 11167 ax-i2m1 11168 ax-1ne0 11169 ax-1rid 11170 ax-rnegex 11171 ax-rrecex 11172 ax-cnre 11173 ax-pre-lttri 11174 ax-pre-lttrn 11175 ax-pre-ltadd 11176 ax-pre-mulgt0 11177 ax-pre-sup 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 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 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7863 df-1st 7986 df-2nd 7987 df-frecs 8278 df-wrecs 8309 df-recs 8358 df-rdg 8397 df-1o 8453 df-er 8694 df-map 8826 df-pm 8827 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-fi 9371 df-sup 9402 df-inf 9403 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11443 df-neg 11444 df-div 11872 df-nn 12234 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12505 df-z 12592 df-dec 12712 df-uz 12863 df-q 12973 df-rp 13017 df-xneg 13137 df-xadd 13138 df-xmul 13139 df-ioo 13376 df-icc 13379 df-fz 13536 df-seq 14038 df-exp 14098 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-struct 17207 df-slot 17242 df-ndx 17254 df-base 17270 df-plusg 17323 df-mulr 17324 df-starv 17325 df-tset 17329 df-ple 17330 df-ds 17332 df-unif 17333 df-rest 17475 df-topn 17476 df-topgen 17496 df-psmet 21483 df-xmet 21484 df-met 21485 df-bl 21486 df-mopn 21487 df-fbas 21488 df-fg 21489 df-cnfld 21492 df-top 23020 df-topon 23037 df-topsp 23059 df-bases 23072 df-cld 23145 df-ntr 23146 df-cls 23147 df-nei 23224 df-lp 23262 df-perf 23263 df-cn 23353 df-cnp 23354 df-haus 23441 df-fil 23972 df-fm 24064 df-flim 24065 df-flf 24066 df-xms 24446 df-ms 24447 df-cncf 25006 df-limc 25994 df-dv 25995 df-dvn 25996

This theorem is referenced by: taylthlem2 26503

W3C validator