islpcn - Mathbox for Glauco Siliprandi

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Theorem islpcn 42281

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 2798	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 23389	. . . 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 23391	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
7	6	islp2 21750	. . 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 1368	. 2 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
9		cnxmet 23378	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
10	9	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (abs ∘ − ) ∈ (∞Met‘ℂ))
11	5	adantr 484	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑃 ∈ ℂ)
12		simpr 488	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → 𝑒 ∈ ℝ⁺)
13	1	cnfldtopn 23387	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blnei 23109	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
15	10, 11, 12, 14	syl3anc 1368	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
16	15	adantlr 714	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
17		simplr 768	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
18		ineq1 4131	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → (𝑛 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
19	18	neeq1d 3046	. . . . . . . . 9 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
20	19	rspcva 3569	. . . . . . . 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 587	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
22		n0 4260	. . . . . . 7 ⊢ (((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
23	21, 22	sylib 221	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
24		elinel2 4123	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
25	24	adantl 485	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
26	4	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑆 ⊆ ℂ)
27	24	eldifad 3893	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
28	27	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ 𝑆)
29	26, 28	sseldd 3916	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑥 ∈ ℂ)
30	5	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
31	29, 30	abssubd 14805	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (abs‘(𝑃 − 𝑥)))
32		eqid 2798	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
33	32	cnmetdval 23376	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
34	30, 29, 33	syl2anc 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
35	31, 34	eqtr4d 2836	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
36	35	adantlr 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
37		elinel1 4122	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
38	37	adantl 485	. . . . . . . . . . . . 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 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑃 ∈ ℂ)
41		rpxr 12386	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ → 𝑒 ∈ ℝ^*)
42	41	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑒 ∈ ℝ^*)
43		elbl 22995	. . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
46	45	simprd 499	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) < 𝑒)
47	36, 46	eqbrtrd 5052	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) < 𝑒)
48	25, 47	jca 515	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
49	48	ex 416	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) → (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)))
50	49	adantlr 714	. . . . . . 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 1918	. . . . . 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 3112	. . . . 5 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ↔ ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
54	52, 53	sylibr 237	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
55	54	ralrimiva 3149	. . 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 23108	. . . . . . . 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 587	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}) ↔ (𝑛 ⊆ ℂ ∧ ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)))
59	58	simplbda 503	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})) → ∃𝑒 ∈ ℝ⁺ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
60	59	adantlr 714	. . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑒𝜑
62		nfra1 3183	. . . . . . . 8 ⊢ Ⅎ𝑒∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
63	61, 62	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑒(𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
64		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑒 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})
65	63, 64	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑒((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
66		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑒(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅
67		simp1l 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝜑)
68		simp2 1134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝑒 ∈ ℝ⁺)
69	67, 68	jca 515	. . . . . . . . 9 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝜑 ∧ 𝑒 ∈ ℝ⁺))
70		rspa 3171	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
71	70	adantll 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
72	71	3adant3 1129	. . . . . . . . 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 1135	. . . . . . . . 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 219	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
75	74	ad2antlr 726	. . . . . . . . . . 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 1915	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑒 ∈ ℝ⁺)
77		nfre1 3265	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
78	76, 77	nfan 1900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
79		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛
80	78, 79	nfan 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
81		simplr 768	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
82	4	adantr 484	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑆 ⊆ ℂ)
83		eldifi 4054	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆)
84	83	adantl 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
85	82, 84	sseldd 3916	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ ℂ)
86	85	adantrr 716	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ ℂ)
87	5	adantr 484	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑃 ∈ ℂ)
88	87, 85, 33	syl2anc 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
89	87, 85	abssubd 14805	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (abs‘(𝑃 − 𝑥)) = (abs‘(𝑥 − 𝑃)))
90	88, 89	eqtrd 2833	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
91	90	adantrr 716	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
92		simprr 772	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (abs‘(𝑥 − 𝑃)) < 𝑒)
93	91, 92	eqbrtrd 5052	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) < 𝑒)
94	86, 93	jca 515	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑥 ∈ ℂ ∧ (𝑃(abs ∘ − )𝑥) < 𝑒))
95	94	adantlr 714	. . . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑃 ∈ ℂ)
98	41	ad2antlr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑒 ∈ ℝ^*)
99	96, 97, 98, 43	syl3anc 1368	. . . . . . . . . . . . . . . . . 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 260	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
101	100	adantlr 714	. . . . . . . . . . . . . . . 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 3916	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ 𝑛)
103		simprl 770	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
104	102, 103	elind 4121	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
105	104	ex 416	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → ((𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒) → 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃}))))
106	105	adantlr 714	. . . . . . . . . . . 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 2214	. . . . . . . . . . 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 4260	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
110	108, 109	sylibr 237	. . . . . . . . 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 836	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
112	111	3exp 1116	. . . . . . 7 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → (𝑒 ∈ ℝ⁺ → ((𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛 → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)))
113	112	adantr 484	. . . . . 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 3276	. . . . 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 3149	. . 3 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
117	55, 116	impbida 800	. 2 ⊢ (𝜑 → (∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))
118	8, 117	bitrd 282	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 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ≠ wne 2987 ∀wral 3106 ∃wrex 3107 ∖ cdif 3878 ∩ cin 3880 ⊆ wss 3881 ∅c0 4243 {csn 4525 class class class wbr 5030 ∘ ccom 5523 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝ^*cxr 10663 < clt 10664 − cmin 10859 ℝ⁺crp 12377 abscabs 14585 TopOpenctopn 16687 ∞Metcxmet 20076 ballcbl 20078 ℂ_fldccnfld 20091 Topctop 21498 neicnei 21702 limPtclp 21739

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-fz 12886 df-seq 13365 df-exp 13426 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-plusg 16570 df-mulr 16571 df-starv 16572 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-topgen 16709 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-cnfld 20092 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-lp 21741 df-xms 22927 df-ms 22928

This theorem is referenced by: limclner 42293

W3C validator