islpcn - Mathbox for Glauco Siliprandi

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Theorem islpcn 45991

Description: A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

**Hypotheses**
Ref	Expression
islpcn.s	⊢ (𝜑 → 𝑆 ⊆ ℂ)
islpcn.p	⊢ (𝜑 → 𝑃 ∈ ℂ)

**Assertion**
Ref	Expression
islpcn	⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))

Distinct variable groups: 𝑃,𝑒,𝑥 𝑆,𝑒,𝑥 𝜑,𝑒,𝑥

Proof of Theorem islpcn Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 24739	. . . 4 ⊢ (TopOpen‘ℂ_fld) ∈ Top
3	2	a1i 11	. . 3 ⊢ (𝜑 → (TopOpen‘ℂ_fld) ∈ Top)
4		islpcn.s	. . 3 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
5		islpcn.p	. . 3 ⊢ (𝜑 → 𝑃 ∈ ℂ)
6		unicntop 24741	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
7	6	islp2 23101	. . 3 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ 𝑆 ⊆ ℂ ∧ 𝑃 ∈ ℂ) → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
8	3, 4, 5, 7	syl3anc 1374	. 2 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
9		cnxmet 24728	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
11	5	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑃 ∈ ℂ)
12		simpr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
13	1	cnfldtopn 24737	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blnei 24458	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
15	10, 11, 12, 14	syl3anc 1374	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
16	15	adantlr 716	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
17		simplr 769	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
18		ineq1 4167	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → (𝑛 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
19	18	neeq1d 2992	. . . . . . . . 9 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
20	19	rspcva 3576	. . . . . . . 8 ⊢ (((𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}) ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
21	16, 17, 20	syl2anc 585	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
22		n0 4307	. . . . . . 7 ⊢ (((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
23	21, 22	sylib 218	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
24		elinel2 4156	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
25	24	adantl 481	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
26	4	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑆 ⊆ ℂ)
27	24	eldifad 3915	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
28	27	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ 𝑆)
29	26, 28	sseldd 3936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ ℂ)
30	5	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
31	29, 30	abssubd 15391	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (abs‘(𝑃 − 𝑥)))
32		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
33	32	cnmetdval 24726	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
34	30, 29, 33	syl2anc 585	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
35	31, 34	eqtr4d 2775	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
36	35	adantlr 716	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
37		elinel1 4155	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
38	37	adantl 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
39	9	a1i 11	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs ∘ − ) ∈ (∞Met‘ℂ))
40	11	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
41		rpxr 12927	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ → 𝑒 ∈ ℝ^*)
42	41	ad2antlr 728	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑒 ∈ ℝ^*)
43		elbl 24344	. . . . . . . . . . . . . 14 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ^*) → (𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒) ↔ (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒)))
44	39, 40, 42, 43	syl3anc 1374	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒) ↔ (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒)))
45	38, 44	mpbid 232	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
46	45	simprd 495	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) < 𝑒)
47	36, 46	eqbrtrd 5122	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) < 𝑒)
48	25, 47	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
49	48	ex 412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)))
50	49	adantlr 716	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)))
51	50	eximdv 1919	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)))
52	23, 51	mpd 15	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
53		df-rex 3063	. . . . 5 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ↔ ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
54	52, 53	sylibr 234	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
55	54	ralrimiva 3130	. . 3 ⊢ ((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) → ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
56	9	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (abs ∘ − ) ∈ (∞Met‘ℂ))
57	13	neibl 24457	. . . . . . . 8 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ) → (𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}) ↔ (𝑛 ⊆ ℂ ∧ ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)))
58	56, 5, 57	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}) ↔ (𝑛 ⊆ ℂ ∧ ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)))
59	58	simplbda 499	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
60	59	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
61		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑒𝜑
62		nfra1 3262	. . . . . . . 8 ⊢ Ⅎ𝑒∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
63	61, 62	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑒(𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
64		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑒 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})
65	63, 64	nfan 1901	. . . . . 6 ⊢ Ⅎ𝑒((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
66		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑒(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅
67		simp1l 1199	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝜑)
68		simp2 1138	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝑒 ∈ ℝ⁺)
69	67, 68	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝜑 ∧ 𝑒 ∈ ℝ⁺))
70		rspa 3227	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
71	70	adantll 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
72	71	3adant3 1133	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
73		simp3 1139	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
74	53	biimpi 216	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
75	74	ad2antlr 728	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
76		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑒 ∈ ℝ⁺)
77		nfre1 3263	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
78	76, 77	nfan 1901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
79		nfv 1916	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛
80	78, 79	nfan 1901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
81		simplr 769	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
82	4	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑆 ⊆ ℂ)
83		eldifi 4085	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆)
84	83	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
85	82, 84	sseldd 3936	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ ℂ)
86	85	adantrr 718	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ ℂ)
87	5	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑃 ∈ ℂ)
88	87, 85, 33	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
89	87, 85	abssubd 15391	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (abs‘(𝑃 − 𝑥)) = (abs‘(𝑥 − 𝑃)))
90	88, 89	eqtrd 2772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
91	90	adantrr 718	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
92		simprr 773	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (abs‘(𝑥 − 𝑃)) < 𝑒)
93	91, 92	eqbrtrd 5122	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) < 𝑒)
94	86, 93	jca 511	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
95	94	adantlr 716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
96	9	a1i 11	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (abs ∘ − ) ∈ (∞Met‘ℂ))
97	11	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑃 ∈ ℂ)
98	41	ad2antlr 728	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑒 ∈ ℝ^*)
99	96, 97, 98, 43	syl3anc 1374	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒) ↔ (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒)))
100	95, 99	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
101	100	adantlr 716	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
102	81, 101	sseldd 3936	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ 𝑛)
103		simprl 771	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
104	102, 103	elind 4154	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
105	104	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ((𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃}))))
106	105	adantlr 716	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ((𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃}))))
107	80, 106	eximd 2224	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒) → ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃}))))
108	75, 107	mpd 15	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
109		n0 4307	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
110	108, 109	sylibr 234	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
111	69, 72, 73, 110	syl21anc 838	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
112	111	3exp 1120	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → (𝑒 ∈ ℝ⁺ → ((𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛 → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))
113	112	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → (𝑒 ∈ ℝ⁺ → ((𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛 → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))
114	65, 66, 113	rexlimd 3245	. . . . 5 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → (∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛 → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
115	60, 114	mpd 15	. . . 4 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
116	115	ralrimiva 3130	. . 3 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
117	55, 116	impbida 801	. 2 ⊢ (𝜑 → (∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))
118	8, 117	bitrd 279	1 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∃wex 1781 ∈ wcel 2114 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 ∖ cdif 3900 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 {csn 4582 class class class wbr 5100 ∘ ccom 5636 ‘cfv 6500 (class class class)co 7368 ℂcc 11036 ℝ^*cxr 11177 < clt 11178 − cmin 11376 ℝ⁺crp 12917 abscabs 15169 TopOpenctopn 17353 ∞Metcxmet 21306 ballcbl 21308 ℂ_fldccnfld 21321 Topctop 22849 neicnei 23053 limPtclp 23090

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 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 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 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-sup 9357 df-inf 9358 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-fz 13436 df-seq 13937 df-exp 13997 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-struct 17086 df-slot 17121 df-ndx 17133 df-base 17149 df-plusg 17202 df-mulr 17203 df-starv 17204 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-rest 17354 df-topn 17355 df-topgen 17375 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-xms 24276 df-ms 24277

This theorem is referenced by: limclner 46003

W3C validator