limcrecl - Mathbox for Glauco Siliprandi

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Theorem limcrecl 41903

Description: If 𝐹 is a real-valued function, 𝐵 is a limit point of its domain, and the limit of 𝐹 at 𝐵 exists, then this limit is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
limcrecl.1	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
limcrecl.2	⊢ (𝜑 → 𝐴 ⊆ ℂ)
limcrecl.3	⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝐴))
limcrecl.4	⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))

**Assertion**
Ref	Expression
limcrecl	⊢ (𝜑 → 𝐿 ∈ ℝ)

Proof of Theorem limcrecl Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		limcrecl.4	. . 3 ⊢ (𝜑 → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
2	1	adantr 483	. 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
3		limccl 24467	. . . . . . . . . 10 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
4	3, 1	sseldi 3964	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℂ)
5	4	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ ℂ)
6		simpr 487	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ 𝐿 ∈ ℝ)
7	5, 6	eldifd 3946	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ (ℂ ∖ ℝ))
8	7	dstregt0 41540	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)))
9		cnxmet 23375	. . . . . . . . . . . . . 14 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
11		limcrecl.2	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
12	11	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ⊆ ℂ)
13	12	ssdifssd 4118	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → (𝐴 ∖ {𝐵}) ⊆ ℂ)
14		limcrecl.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝐴))
15		eqid 2821	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16	15	cnfldtop 23386	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ Top
17	16	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
18		unicntop 23388	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
19	11, 18	sseqtrdi 4016	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ⊆ ∪ (TopOpen‘ℂ_fld))
20		eqid 2821	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
21	20	lpdifsn 21745	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐴 ⊆ ∪ (TopOpen‘ℂ_fld)) → (𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ↔ 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵}))))
22	17, 19, 21	syl2anc 586	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ↔ 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵}))))
23	14, 22	mpbid 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵})))
24	23	ad4antr 730	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵})))
25		simpr 487	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
26	15	cnfldtopn 23384	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
27	26	lpbl 23107	. . . . . . . . . . . . 13 ⊢ ((((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ (𝐴 ∖ {𝐵}) ⊆ ℂ ∧ 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵}))) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ (𝐴 ∖ {𝐵})𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
28	10, 13, 24, 25, 27	syl31anc 1369	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ (𝐴 ∖ {𝐵})𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
29		eldif 3945	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}))
30	29	anbi1i 625	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
31		anass 471	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))))
32	30, 31	bitri 277	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))))
33	32	rexbii2 3245	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (𝐴 ∖ {𝐵})𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
34	28, 33	sylib 220	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
35		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → ¬ 𝑧 ∈ {𝐵})
36		velsn 4576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
37	36	necon3bbii 3063	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ {𝐵} ↔ 𝑧 ≠ 𝐵)
38	35, 37	sylib 220	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝑧 ≠ 𝐵)
39		simp-5l 783	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝜑)
40		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝑦 ∈ ℝ⁺)
41		simprr 771	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
42		simp3 1134	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
43	9	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
44	18	lpss 21744	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐴 ⊆ ℂ) → ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ⊆ ℂ)
45	17, 11, 44	syl2anc 586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ⊆ ℂ)
46	45, 14	sseldd 3967	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ ℂ)
47	46	3ad2ant1 1129	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝐵 ∈ ℂ)
48		rpxr 12392	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
49	48	3ad2ant2 1130	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝑦 ∈ ℝ^*)
50		elbl 22992	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝐵 ∈ ℂ ∧ 𝑦 ∈ ℝ^*) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝐵(abs ∘ − )𝑧) < 𝑦)))
51	43, 47, 49, 50	syl3anc 1367	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝐵(abs ∘ − )𝑧) < 𝑦)))
52	42, 51	mpbid 234	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝑧 ∈ ℂ ∧ (𝐵(abs ∘ − )𝑧) < 𝑦))
53	52	simpld 497	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝑧 ∈ ℂ)
54	53, 47	abssubd 14807	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝑧 − 𝐵)) = (abs‘(𝐵 − 𝑧)))
55		eqid 2821	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs ∘ − ) = (abs ∘ − )
56	55	cnmetdval 23373	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐵(abs ∘ − )𝑧) = (abs‘(𝐵 − 𝑧)))
57	47, 53, 56	syl2anc 586	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐵(abs ∘ − )𝑧) = (abs‘(𝐵 − 𝑧)))
58	52	simprd 498	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐵(abs ∘ − )𝑧) < 𝑦)
59	57, 58	eqbrtrrd 5082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝐵 − 𝑧)) < 𝑦)
60	54, 59	eqbrtrd 5080	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝑧 − 𝐵)) < 𝑦)
61	39, 40, 41, 60	syl3anc 1367	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → (abs‘(𝑧 − 𝐵)) < 𝑦)
62	38, 61	jca 514	. . . . . . . . . . . . . . 15 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → (𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦))
63	62	adantlr 713	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → (𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦))
64	39	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝜑)
65		simplr 767	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝑧 ∈ 𝐴)
66	64, 65	jca 514	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → (𝜑 ∧ 𝑧 ∈ 𝐴))
67		simp-5r 784	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝑥 ∈ ℝ⁺)
68		simp-4r 782	. . . . . . . . . . . . . . 15 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)))
69		rpre 12391	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
70	69	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝑥 ∈ ℝ)
71		limcrecl.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
72	71	ffvelrnda 6845	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ)
73	72	recnd 10663	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
74	73	ad2antrr 724	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝐹‘𝑧) ∈ ℂ)
75	4	ad3antrrr 728	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝐿 ∈ ℂ)
76	74, 75	subcld 10991	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ((𝐹‘𝑧) − 𝐿) ∈ ℂ)
77	76	abscld 14790	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (abs‘((𝐹‘𝑧) − 𝐿)) ∈ ℝ)
78	72	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝐹‘𝑧) ∈ ℝ)
79		nfv 1911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤𝜑
80		nfra1 3219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))
81	79, 80	nfan 1896	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑤(𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)))
82		rspa 3206	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝐿 − 𝑤)))
83	82	adantll 712	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝐿 − 𝑤)))
84	4	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝐿 ∈ ℂ)
85		ax-resscn 10588	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ⊆ ℂ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ℝ ⊆ ℂ)
87	86	sselda 3966	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℂ)
88	84, 87	abssubd 14807	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (abs‘(𝐿 − 𝑤)) = (abs‘(𝑤 − 𝐿)))
89	88	adantlr 713	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → (abs‘(𝐿 − 𝑤)) = (abs‘(𝑤 − 𝐿)))
90	83, 89	breqtrd 5084	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝑤 − 𝐿)))
91	90	ex 415	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝑤 ∈ ℝ → 𝑥 < (abs‘(𝑤 − 𝐿))))
92	81, 91	ralrimi 3216	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝑤 − 𝐿)))
93	92	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝑤 − 𝐿)))
94		fvoveq1 7173	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (𝐹‘𝑧) → (abs‘(𝑤 − 𝐿)) = (abs‘((𝐹‘𝑧) − 𝐿)))
95	94	breq2d 5070	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = (𝐹‘𝑧) → (𝑥 < (abs‘(𝑤 − 𝐿)) ↔ 𝑥 < (abs‘((𝐹‘𝑧) − 𝐿))))
96	95	rspcv 3617	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑧) ∈ ℝ → (∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝑤 − 𝐿)) → 𝑥 < (abs‘((𝐹‘𝑧) − 𝐿))))
97	78, 93, 96	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝑥 < (abs‘((𝐹‘𝑧) − 𝐿)))
98	97	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝑥 < (abs‘((𝐹‘𝑧) − 𝐿)))
99	70, 77, 98	ltnsymd 10783	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ¬ (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)
100	66, 67, 68, 99	syl21anc 835	. . . . . . . . . . . . . 14 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → ¬ (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)
101	63, 100	jcnd 165	. . . . . . . . . . . . 13 ⊢ (((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → ¬ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
102	101	ex 415	. . . . . . . . . . . 12 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ 𝑧 ∈ 𝐴) → ((¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → ¬ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)))
103	102	reximdva 3274	. . . . . . . . . . 11 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → (∃𝑧 ∈ 𝐴 (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → ∃𝑧 ∈ 𝐴 ¬ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)))
104	34, 103	mpd 15	. . . . . . . . . 10 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ 𝐴 ¬ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
105		rexnal 3238	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐴 ¬ ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥) ↔ ¬ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
106	104, 105	sylib 220	. . . . . . . . 9 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → ¬ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
107	106	nrexdv 3270	. . . . . . . 8 ⊢ ((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ¬ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
108	107	ex 415	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) → (∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)) → ¬ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)))
109	108	reximdva 3274	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → (∃𝑥 ∈ ℝ⁺ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)) → ∃𝑥 ∈ ℝ⁺ ¬ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)))
110	8, 109	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ¬ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
111		rexnal 3238	. . . . 5 ⊢ (∃𝑥 ∈ ℝ⁺ ¬ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥) ↔ ¬ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
112	110, 111	sylib 220	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
113	112	intnand 491	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ (𝐿 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)))
114	71, 86	fssd 6522	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
115	114, 11, 46	ellimc3 24471	. . . 4 ⊢ (𝜑 → (𝐿 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐿 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))))
116	115	adantr 483	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → (𝐿 ∈ (𝐹 lim_ℂ 𝐵) ↔ (𝐿 ∈ ℂ ∧ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))))
117	113, 116	mtbird 327	. 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
118	2, 117	condan 816	1 ⊢ (𝜑 → 𝐿 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3932 ⊆ wss 3935 {csn 4560 ∪ cuni 4831 class class class wbr 5058 ∘ ccom 5553 ⟶wf 6345 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 ℝcr 10530 ℝ^*cxr 10668 < clt 10669 − cmin 10864 ℝ⁺crp 12383 abscabs 14587 TopOpenctopn 16689 ∞Metcxmet 20524 ballcbl 20526 ℂ_fldccnfld 20539 Topctop 21495 limPtclp 21736 lim_ℂ climc 24454

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-iin 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fi 8869 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-fz 12887 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-plusg 16572 df-mulr 16573 df-starv 16574 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-rest 16690 df-topn 16691 df-topgen 16711 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-cnp 21830 df-xms 22924 df-ms 22925 df-limc 24458

This theorem is referenced by: cncfiooiccre 42171 fourierdlem60 42445 fourierdlem61 42446 fourierdlem74 42459 fourierdlem75 42460 fourierdlem85 42470 fourierdlem88 42473 fourierdlem95 42480 fourierdlem103 42488 fourierdlem104 42489

W3C validator