islpcn - Mathbox for Glauco Siliprandi

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Theorem islpcn 43070

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 2738	. . . . 5 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
2	1	cnfldtop 23853	. . . 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 23855	. . . 4 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
7	6	islp2 22204	. . 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 1369	. 2 ⊢ (𝜑 → (𝑃 ∈ ((limPt‘(TopOpen‘ℂ_fld))‘𝑆) ↔ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
9		cnxmet 23842	. . . . . . . . . . 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 23851	. . . . . . . . . . 11 ⊢ (TopOpen‘ℂ_fld) = (MetOpen‘(abs ∘ − ))
14	13	blnei 23564	. . . . . . . . . 10 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 𝑃 ∈ ℂ ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
15	10, 11, 12, 14	syl3anc 1369	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
16	15	adantlr 711	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → (𝑃(ball‘(abs ∘ − ))𝑒) ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
17		simplr 765	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
18		ineq1 4136	. . . . . . . . . 10 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → (𝑛 ∩ (𝑆 ∖ {𝑃})) = ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
19	18	neeq1d 3002	. . . . . . . . 9 ⊢ (𝑛 = (𝑃(ball‘(abs ∘ − ))𝑒) → ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅))
20	19	rspcva 3550	. . . . . . . 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 583	. . . . . . 7 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
22		n0 4277	. . . . . . 7 ⊢ (((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
23	21, 22	sylib 217	. . . . . 6 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃})))
24		elinel2 4126	. . . . . . . . . . 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 3895	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 15093	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (abs‘(𝑃 − 𝑥)))
32		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
33	32	cnmetdval 23840	. . . . . . . . . . . . . 14 ⊢ ((𝑃 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
34	30, 29, 33	syl2anc 583	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
35	31, 34	eqtr4d 2781	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
36	35	adantlr 711	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → (abs‘(𝑥 − 𝑃)) = (𝑃(abs ∘ − )𝑥))
37		elinel1 4125	. . . . . . . . . . . . . 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 12668	. . . . . . . . . . . . . . 15 ⊢ (𝑒 ∈ ℝ⁺ → 𝑒 ∈ ℝ^*)
42	41	ad2antlr 723	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ 𝑥 ∈ ((𝑃(ball‘(abs ∘ − ))𝑒) ∩ (𝑆 ∖ {𝑃}))) → 𝑒 ∈ ℝ^*)
43		elbl 23449	. . . . . . . . . . . . . 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 1369	. . . . . . . . . . . . 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 231	. . . . . . . . . . . 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 5092	. . . . . . . . . 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 711	. . . . . . 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 1921	. . . . . 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 3069	. . . . 5 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ↔ ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
54	52, 53	sylibr 233	. . . 4 ⊢ (((𝜑 ∧ ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
55	54	ralrimiva 3107	. . 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 23563	. . . . . . . 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 583	. . . . . . 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 711	. . . . 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 1918	. . . . . . . 8 ⊢ Ⅎ𝑒𝜑
62		nfra1 3142	. . . . . . . 8 ⊢ Ⅎ𝑒∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
63	61, 62	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑒(𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
64		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑒 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})
65	63, 64	nfan 1903	. . . . . 6 ⊢ Ⅎ𝑒((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃}))
66		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑒(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅
67		simp1l 1195	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → 𝜑)
68		simp2 1135	. . . . . . . . . 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 3130	. . . . . . . . . . 11 ⊢ ((∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
71	70	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺) → ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
72	71	3adant3 1130	. . . . . . . . 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 1136	. . . . . . . . 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 215	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒 → ∃𝑥(𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒))
75	74	ad2antlr 723	. . . . . . . . . . 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 1918	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑒 ∈ ℝ⁺)
77		nfre1 3234	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒
78	76, 77	nfan 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒)
79		nfv 1918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥(𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛
80	78, 79	nfan 1903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛)
81		simplr 765	. . . . . . . . . . . . . . . 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 4057	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ (𝑆 ∖ {𝑃}) → 𝑥 ∈ 𝑆)
84	83	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ 𝑆)
85	82, 84	sseldd 3918	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → 𝑥 ∈ ℂ)
86	85	adantrr 713	. . . . . . . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑃 − 𝑥)))
89	87, 85	abssubd 15093	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (abs‘(𝑃 − 𝑥)) = (abs‘(𝑥 − 𝑃)))
90	88, 89	eqtrd 2778	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑆 ∖ {𝑃})) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
91	90	adantrr 713	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (𝑃(abs ∘ − )𝑥) = (abs‘(𝑥 − 𝑃)))
92		simprr 769	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → (abs‘(𝑥 − 𝑃)) < 𝑒)
93	91, 92	eqbrtrd 5092	. . . . . . . . . . . . . . . . . . . 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 711	. . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑒 ∈ ℝ^*)
99	96, 97, 98, 43	syl3anc 1369	. . . . . . . . . . . . . . . . . 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 256	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑃(ball‘(abs ∘ − ))𝑒))
101	100	adantlr 711	. . . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ 𝑛)
103		simprl 767	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ⁺) ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) ∧ (𝑥 ∈ (𝑆 ∖ {𝑃}) ∧ (abs‘(𝑥 − 𝑃)) < 𝑒)) → 𝑥 ∈ (𝑆 ∖ {𝑃}))
104	102, 103	elind 4124	. . . . . . . . . . . . . 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 711	. . . . . . . . . . . 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 2212	. . . . . . . . . . 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 4277	. . . . . . . . . 10 ⊢ ((𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (𝑛 ∩ (𝑆 ∖ {𝑃})))
110	108, 109	sylibr 233	. . . . . . . . 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 834	. . . . . . . 8 ⊢ (((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) ∧ 𝑒 ∈ ℝ⁺ ∧ (𝑃(ball‘(abs ∘ − ))𝑒) ⊆ 𝑛) → (𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
112	111	3exp 1117	. . . . . . 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 3107	. . 3 ⊢ ((𝜑 ∧ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒) → ∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅)
117	55, 116	impbida 797	. 2 ⊢ (𝜑 → (∀𝑛 ∈ ((nei‘(TopOpen‘ℂ_fld))‘{𝑃})(𝑛 ∩ (𝑆 ∖ {𝑃})) ≠ ∅ ↔ ∀𝑒 ∈ ℝ⁺ ∃𝑥 ∈ (𝑆 ∖ {𝑃})(abs‘(𝑥 − 𝑃)) < 𝑒))
118	8, 117	bitrd 278	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4253 {csn 4558 class class class wbr 5070 ∘ ccom 5584 ‘cfv 6418 (class class class)co 7255 ℂcc 10800 ℝ^*cxr 10939 < clt 10940 − cmin 11135 ℝ⁺crp 12659 abscabs 14873 TopOpenctopn 17049 ∞Metcxmet 20495 ballcbl 20497 ℂ_fldccnfld 20510 Topctop 21950 neicnei 22156 limPtclp 22193

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-pre-sup 10880

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-sup 9131 df-inf 9132 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-7 11971 df-8 11972 df-9 11973 df-n0 12164 df-z 12250 df-dec 12367 df-uz 12512 df-q 12618 df-rp 12660 df-xneg 12777 df-xadd 12778 df-xmul 12779 df-fz 13169 df-seq 13650 df-exp 13711 df-cj 14738 df-re 14739 df-im 14740 df-sqrt 14874 df-abs 14875 df-struct 16776 df-slot 16811 df-ndx 16823 df-base 16841 df-plusg 16901 df-mulr 16902 df-starv 16903 df-tset 16907 df-ple 16908 df-ds 16910 df-unif 16911 df-rest 17050 df-topn 17051 df-topgen 17071 df-psmet 20502 df-xmet 20503 df-met 20504 df-bl 20505 df-mopn 20506 df-cnfld 20511 df-top 21951 df-topon 21968 df-topsp 21990 df-bases 22004 df-cld 22078 df-ntr 22079 df-cls 22080 df-nei 22157 df-lp 22195 df-xms 23381 df-ms 23382

This theorem is referenced by: limclner 43082

W3C validator