islpcn - Mathbox for Glauco Siliprandi

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Theorem islpcn 46151

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 2752	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 24812	. . . 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 24814	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
7	6	islp2 23174	. . 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 1382	. 2 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
9		cnxmet 24801	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
11	5	adantr 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑃 ∈ ℂ)
12		simpr 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
13	1	cnfldtopn 24810	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blnei 24531	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
15	10, 11, 12, 14	syl3anc 1382	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
16	15	adantlr 723	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
17		simplr 776	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
18		ineq1 4156	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → (𝑛 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
19	18	neeq1d 3006	. . . . . . . . 9 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
20	19	rspcva 3570	. . . . . . . 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 592	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
22		n0 4296	. . . . . . 7 ⊢ (((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
23	21, 22	sylib 220	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
24		elinel2 4145	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
25	24	adantl 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
26	4	adantr 483	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑆 ⊆ ℂ)
27	24	eldifad 3907	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
28	27	adantl 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ 𝑆)
29	26, 28	sseldd 3928	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ ℂ)
30	5	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
31	29, 30	abssubd 15455	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (abs‘(𝑃 − 𝑥)))
32		eqid 2752	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
33	32	cnmetdval 24799	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
34	30, 29, 33	syl2anc 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
35	31, 34	eqtr4d 2790	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
36	35	adantlr 723	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
37		elinel1 4144	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
38	37	adantl 484	. . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
41		rpxr 12989	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ → 𝑒 ∈ ℝ^*)
42	41	ad2antlr 735	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑒 ∈ ℝ^*)
43		elbl 24417	. . . . . . . . . . . . . 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 1382	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
46	45	simprd 498	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) < 𝑒)
47	36, 46	eqbrtrd 5112	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) < 𝑒)
48	25, 47	jca 518	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
49	48	ex 415	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)))
50	49	adantlr 723	. . . . . . 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 1927	. . . . . 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 3077	. . . . 5 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ↔ ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
54	52, 53	sylibr 236	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
55	54	ralrimiva 3144	. . 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 24530	. . . . . . . 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 592	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}) ↔ (𝑛 ⊆ ℂ ∧ ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)))
59	58	simplbda 502	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
60	59	adantlr 723	. . . . 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 1924	. . . . . . . 8 ⊢ Ⅎ𝑒𝜑
62		nfra1 3276	. . . . . . . 8 ⊢ Ⅎ𝑒∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
63	61, 62	nfan 1909	. . . . . . 7 ⊢ Ⅎ𝑒(𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
64		nfv 1924	. . . . . . 7 ⊢ Ⅎ𝑒 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})
65	63, 64	nfan 1909	. . . . . 6 ⊢ Ⅎ𝑒((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
66		nfv 1924	. . . . . 6 ⊢ Ⅎ𝑒(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅
67		simp1l 1207	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝜑)
68		simp2 1146	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝑒 ∈ ℝ⁺)
69	67, 68	jca 518	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝜑 ∧ 𝑒 ∈ ℝ⁺))
70		rspa 3241	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
71	70	adantll 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
72	71	3adant3 1141	. . . . . . . . 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 1147	. . . . . . . . 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 218	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
75	74	ad2antlr 735	. . . . . . . . . . 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 1924	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑒 ∈ ℝ⁺)
77		nfre1 3277	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
78	76, 77	nfan 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
79		nfv 1924	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛
80	78, 79	nfan 1909	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
81		simplr 776	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
82	4	adantr 483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑆 ⊆ ℂ)
83		eldifi 4075	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆)
84	83	adantl 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
85	82, 84	sseldd 3928	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ ℂ)
86	85	adantrr 725	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ ℂ)
87	5	adantr 483	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑃 ∈ ℂ)
88	87, 85, 33	syl2anc 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
89	87, 85	abssubd 15455	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (abs‘(𝑃 − 𝑥)) = (abs‘(𝑥 − 𝑃)))
90	88, 89	eqtrd 2787	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
91	90	adantrr 725	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
92		simprr 780	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (abs‘(𝑥 − 𝑃)) < 𝑒)
93	91, 92	eqbrtrd 5112	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) < 𝑒)
94	86, 93	jca 518	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
95	94	adantlr 723	. . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑃 ∈ ℂ)
98	41	ad2antlr 735	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑒 ∈ ℝ^*)
99	96, 97, 98, 43	syl3anc 1382	. . . . . . . . . . . . . . . . . 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 259	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
101	100	adantlr 723	. . . . . . . . . . . . . . . 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 3928	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ 𝑛)
103		simprl 778	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
104	102, 103	elind 4143	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
105	104	ex 415	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ((𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃}))))
106	105	adantlr 723	. . . . . . . . . . . 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 2241	. . . . . . . . . . 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 4296	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
110	108, 109	sylibr 236	. . . . . . . . 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 846	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
112	111	3exp 1128	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → (𝑒 ∈ ℝ⁺ → ((𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛 → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))
113	112	adantr 483	. . . . . 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 3259	. . . . 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 3144	. . 3 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
117	55, 116	impbida 808	. 2 ⊢ (𝜑 → (∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))
118	8, 117	bitrd 281	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 208 ∧ wa 398 ∧ w3a 1095 = wceq 1550 ∃wex 1789 ∈ wcel 2132 ≠ wne 2947 ∀wral 3066 ∃wrex 3076 ∖ cdif 3892 ∩ cin 3894 ⊆ wss 3895 ∅c0 4276 {csn 4572 class class class wbr 5090 ∘ ccom 5640 ‘cfv 6506 (class class class)co 7381 ℂcc 11057 ℝ^*cxr 11201 < clt 11202 − cmin 11400 ℝ⁺crp 12979 abscabs 15233 TopOpenctopn 17422 ∞Metcxmet 21378 ballcbl 21380 ℂ_fldccnfld 21393 Topctop 22922 neicnei 23126 limPtclp 23163

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137

This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3or 1096 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-nel 3052 df-ral 3067 df-rex 3077 df-rmo 3357 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-pss 3915 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4856 df-int 4896 df-iun 4941 df-iin 4942 df-br 5091 df-opab 5153 df-mpt 5172 df-tr 5198 df-id 5531 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5589 df-we 5591 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-pred 6273 df-ord 6334 df-on 6335 df-lim 6336 df-suc 6337 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-riota 7338 df-ov 7384 df-oprab 7385 df-mpo 7386 df-om 7832 df-1st 7955 df-2nd 7956 df-frecs 8246 df-wrecs 8277 df-recs 8326 df-rdg 8365 df-1o 8421 df-er 8662 df-map 8794 df-en 8913 df-dom 8914 df-sdom 8915 df-fin 8916 df-sup 9374 df-inf 9375 df-pnf 11204 df-mnf 11205 df-xr 11206 df-ltxr 11207 df-le 11208 df-sub 11402 df-neg 11403 df-div 11831 df-nn 12197 df-2 12266 df-3 12267 df-4 12268 df-5 12269 df-6 12270 df-7 12271 df-8 12272 df-9 12273 df-n0 12468 df-z 12555 df-dec 12675 df-uz 12826 df-q 12936 df-rp 12980 df-xneg 13100 df-xadd 13101 df-xmul 13102 df-fz 13499 df-seq 14001 df-exp 14061 df-cj 15098 df-re 15099 df-im 15100 df-sqrt 15234 df-abs 15235 df-struct 17155 df-slot 17190 df-ndx 17202 df-base 17218 df-plusg 17271 df-mulr 17272 df-starv 17273 df-tset 17277 df-ple 17278 df-ds 17280 df-unif 17281 df-rest 17423 df-topn 17424 df-topgen 17444 df-psmet 21385 df-xmet 21386 df-met 21387 df-bl 21388 df-mopn 21389 df-cnfld 21394 df-top 22923 df-topon 22940 df-topsp 22962 df-bases 22975 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-xms 24349 df-ms 24350

This theorem is referenced by: limclner 46163

W3C validator