limcrecl - Mathbox for Glauco Siliprandi

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Theorem limcrecl 45584

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 480	. 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
3		limccl 25924	. . . . . . . . . 10 ⊢ (𝐹 lim_ℂ 𝐵) ⊆ ℂ
4	3, 1	sselid 3992	. . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ ℂ)
5	4	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ ℂ)
6		simpr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ 𝐿 ∈ ℝ)
7	5, 6	eldifd 3973	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → 𝐿 ∈ (ℂ ∖ ℝ))
8	7	dstregt0 45231	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ∃𝑥 ∈ ℝ⁺ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)))
9		cnxmet 24808	. . . . . . . . . . . . . 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 732	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝐴 ⊆ ℂ)
13	12	ssdifssd 4156	. . . . . . . . . . . . 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 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
16	15	cnfldtop 24819	. . . . . . . . . . . . . . . . 17 ⊢ (TopOpen‘ℂ_fld) ∈ Top
17	16	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
18		unicntop 24821	. . . . . . . . . . . . . . . . 17 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
19	11, 18	sseqtrdi 4045	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ⊆ ∪ (TopOpen‘ℂ_fld))
20		eqid 2734	. . . . . . . . . . . . . . . . 17 ⊢ ∪ (TopOpen‘ℂ_fld) = ∪ (TopOpen‘ℂ_fld)
21	20	lpdifsn 23166	. . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ↔ 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵}))))
23	14, 22	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵})))
24	23	ad4antr 732	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝐵 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘(𝐴 ∖ {𝐵})))
25		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → 𝑦 ∈ ℝ⁺)
26	15	cnfldtopn 24817	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
27	26	lpbl 24531	. . . . . . . . . . . . 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 1372	. . . . . . . . . . . 12 ⊢ (((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) → ∃𝑧 ∈ (𝐴 ∖ {𝐵})𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))
29		eldif 3972	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ (𝐴 ∖ {𝐵}) ↔ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}))
30	29	anbi1i 624	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
31		anass 468	. . . . . . . . . . . . . 14 ⊢ (((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ∈ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))))
32	30, 31	bitri 275	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ (𝐴 ∖ {𝐵}) ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) ↔ (𝑧 ∈ 𝐴 ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))))
33	32	rexbii2 3087	. . . . . . . . . . . 12 ⊢ (∃𝑧 ∈ (𝐴 ∖ {𝐵})𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ ∃𝑧 ∈ 𝐴 (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)))
34	28, 33	sylib 218	. . . . . . . . . . 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 771	. . . . . . . . . . . . . . . . 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 4646	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ {𝐵} ↔ 𝑧 = 𝐵)
37	36	necon3bbii 2985	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑧 ∈ {𝐵} ↔ 𝑧 ≠ 𝐵)
38	35, 37	sylib 218	. . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝜑 ∧ ¬ 𝐿 ∈ ℝ) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑦 ∈ ℝ⁺) ∧ (¬ 𝑧 ∈ {𝐵} ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦))) → 𝜑)
40		simplr 769	. . . . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . 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 1137	. . . . . . . . . . . . . . . . . . . . 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 23165	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝐴 ⊆ ℂ) → ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ⊆ ℂ)
45	17, 11, 44	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → ((limPt‘(TopOpen‘ℂ_fld))‘𝐴) ⊆ ℂ)
46	45, 14	sseldd 3995	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → 𝐵 ∈ ℂ)
47	46	3ad2ant1 1132	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝐵 ∈ ℂ)
48		rpxr 13041	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ ℝ⁺ → 𝑦 ∈ ℝ^*)
49	48	3ad2ant2 1133	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝑦 ∈ ℝ^*)
50		elbl 24413	. . . . . . . . . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦) ↔ (𝑧 ∈ ℂ ∧ (𝐵(abs ∘ − )𝑧) < 𝑦)))
52	42, 51	mpbid 232	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝑧 ∈ ℂ ∧ (𝐵(abs ∘ − )𝑧) < 𝑦))
53	52	simpld 494	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → 𝑧 ∈ ℂ)
54	53, 47	abssubd 15488	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝑧 − 𝐵)) = (abs‘(𝐵 − 𝑧)))
55		eqid 2734	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (abs ∘ − ) = (abs ∘ − )
56	55	cnmetdval 24806	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐵 ∈ ℂ ∧ 𝑧 ∈ ℂ) → (𝐵(abs ∘ − )𝑧) = (abs‘(𝐵 − 𝑧)))
57	47, 53, 56	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐵(abs ∘ − )𝑧) = (abs‘(𝐵 − 𝑧)))
58	52	simprd 495	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (𝐵(abs ∘ − )𝑧) < 𝑦)
59	57, 58	eqbrtrrd 5171	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝐵 − 𝑧)) < 𝑦)
60	54, 59	eqbrtrd 5169	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ⁺ ∧ 𝑧 ∈ (𝐵(ball‘(abs ∘ − ))𝑦)) → (abs‘(𝑧 − 𝐵)) < 𝑦)
61	39, 40, 41, 60	syl3anc 1370	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . 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 511	. . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . 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 13040	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ ℝ⁺ → 𝑥 ∈ ℝ)
70	69	ad2antlr 727	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝑥 ∈ ℝ)
71		limcrecl.1	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
72	71	ffvelcdmda 7103	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℝ)
73	72	recnd 11286	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ ℂ)
74	73	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝐹‘𝑧) ∈ ℂ)
75	4	ad3antrrr 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝐿 ∈ ℂ)
76	74, 75	subcld 11617	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ((𝐹‘𝑧) − 𝐿) ∈ ℂ)
77	76	abscld 15471	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (abs‘((𝐹‘𝑧) − 𝐿)) ∈ ℝ)
78	72	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝐹‘𝑧) ∈ ℝ)
79		nfv 1911	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤𝜑
80		nfra1 3281	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑤∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))
81	79, 80	nfan 1896	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑤(𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)))
82		rspa 3245	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤)) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝐿 − 𝑤)))
83	82	adantll 714	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝐿 − 𝑤)))
84	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝐿 ∈ ℂ)
85		ax-resscn 11209	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ℝ ⊆ ℂ
86	85	a1i 11	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → ℝ ⊆ ℂ)
87	86	sselda 3994	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → 𝑤 ∈ ℂ)
88	84, 87	abssubd 15488	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (abs‘(𝐿 − 𝑤)) = (abs‘(𝑤 − 𝐿)))
89	88	adantlr 715	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → (abs‘(𝐿 − 𝑤)) = (abs‘(𝑤 − 𝐿)))
90	83, 89	breqtrd 5173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) ∧ 𝑤 ∈ ℝ) → 𝑥 < (abs‘(𝑤 − 𝐿)))
91	90	ex 412	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → (𝑤 ∈ ℝ → 𝑥 < (abs‘(𝑤 − 𝐿))))
92	81, 91	ralrimi 3254	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝑤 − 𝐿)))
93	92	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝑤 − 𝐿)))
94		fvoveq1 7453	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = (𝐹‘𝑧) → (abs‘(𝑤 − 𝐿)) = (abs‘((𝐹‘𝑧) − 𝐿)))
95	94	breq2d 5159	. . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → 𝑥 < (abs‘((𝐹‘𝑧) − 𝐿)))
99	70, 77, 98	ltnsymd 11407	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑧 ∈ 𝐴) ∧ 𝑥 ∈ ℝ⁺) ∧ ∀𝑤 ∈ ℝ 𝑥 < (abs‘(𝐿 − 𝑤))) → ¬ (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥)
100	66, 67, 68, 99	syl21anc 838	. . . . . . . . . . . . . 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 163	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 3165	. . . . . . . . . . 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 3097	. . . . . . . . . 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 218	. . . . . . . . 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 3146	. . . . . . . 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 412	. . . . . . 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 3165	. . . . . 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 3097	. . . . 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 218	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ ∀𝑥 ∈ ℝ⁺ ∃𝑦 ∈ ℝ⁺ ∀𝑧 ∈ 𝐴 ((𝑧 ≠ 𝐵 ∧ (abs‘(𝑧 − 𝐵)) < 𝑦) → (abs‘((𝐹‘𝑧) − 𝐿)) < 𝑥))
113	112	intnand 488	. . 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 6753	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
115	114, 11, 46	ellimc3 25928	. . . 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 480	. . 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 325	. 2 ⊢ ((𝜑 ∧ ¬ 𝐿 ∈ ℝ) → ¬ 𝐿 ∈ (𝐹 lim_ℂ 𝐵))
118	2, 117	condan 818	1 ⊢ (𝜑 → 𝐿 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ∖ cdif 3959 ⊆ wss 3962 {csn 4630 ∪ cuni 4911 class class class wbr 5147 ∘ ccom 5692 ⟶wf 6558 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 ℝ^*cxr 11291 < clt 11292 − cmin 11489 ℝ⁺crp 13031 abscabs 15269 TopOpenctopn 17467 ∞Metcxmet 21366 ballcbl 21368 ℂ_fldccnfld 21381 Topctop 22914 limPtclp 23157 lim_ℂ climc 25911

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-iin 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-map 8866 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-fi 9448 df-sup 9479 df-inf 9480 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-q 12988 df-rp 13032 df-xneg 13151 df-xadd 13152 df-xmul 13153 df-fz 13544 df-seq 14039 df-exp 14099 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-struct 17180 df-slot 17215 df-ndx 17227 df-base 17245 df-plusg 17310 df-mulr 17311 df-starv 17312 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-rest 17468 df-topn 17469 df-topgen 17489 df-psmet 21373 df-xmet 21374 df-met 21375 df-bl 21376 df-mopn 21377 df-cnfld 21382 df-top 22915 df-topon 22932 df-topsp 22954 df-bases 22968 df-cld 23042 df-ntr 23043 df-cls 23044 df-nei 23121 df-lp 23159 df-cnp 23251 df-xms 24345 df-ms 24346 df-limc 25915

This theorem is referenced by: cncfiooiccre 45850 fourierdlem60 46121 fourierdlem61 46122 fourierdlem74 46135 fourierdlem75 46136 fourierdlem85 46146 fourierdlem88 46149 fourierdlem95 46156 fourierdlem103 46164 fourierdlem104 46165

W3C validator