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Theorem cnheibor 25215

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 25042	. . . 4 ⊢ 𝐽 ∈ Haus
3		simpl 488	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4		cnheibor.3	. . . . 5 ⊢ 𝑇 = (𝐽 ↾_t 𝑋)
5		simpr 490	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
6	4, 5	eqeltrrid 2865	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽 ↾_t 𝑋) ∈ Comp)
7	1	cnfldtopon 25040	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
8	7	toponunii 23173	. . . . 5 ⊢ ℂ = ∪ 𝐽
9	8	hauscmp 23664	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽 ↾_t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
10	2, 3, 6, 9	mp3an2i 1495	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
11	1	cnfldtop 25041	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12	8	restuni 23419	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
13	11, 3, 12	sylancr 599	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
14	4	unieqi 4879	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾_t 𝑋)
15	13, 14	eqtr4di 2813	. . . . . . . . 9 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ 𝑇)
16	15	eleq2d 2846	. . . . . . . 8 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑇))
17	16	biimpar 483	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → 𝑥 ∈ 𝑋)
18		cnex 11230	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
19		ssexg 5284	. . . . . . . . . . . 12 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
20	3, 18, 19	sylancl 598	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
21	20	adantr 486	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
22		cnxmet 25030	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
23		0cnd 11248	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ)
24	3	sselda 3931	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
25	24	abscld 15551	. . . . . . . . . . . . 13 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
26		peano2re 11432	. . . . . . . . . . . . 13 ⊢ ((abs‘𝑥) ∈ ℝ → ((abs‘𝑥) + 1) ∈ ℝ)
27	25, 26	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ)
28	27	rexrd 11308	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ^*)
29	1	cnfldtopn 25039	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
30	29	blopn 24758	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
31	22, 23, 28, 30	mp3an2i 1495	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32		elrestr 17538	. . . . . . . . . 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 1495	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
34	33, 4	eleqtrrdi 2871	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35		0cn 11247	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
36		eqid 2760	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
37	36	cnmetdval 25028	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
38	35, 37	mpan 703	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39		df-neg 11493	. . . . . . . . . . . . . . 15 ⊢ -𝑥 = (0 − 𝑥)
40	39	fveq2i 6884	. . . . . . . . . . . . . 14 ⊢ (abs‘-𝑥) = (abs‘(0 − 𝑥))
41		absneg 15389	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
42	40, 41	eqtr3id 2809	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
43	38, 42	eqtrd 2795	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘𝑥))
44	24, 43	syl 18	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) = (abs‘𝑥))
45	25	ltp1d 12194	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
46	44, 45	eqbrtrd 5127	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47		elbl 24646	. . . . . . . . . . 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 1495	. . . . . . . . . 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 726	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50		simpr 490	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
51	49, 50	elind 4146	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
52	24	absge0d 15559	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
53	25, 52	ge0p1rpd 13141	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ⁺)
54		eqid 2760	. . . . . . . . 9 ⊢ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55		oveq2 7424	. . . . . . . . . . 11 ⊢ (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
56	55	ineq1d 4165	. . . . . . . . . 10 ⊢ (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
57	56	rspceeqv 3599	. . . . . . . . 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 598	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60		eqeq1 2764	. . . . . . . . . . 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 3186	. . . . . . . . . 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 644	. . . . . . . . 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 3576	. . . . . . . 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 850	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
65	17, 64	syldan 603	. . . . . 6 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
66	65	ralrimiva 3154	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67		eqid 2760	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
68		oveq2 7424	. . . . . . . 8 ⊢ (𝑟 = (𝑓‘𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓‘𝑢)))
69	68	ineq1d 4165	. . . . . . 7 ⊢ (𝑟 = (𝑓‘𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
70	69	eqeq2d 2771	. . . . . 6 ⊢ (𝑟 = (𝑓‘𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)))
71	67, 70	cmpcovf 23648	. . . . 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 596	. . . 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 745	. . . . . . . . . . . . . 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 788	. . . . . . . . . . . . . 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 2795	. . . . . . . . . . . . 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 2846	. . . . . . . . . . . 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 290	. . . . . . . . . . 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 741	. . . . . . . . . . . . . . . . 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 486	. . . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . . 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 799	. . . . . . . . . . . . . . . 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 3941	. . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . 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 3932	. . . . . . . . . . . . . 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 15551	. . . . . . . . . . . . 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 789	. . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . 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 7081	. . . . . . . . . . . . . . 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 13111	. . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → 𝑢 = 𝑧)
96		fveq2 6881	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
97	96	oveq2d 7432	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓‘𝑢)) = (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
98	97	ineq1d 4165	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
99	95, 98	eqeq12d 2776	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋)))
100		simprr 785	. . . . . . . . . . . . . . . . . . . . 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 739	. . . . . . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . . . . 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 4150	. . . . . . . . . . . . . . . . 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 11248	. . . . . . . . . . . . . . . . . 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 13112	. . . . . . . . . . . . . . . . . 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 24646	. . . . . . . . . . . . . . . . . 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 1495	. . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . 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 501	. . . . . . . . . . . . . . 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 5129	. . . . . . . . . . . . . 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 5113	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) ≤ 𝑟 ↔ (𝑓‘𝑧) ≤ 𝑟))
113		simplrr 790	. . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . 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 11419	. . . . . . . . . . . . 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 11407	. . . . . . . . . . . 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 3164	. . . . . . . . . . 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 243	. . . . . . . . . 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 3153	. . . . . . . . 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 788	. . . . . . . . . . 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 4151	. . . . . . . . . 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 7077	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ⁺)
123	122	rpred 13111	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ)
124	123	ralrimiva 3154	. . . . . . . . . . 11 ⊢ (𝑓:𝑠⟶ℝ⁺ → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ)
125	124	ad2antrl 741	. . . . . . . . . 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 12210	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Fin ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
127	121, 125, 126	syl2anc 596	. . . . . . . . 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 3178	. . . . . . . 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 418	. . . . . . 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 1966	. . . . . 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 459	. . . . 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 3163	. . . 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 16	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)
134	10, 133	jca 521	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
135		eqid 2760	. . . . . 6 ⊢ (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136		eqid 2760	. . . . . 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 25214	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138	137	rexlimdvaa 3164	. . . 4 ⊢ (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟 → 𝑇 ∈ Comp))
139	138	imp 412	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140	139	adantl 487	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141	134, 140	impbida 813	1 ⊢ (𝑋 ⊆ ℂ → (𝑇 ∈ Comp ↔ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∃wex 1812 ∈ wcel 2145 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∩ cin 3898 ⊆ wss 3899 𝒫 cpw 4557 ∪ cuni 4867 class class class wbr 5103 × cxp 5653 “ cima 5658 ∘ ccom 5659 ⟶wf 6531 ‘cfv 6535 (class class class)co 7416 ∈ cmpo 7418 Fincfn 8959 ℂcc 11147 ℝcr 11148 0cc0 11149 1c1 11150 ici 11151 + caddc 11152 · cmul 11154 ℝ^*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11490 -cneg 11491 ℝ⁺crp 13067 [,]cicc 13426 abscabs 15346 ↾_t crest 17530 TopOpenctopn 17531 ∞Metcxmet 21602 ballcbl 21604 ℂ_fldccnfld 21617 Topctop 23150 Clsdccld 23273 Hauscha 23565 Compccmp 23643

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 ax-pre-sup 11227 ax-addf 11228

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 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 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6301 df-ord 6362 df-on 6363 df-lim 6364 df-suc 6365 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-isom 6544 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-1o 8462 df-2o 8463 df-er 8703 df-map 8835 df-ixp 8912 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-fsupp 9339 df-fi 9388 df-sup 9419 df-inf 9420 df-oi 9489 df-card 9969 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11492 df-neg 11493 df-div 11921 df-nn 12283 df-2 12352 df-3 12353 df-4 12354 df-5 12355 df-6 12356 df-7 12357 df-8 12358 df-9 12359 df-n0 12554 df-z 12641 df-dec 12762 df-uz 12913 df-q 13023 df-rp 13068 df-xneg 13188 df-xadd 13189 df-xmul 13190 df-ioo 13427 df-icc 13430 df-fz 13587 df-fzo 13735 df-seq 14091 df-exp 14151 df-hash 14420 df-cj 15211 df-re 15212 df-im 15213 df-sqrt 15347 df-abs 15348 df-struct 17264 df-sets 17281 df-slot 17299 df-ndx 17311 df-base 17327 df-ress 17348 df-plusg 17380 df-mulr 17381 df-starv 17382 df-sca 17383 df-vsca 17384 df-ip 17385 df-tset 17386 df-ple 17387 df-ds 17389 df-unif 17390 df-hom 17391 df-cco 17392 df-rest 17532 df-topn 17533 df-0g 17551 df-gsum 17552 df-topgen 17553 df-pt 17554 df-prds 17557 df-xrs 17613 df-qtop 17618 df-imas 17619 df-xps 17621 df-mre 17695 df-mrc 17696 df-acs 17698 df-mgm 18755 df-sgrp 18847 df-mnd 18863 df-submnd 18918 df-mulg 19217 df-cntz 19470 df-cmn 19935 df-psmet 21609 df-xmet 21610 df-met 21611 df-bl 21612 df-mopn 21613 df-cnfld 21618 df-top 23151 df-topon 23168 df-topsp 23190 df-bases 23203 df-cld 23276 df-cls 23278 df-cn 23484 df-cnp 23485 df-haus 23572 df-cmp 23644 df-tx 23820 df-hmeo 24013 df-xms 24578 df-ms 24579 df-tms 24580 df-cncf 25138

This theorem is used by: cnllycmp 25216 cncmet 25582 ftalem3 27343

W3C validator