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Theorem cnheibor 23559

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 23393	. . . 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 2918	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽 ↾_t 𝑋) ∈ Comp)
7	1	cnfldtopon 23391	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
8	7	toponunii 21524	. . . . 5 ⊢ ℂ = ∪ 𝐽
9	8	hauscmp 22015	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽 ↾_t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
10	2, 3, 6, 9	mp3an2i 1462	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
11	1	cnfldtop 23392	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12	8	restuni 21770	. . . . . . . . . . 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 4851	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾_t 𝑋)
15	13, 14	syl6eqr 2874	. . . . . . . . 9 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ 𝑇)
16	15	eleq2d 2898	. . . . . . . 8 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑇))
17	16	biimpar 480	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → 𝑥 ∈ 𝑋)
18		cnex 10618	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
19		ssexg 5227	. . . . . . . . . . . 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 23381	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
23		0cnd 10634	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ)
24	3	sselda 3967	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
25	24	abscld 14796	. . . . . . . . . . . . 13 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
26		peano2re 10813	. . . . . . . . . . . . 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 10691	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ^*)
29	1	cnfldtopn 23390	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
30	29	blopn 23110	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
31	22, 23, 28, 30	mp3an2i 1462	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32		elrestr 16702	. . . . . . . . . 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 1462	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
34	33, 4	eleqtrrdi 2924	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35		0cn 10633	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
36		eqid 2821	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
37	36	cnmetdval 23379	. . . . . . . . . . . . . 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 10873	. . . . . . . . . . . . . . 15 ⊢ -𝑥 = (0 − 𝑥)
40	39	fveq2i 6673	. . . . . . . . . . . . . 14 ⊢ (abs‘-𝑥) = (abs‘(0 − 𝑥))
41		absneg 14637	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
42	40, 41	syl5eqr 2870	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
43	38, 42	eqtrd 2856	. . . . . . . . . . . 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 11570	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
46	44, 45	eqbrtrd 5088	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47		elbl 22998	. . . . . . . . . . 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 1462	. . . . . . . . . 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 4171	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
52	24	absge0d 14804	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
53	25, 52	ge0p1rpd 12462	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ⁺)
54		eqid 2821	. . . . . . . . 9 ⊢ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55		oveq2 7164	. . . . . . . . . . 11 ⊢ (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
56	55	ineq1d 4188	. . . . . . . . . 10 ⊢ (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
57	56	rspceeqv 3638	. . . . . . . . 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 2901	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60		eqeq1 2825	. . . . . . . . . . 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 3297	. . . . . . . . . 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 3623	. . . . . . . 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 3182	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67		eqid 2821	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
68		oveq2 7164	. . . . . . . 8 ⊢ (𝑟 = (𝑓‘𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓‘𝑢)))
69	68	ineq1d 4188	. . . . . . 7 ⊢ (𝑟 = (𝑓‘𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
70	69	eqeq2d 2832	. . . . . 6 ⊢ (𝑟 = (𝑓‘𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)))
71	67, 70	cmpcovf 21999	. . . . 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 2856	. . . . . . . . . . . . 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 2898	. . . . . . . . . . . 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 4842	. . . . . . . . . . . 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 4868	. . . . . . . . . . . . . . . . . 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 4008	. . . . . . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . 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 14796	. . . . . . . . . . . . 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 6852	. . . . . . . . . . . . . . 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 12432	. . . . . . . . . . . . . 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 6670	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
97	96	oveq2d 7172	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓‘𝑢)) = (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
98	97	ineq1d 4188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
99	95, 98	eqeq12d 2837	. . . . . . . . . . . . . . . . . . . 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 3625	. . . . . . . . . . . . . . . . . . 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 2915	. . . . . . . . . . . . . . . . . 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 4175	. . . . . . . . . . . . . . . . 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 10634	. . . . . . . . . . . . . . . . . 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 12433	. . . . . . . . . . . . . . . . . 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 22998	. . . . . . . . . . . . . . . . . 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 1462	. . . . . . . . . . . . . . . . 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 5090	. . . . . . . . . . . . . 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 5076	. . . . . . . . . . . . . . 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 3625	. . . . . . . . . . . . . 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 10800	. . . . . . . . . . . . 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 10788	. . . . . . . . . . . 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 3285	. . . . . . . . . . 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 3181	. . . . . . . . 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 4176	. . . . . . . . . 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 6849	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ⁺)
123	122	rpred 12432	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ)
124	123	ralrimiva 3182	. . . . . . . . . . 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 11587	. . . . . . . . . 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 3275	. . . . . . . 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 1934	. . . . . 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 3284	. . . 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 2821	. . . . . 6 ⊢ (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136		eqid 2821	. . . . . 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 23558	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138	137	rexlimdvaa 3285	. . . 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 1537 ∃wex 1780 ∈ wcel 2114 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ∩ cin 3935 ⊆ wss 3936 𝒫 cpw 4539 ∪ cuni 4838 class class class wbr 5066 × cxp 5553 “ cima 5558 ∘ ccom 5559 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ∈ cmpo 7158 Fincfn 8509 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 ici 10539 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 ℝ⁺crp 12390 [,]cicc 12742 abscabs 14593 ↾_t crest 16694 TopOpenctopn 16695 ∞Metcxmet 20530 ballcbl 20532 ℂ_fldccnfld 20545 Topctop 21501 Clsdccld 21624 Hauscha 21916 Compccmp 21994

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-cls 21629 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486

This theorem is referenced by: cnllycmp 23560 cncmet 23925 ftalem3 25652

W3C validator