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Theorem cnheibor 25019

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 24846	. . . 4 ⊢ 𝐽 ∈ Haus
3		simpl 486	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4		cnheibor.3	. . . . 5 ⊢ 𝑇 = (𝐽 ↾_t 𝑋)
5		simpr 488	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
6	4, 5	eqeltrrid 2869	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽 ↾_t 𝑋) ∈ Comp)
7	1	cnfldtopon 24844	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
8	7	toponunii 22978	. . . . 5 ⊢ ℂ = ∪ 𝐽
9	8	hauscmp 23469	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽 ↾_t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
10	2, 3, 6, 9	mp3an2i 1489	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
11	1	cnfldtop 24845	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12	8	restuni 23224	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
13	11, 3, 12	sylancr 596	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
14	4	unieqi 4879	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾_t 𝑋)
15	13, 14	eqtr4di 2817	. . . . . . . . 9 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ 𝑇)
16	15	eleq2d 2850	. . . . . . . 8 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑇))
17	16	biimpar 481	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → 𝑥 ∈ 𝑋)
18		cnex 11156	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
19		ssexg 5281	. . . . . . . . . . . 12 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
20	3, 18, 19	sylancl 595	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
21	20	adantr 484	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
22		cnxmet 24834	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
23		0cnd 11174	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ)
24	3	sselda 3938	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
25	24	abscld 15468	. . . . . . . . . . . . 13 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
26		peano2re 11358	. . . . . . . . . . . . 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 11234	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ^*)
29	1	cnfldtopn 24843	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
30	29	blopn 24562	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
31	22, 23, 28, 30	mp3an2i 1489	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32		elrestr 17459	. . . . . . . . . 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 1489	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
34	33, 4	eleqtrrdi 2875	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35		0cn 11173	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
36		eqid 2764	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
37	36	cnmetdval 24832	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
38	35, 37	mpan 700	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39		df-neg 11419	. . . . . . . . . . . . . . 15 ⊢ -𝑥 = (0 − 𝑥)
40	39	fveq2i 6872	. . . . . . . . . . . . . 14 ⊢ (abs‘-𝑥) = (abs‘(0 − 𝑥))
41		absneg 15306	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
42	40, 41	eqtr3id 2813	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
43	38, 42	eqtrd 2799	. . . . . . . . . . . 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 12124	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
46	44, 45	eqbrtrd 5124	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47		elbl 24450	. . . . . . . . . . 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 1489	. . . . . . . . . 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 723	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50		simpr 488	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
51	49, 50	elind 4154	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
52	24	absge0d 15476	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
53	25, 52	ge0p1rpd 13069	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ⁺)
54		eqid 2764	. . . . . . . . 9 ⊢ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55		oveq2 7406	. . . . . . . . . . 11 ⊢ (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
56	55	ineq1d 4173	. . . . . . . . . 10 ⊢ (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
57	56	rspceeqv 3606	. . . . . . . . 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 595	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59		eleq2 2853	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60		eqeq1 2768	. . . . . . . . . . 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 3188	. . . . . . . . . 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 641	. . . . . . . . 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 3583	. . . . . . . 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 847	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
65	17, 64	syldan 600	. . . . . 6 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
66	65	ralrimiva 3156	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67		eqid 2764	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
68		oveq2 7406	. . . . . . . 8 ⊢ (𝑟 = (𝑓‘𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓‘𝑢)))
69	68	ineq1d 4173	. . . . . . 7 ⊢ (𝑟 = (𝑓‘𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
70	69	eqeq2d 2775	. . . . . 6 ⊢ (𝑟 = (𝑓‘𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)))
71	67, 70	cmpcovf 23453	. . . . 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 593	. . . 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 742	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . 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 2799	. . . . . . . . . . . . 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 2850	. . . . . . . . . . . 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 4871	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑠 ↔ ∃𝑧 ∈ 𝑠 𝑥 ∈ 𝑧)
78	76, 77	bitrdi 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 4899	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑠 → 𝑧 ⊆ ∪ 𝑠)
80	79	ad2antrl 738	. . . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . . . . 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 796	. . . . . . . . . . . . . . . 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 3948	. . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . 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 3939	. . . . . . . . . . . . . 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 15468	. . . . . . . . . . . . 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 786	. . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . 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 780	. . . . . . . . . . . . . . . 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	ffvelcdmd 7068	. . . . . . . . . . . . . . 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 13039	. . . . . . . . . . . . . 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 6869	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
97	96	oveq2d 7414	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓‘𝑢)) = (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
98	97	ineq1d 4173	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
99	95, 98	eqeq12d 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋)))
100		simprr 782	. . . . . . . . . . . . . . . . . . . . 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 736	. . . . . . . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . . . . . . 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 2866	. . . . . . . . . . . . . . . . . 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 4158	. . . . . . . . . . . . . . . . 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 11174	. . . . . . . . . . . . . . . . . 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 13040	. . . . . . . . . . . . . . . . . 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 24450	. . . . . . . . . . . . . . . . . 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 1489	. . . . . . . . . . . . . . . . 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 499	. . . . . . . . . . . . . . 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 5126	. . . . . . . . . . . . . 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 5112	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) ≤ 𝑟 ↔ (𝑓‘𝑧) ≤ 𝑟))
113		simplrr 787	. . . . . . . . . . . . . . 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 3584	. . . . . . . . . . . . . 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 11345	. . . . . . . . . . . . 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 11333	. . . . . . . . . . . 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 3166	. . . . . . . . . . 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 3155	. . . . . . . . 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 785	. . . . . . . . . . 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 4159	. . . . . . . . . 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		ffvelcdm 7064	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ⁺)
123	122	rpred 13039	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ)
124	123	ralrimiva 3156	. . . . . . . . . . 11 ⊢ (𝑓:𝑠⟶ℝ⁺ → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ)
125	124	ad2antrl 738	. . . . . . . . . 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 12140	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Fin ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
127	121, 125, 126	syl2anc 593	. . . . . . . . 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 3180	. . . . . . . 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 416	. . . . . . 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 1955	. . . . . 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 457	. . . . 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 3165	. . . 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 519	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
135		eqid 2764	. . . . . 6 ⊢ (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136		eqid 2764	. . . . . 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 25018	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138	137	rexlimdvaa 3166	. . . 4 ⊢ (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟 → 𝑇 ∈ Comp))
139	138	imp 410	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140	139	adantl 485	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141	134, 140	impbida 810	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 399 = wceq 1562 ∃wex 1801 ∈ wcel 2144 ∀wral 3078 ∃wrex 3088 Vcvv 3456 ∩ cin 3905 ⊆ wss 3906 𝒫 cpw 4557 ∪ cuni 4867 class class class wbr 5102 × cxp 5647 “ cima 5652 ∘ ccom 5653 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ∈ cmpo 7400 Fincfn 8929 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 ici 11077 + caddc 11078 · cmul 11080 ℝ^*cxr 11217 < clt 11218 ≤ cle 11219 − cmin 11416 -cneg 11417 ℝ⁺crp 12995 [,]cicc 13354 abscabs 15263 ↾_t crest 17451 TopOpenctopn 17452 ∞Metcxmet 21411 ballcbl 21413 ℂ_fldccnfld 21426 Topctop 22955 Clsdccld 23078 Hauscha 23370 Compccmp 23448

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-se 5603 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-isom 6532 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-of 7662 df-om 7849 df-1st 7972 df-2nd 7973 df-supp 8143 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-2o 8440 df-er 8680 df-map 8812 df-ixp 8882 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fsupp 9310 df-fi 9359 df-sup 9390 df-inf 9391 df-oi 9460 df-card 9899 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-ioo 13355 df-icc 13358 df-fz 13515 df-fzo 13662 df-seq 14017 df-exp 14077 df-hash 14346 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17248 df-ress 17269 df-plusg 17301 df-mulr 17302 df-starv 17303 df-sca 17304 df-vsca 17305 df-ip 17306 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-hom 17312 df-cco 17313 df-rest 17453 df-topn 17454 df-0g 17472 df-gsum 17473 df-topgen 17474 df-pt 17475 df-prds 17478 df-xrs 17534 df-qtop 17539 df-imas 17540 df-xps 17542 df-mre 17616 df-mrc 17617 df-acs 17619 df-mgm 18676 df-sgrp 18755 df-mnd 18771 df-submnd 18820 df-mulg 19112 df-cntz 19359 df-cmn 19824 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-cls 23083 df-cn 23289 df-cnp 23290 df-haus 23377 df-cmp 23449 df-tx 23624 df-hmeo 23817 df-xms 24382 df-ms 24383 df-tms 24384 df-cncf 24942

This theorem is referenced by: cnllycmp 25020 cncmet 25386 ftalem3 27141

W3C validator