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Theorem cnheibor 23551

Description: Heine-Borel theorem for complex numbers. A subset of ℂ is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)

**Hypotheses**
Ref	Expression
cnheibor.2	⊢ 𝐽 = (TopOpen‘ℂ_fld)
cnheibor.3	⊢ 𝑇 = (𝐽 ↾_t 𝑋)

**Assertion**
Ref	Expression
cnheibor	⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)))

Distinct variable groups: 𝑥,𝑟,𝑇 𝐽,𝑟,𝑥 𝑋,𝑟,𝑥

Proof of Theorem cnheibor Dummy variables 𝑧 𝑢 𝑓 𝑠 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnheibor.2	. . . . 5 ⊢ 𝐽 = (TopOpen‘ℂ_fld)
2	1	cnfldhaus 23385	. . . 4 ⊢ 𝐽 ∈ Haus
3		simpl 485	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4		cnheibor.3	. . . . 5 ⊢ 𝑇 = (𝐽 ↾_t 𝑋)
5		simpr 487	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
6	4, 5	eqeltrrid 2916	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽 ↾_t 𝑋) ∈ Comp)
7	1	cnfldtopon 23383	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
8	7	toponunii 21516	. . . . 5 ⊢ ℂ = ∪ 𝐽
9	8	hauscmp 22007	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽 ↾_t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
10	2, 3, 6, 9	mp3an2i 1460	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
11	1	cnfldtop 23384	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12	8	restuni 21762	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
13	11, 3, 12	sylancr 589	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
14	4	unieqi 4839	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾_t 𝑋)
15	13, 14	syl6eqr 2872	. . . . . . . . 9 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ 𝑇)
16	15	eleq2d 2896	. . . . . . . 8 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑇))
17	16	biimpar 480	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → 𝑥 ∈ 𝑋)
18		cnex 10610	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
19		ssexg 5218	. . . . . . . . . . . 12 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
20	3, 18, 19	sylancl 588	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
21	20	adantr 483	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
22		cnxmet 23373	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
23		0cnd 10626	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ)
24	3	sselda 3965	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
25	24	abscld 14788	. . . . . . . . . . . . 13 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
26		peano2re 10805	. . . . . . . . . . . . 13 ⊢ ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
27	25, 26	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
28	27	rexrd 10683	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ^*)
29	1	cnfldtopn 23382	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
30	29	blopn 23102	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
31	22, 23, 28, 30	mp3an2i 1460	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32		elrestr 16694	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ V ∧ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
33	11, 21, 31, 32	mp3an2i 1460	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
34	33, 4	eleqtrrdi 2922	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35		0cn 10625	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
36		eqid 2819	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
37	36	cnmetdval 23371	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
38	35, 37	mpan 688	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39		df-neg 10865	. . . . . . . . . . . . . . 15 ⊢ -𝑥 = (0 − 𝑥)
40	39	fveq2i 6666	. . . . . . . . . . . . . 14 ⊢ (abs‘-𝑥) = (abs‘(0 − 𝑥))
41		absneg 14629	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
42	40, 41	syl5eqr 2868	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
43	38, 42	eqtrd 2854	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
44	24, 43	syl 17	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
45	25	ltp1d 11562	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
46	44, 45	eqbrtrd 5079	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47		elbl 22990	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
48	22, 23, 28, 47	mp3an2i 1460	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))))
49	24, 46, 48	mpbir2and 711	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50		simpr 487	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
51	49, 50	elind 4169	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
52	24	absge0d 14796	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
53	25, 52	ge0p1rpd 12453	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ⁺)
54		eqid 2819	. . . . . . . . 9 ⊢ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55		oveq2 7156	. . . . . . . . . . 11 ⊢ (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
56	55	ineq1d 4186	. . . . . . . . . 10 ⊢ (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
57	56	rspceeqv 3636	. . . . . . . . 9 ⊢ ((((abs‘𝑥) + 1) ∈ ℝ⁺ ∧ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
58	53, 54, 57	sylancl 588	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59		eleq2 2899	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60		eqeq1 2823	. . . . . . . . . . 11 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
61	60	rexbidv 3295	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
62	59, 61	anbi12d 632	. . . . . . . . 9 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → ((𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)) ↔ (𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∧ ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))))
63	62	rspcev 3621	. . . . . . . 8 ⊢ ((((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇 ∧ (𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∧ ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
64	34, 51, 58, 63	syl12anc 834	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
65	17, 64	syldan 593	. . . . . 6 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
66	65	ralrimiva 3180	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67		eqid 2819	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
68		oveq2 7156	. . . . . . . 8 ⊢ (𝑟 = (𝑓‘𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓‘𝑢)))
69	68	ineq1d 4186	. . . . . . 7 ⊢ (𝑟 = (𝑓‘𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
70	69	eqeq2d 2830	. . . . . 6 ⊢ (𝑟 = (𝑓‘𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)))
71	67, 70	cmpcovf 21991	. . . . 5 ⊢ ((𝑇 ∈ Comp ∧ ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)(∪ 𝑇 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))))
72	5, 66, 71	syl2anc 586	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)(∪ 𝑇 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))))
73	15	ad4antr 730	. . . . . . . . . . . . . 14 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → 𝑋 = ∪ 𝑇)
74		simpllr 774	. . . . . . . . . . . . . 14 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → ∪ 𝑇 = ∪ 𝑠)
75	73, 74	eqtrd 2854	. . . . . . . . . . . . 13 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → 𝑋 = ∪ 𝑠)
76	75	eleq2d 2896	. . . . . . . . . . . 12 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑠))
77		eluni2 4834	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑠 ↔ ∃𝑧 ∈ 𝑠 𝑥 ∈ 𝑧)
78	76, 77	syl6bb 289	. . . . . . . . . . 11 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → (𝑥 ∈ 𝑋 ↔ ∃𝑧 ∈ 𝑠 𝑥 ∈ 𝑧))
79		elssuni 4859	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑠 → 𝑧 ⊆ ∪ 𝑠)
80	79	ad2antrl 726	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ⊆ ∪ 𝑠)
81	75	adantr 483	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑋 = ∪ 𝑠)
82	80, 81	sseqtrrd 4006	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ⊆ 𝑋)
83		simp-6l 785	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑋 ⊆ ℂ)
84	82, 83	sstrd 3975	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ⊆ ℂ)
85		simprr 771	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ 𝑧)
86	84, 85	sseldd 3966	. . . . . . . . . . . . . 14 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ ℂ)
87	86	abscld 14788	. . . . . . . . . . . . 13 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (abs‘𝑥) ∈ ℝ)
88		simplrl 775	. . . . . . . . . . . . 13 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑟 ∈ ℝ)
89		simprl 769	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → 𝑓:𝑠⟶ℝ⁺)
90	89	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑓:𝑠⟶ℝ⁺)
91		simprl 769	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑧 ∈ 𝑠)
92	90, 91	ffvelrnd 6845	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑓‘𝑧) ∈ ℝ⁺)
93	92	rpred 12423	. . . . . . . . . . . . . 14 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑓‘𝑧) ∈ ℝ)
94	86, 43	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
95		id 22	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
96		fveq2 6663	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
97	96	oveq2d 7164	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓‘𝑢)) = (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
98	97	ineq1d 4186	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
99	95, 98	eqeq12d 2835	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋)))
100		simprr 771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
101	100	ad2antrr 724	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
102	99, 101, 91	rspcdva 3623	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
103	85, 102	eleqtrd 2913	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
104	103	elin1d 4173	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
105		0cnd 10626	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → 0 ∈ ℂ)
106	92	rpxrd 12424	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑓‘𝑧) ∈ ℝ^*)
107		elbl 22990	. . . . . . . . . . . . . . . . . 18 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ (𝑓‘𝑧) ∈ ℝ^*) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓‘𝑧))))
108	22, 105, 106, 107	mp3an2i 1460	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑥 ∈ (0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ↔ (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓‘𝑧))))
109	104, 108	mpbid 234	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑥 ∈ ℂ ∧ (0(abs ∘ − )𝑥) < (𝑓‘𝑧)))
110	109	simprd 498	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (0(abs ∘ − )𝑥) < (𝑓‘𝑧))
111	94, 110	eqbrtrrd 5081	. . . . . . . . . . . . . 14 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (abs‘𝑥) < (𝑓‘𝑧))
112	96	breq1d 5067	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) ≤ 𝑟 ↔ (𝑓‘𝑧) ≤ 𝑟))
113		simplrr 776	. . . . . . . . . . . . . . 15 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
114	112, 113, 91	rspcdva 3623	. . . . . . . . . . . . . 14 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (𝑓‘𝑧) ≤ 𝑟)
115	87, 93, 88, 111, 114	ltletrd 10792	. . . . . . . . . . . . 13 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (abs‘𝑥) < 𝑟)
116	87, 88, 115	ltled 10780	. . . . . . . . . . . 12 ⊢ (((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) ∧ (𝑧 ∈ 𝑠 ∧ 𝑥 ∈ 𝑧)) → (abs‘𝑥) ≤ 𝑟)
117	116	rexlimdvaa 3283	. . . . . . . . . . 11 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → (∃𝑧 ∈ 𝑠 𝑥 ∈ 𝑧 → (abs‘𝑥) ≤ 𝑟))
118	78, 117	sylbid 242	. . . . . . . . . 10 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → (𝑥 ∈ 𝑋 → (abs‘𝑥) ≤ 𝑟))
119	118	ralrimiv 3179	. . . . . . . . 9 ⊢ ((((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)) → ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)
120		simpllr 774	. . . . . . . . . . 11 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → 𝑠 ∈ (𝒫 𝑇 ∩ Fin))
121	120	elin2d 4174	. . . . . . . . . 10 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → 𝑠 ∈ Fin)
122		ffvelrn 6842	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ⁺)
123	122	rpred 12423	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ)
124	123	ralrimiva 3180	. . . . . . . . . . 11 ⊢ (𝑓:𝑠⟶ℝ⁺ → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ)
125	124	ad2antrl 726	. . . . . . . . . 10 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ)
126		fimaxre3 11579	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Fin ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
127	121, 125, 126	syl2anc 586	. . . . . . . . 9 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
128	119, 127	reximddv 3273	. . . . . . . 8 ⊢ (((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) ∧ (𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)
129	128	ex 415	. . . . . . 7 ⊢ ((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) → ((𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
130	129	exlimdv 1928	. . . . . 6 ⊢ ((((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) ∧ ∪ 𝑇 = ∪ 𝑠) → (∃𝑓(𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
131	130	expimpd 456	. . . . 5 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑠 ∈ (𝒫 𝑇 ∩ Fin)) → ((∪ 𝑇 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
132	131	rexlimdva 3282	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (∃𝑠 ∈ (𝒫 𝑇 ∩ Fin)(∪ 𝑇 = ∪ 𝑠 ∧ ∃𝑓(𝑓:𝑠⟶ℝ⁺ ∧ ∀𝑢 ∈ 𝑠 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
133	72, 132	mpd 15	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)
134	10, 133	jca 514	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
135		eqid 2819	. . . . . 6 ⊢ (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136		eqid 2819	. . . . . 6 ⊢ ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟))) = ((𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) “ ((-𝑟[,]𝑟) × (-𝑟[,]𝑟)))
137	1, 4, 135, 136	cnheiborlem 23550	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138	137	rexlimdvaa 3283	. . . 4 ⊢ (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟 → 𝑇 ∈ Comp))
139	138	imp 409	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140	139	adantl 484	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141	134, 140	impbida 799	1 ⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1531 ∃wex 1774 ∈ wcel 2108 ∀wral 3136 ∃wrex 3137 Vcvv 3493 ∩ cin 3933 ⊆ wss 3934 𝒫 cpw 4537 ∪ cuni 4830 class class class wbr 5057 × cxp 5546 “ cima 5551 ∘ ccom 5552 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 ∈ cmpo 7150 Fincfn 8501 ℂcc 10527 ℝcr 10528 0cc0 10529 1c1 10530 ici 10531 + caddc 10532 · cmul 10534 ℝ^*cxr 10666 < clt 10667 ≤ cle 10668 − cmin 10862 -cneg 10863 ℝ⁺crp 12381 [,]cicc 12733 abscabs 14585 ↾_t crest 16686 TopOpenctopn 16687 ∞Metcxmet 20522 ballcbl 20524 ℂ_fldccnfld 20537 Topctop 21493 Clsdccld 21616 Hauscha 21908 Compccmp 21986

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 ax-pre-sup 10607 ax-addf 10608 ax-mulf 10609

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-se 5508 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-isom 6357 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-of 7401 df-om 7573 df-1st 7681 df-2nd 7682 df-supp 7823 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-map 8400 df-ixp 8454 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-fsupp 8826 df-fi 8867 df-sup 8898 df-inf 8899 df-oi 8966 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-div 11290 df-nn 11631 df-2 11692 df-3 11693 df-4 11694 df-5 11695 df-6 11696 df-7 11697 df-8 11698 df-9 11699 df-n0 11890 df-z 11974 df-dec 12091 df-uz 12236 df-q 12341 df-rp 12382 df-xneg 12499 df-xadd 12500 df-xmul 12501 df-ioo 12734 df-icc 12737 df-fz 12885 df-fzo 13026 df-seq 13362 df-exp 13422 df-hash 13683 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-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-mulg 18217 df-cntz 18439 df-cmn 18900 df-psmet 20529 df-xmet 20530 df-met 20531 df-bl 20532 df-mopn 20533 df-cnfld 20538 df-top 21494 df-topon 21511 df-topsp 21533 df-bases 21546 df-cld 21619 df-cls 21621 df-cn 21827 df-cnp 21828 df-haus 21915 df-cmp 21987 df-tx 22162 df-hmeo 22355 df-xms 22922 df-ms 22923 df-tms 22924 df-cncf 23478

This theorem is referenced by: cnllycmp 23552 cncmet 23917 ftalem3 25644

W3C validator