islpcn - Mathbox for Glauco Siliprandi

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Theorem islpcn 45637

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 2729	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 24671	. . . 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 24673	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
7	6	islp2 23032	. . 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 1373	. 2 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
9		cnxmet 24660	. . . . . . . . . . 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 24669	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blnei 24390	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
15	10, 11, 12, 14	syl3anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
16	15	adantlr 715	. . . . . . . 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 4176	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → (𝑛 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
19	18	neeq1d 2984	. . . . . . . . 9 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
20	19	rspcva 3586	. . . . . . . 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 584	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
22		n0 4316	. . . . . . 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 4165	. . . . . . . . . . 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 3926	. . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . 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 15422	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (abs‘(𝑃 − 𝑥)))
32		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
33	32	cnmetdval 24658	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
34	30, 29, 33	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
35	31, 34	eqtr4d 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
36	35	adantlr 715	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
37		elinel1 4164	. . . . . . . . . . . . . 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 12961	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ → 𝑒 ∈ ℝ^*)
42	41	ad2antlr 727	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑒 ∈ ℝ^*)
43		elbl 24276	. . . . . . . . . . . . . 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 1373	. . . . . . . . . . . . 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 5129	. . . . . . . . . 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 715	. . . . . . 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 1917	. . . . . 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 3054	. . . . 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 3125	. . 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 24389	. . . . . . . 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 584	. . . . . . 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 715	. . . . 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 1914	. . . . . . . 8 ⊢ Ⅎ𝑒𝜑
62		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑒∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
63	61, 62	nfan 1899	. . . . . . 7 ⊢ Ⅎ𝑒(𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
64		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑒 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})
65	63, 64	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑒((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
66		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑒(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅
67		simp1l 1198	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝜑)
68		simp2 1137	. . . . . . . . . 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 3226	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
71	70	adantll 714	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
72	71	3adant3 1132	. . . . . . . . 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 1138	. . . . . . . . 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 727	. . . . . . . . . . 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 1914	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑒 ∈ ℝ⁺)
77		nfre1 3262	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
78	76, 77	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
79		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛
80	78, 79	nfan 1899	. . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑆 ⊆ ℂ)
83		eldifi 4094	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆)
84	83	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
85	82, 84	sseldd 3947	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ ℂ)
86	85	adantrr 717	. . . . . . . . . . . . . . . . . . . 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 584	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
89	87, 85	abssubd 15422	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (abs‘(𝑃 − 𝑥)) = (abs‘(𝑥 − 𝑃)))
90	88, 89	eqtrd 2764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
91	90	adantrr 717	. . . . . . . . . . . . . . . . . . . . 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 5129	. . . . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . . . 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 727	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑒 ∈ ℝ^*)
99	96, 97, 98, 43	syl3anc 1373	. . . . . . . . . . . . . . . . . 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 715	. . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . 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 4163	. . . . . . . . . . . . . 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 715	. . . . . . . . . . . 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 2217	. . . . . . . . . . 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 4316	. . . . . . . . . 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 837	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
112	111	3exp 1119	. . . . . . 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 3244	. . . . 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 3125	. . 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 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 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 ∖ cdif 3911 ∩ cin 3913 ⊆ wss 3914 ∅c0 4296 {csn 4589 class class class wbr 5107 ∘ ccom 5642 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝ^*cxr 11207 < clt 11208 − cmin 11405 ℝ⁺crp 12951 abscabs 15200 TopOpenctopn 17384 ∞Metcxmet 21249 ballcbl 21251 ℂ_fldccnfld 21264 Topctop 22780 neicnei 22984 limPtclp 23021

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4872 df-int 4911 df-iun 4957 df-iin 4958 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-1st 7968 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-1o 8434 df-er 8671 df-map 8801 df-en 8919 df-dom 8920 df-sdom 8921 df-fin 8922 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-4 12251 df-5 12252 df-6 12253 df-7 12254 df-8 12255 df-9 12256 df-n0 12443 df-z 12530 df-dec 12650 df-uz 12794 df-q 12908 df-rp 12952 df-xneg 13072 df-xadd 13073 df-xmul 13074 df-fz 13469 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-struct 17117 df-slot 17152 df-ndx 17164 df-base 17180 df-plusg 17233 df-mulr 17234 df-starv 17235 df-tset 17239 df-ple 17240 df-ds 17242 df-unif 17243 df-rest 17385 df-topn 17386 df-topgen 17406 df-psmet 21256 df-xmet 21257 df-met 21258 df-bl 21259 df-mopn 21260 df-cnfld 21265 df-top 22781 df-topon 22798 df-topsp 22820 df-bases 22833 df-cld 22906 df-ntr 22907 df-cls 22908 df-nei 22985 df-lp 23023 df-xms 24208 df-ms 24209

This theorem is referenced by: limclner 45649

W3C validator