cnheibor - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > cnheibor		Structured version Visualization version GIF version

Theorem cnheibor 24118

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 23948	. . . 4 ⊢ 𝐽 ∈ Haus
3		simpl 483	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ⊆ ℂ)
4		cnheibor.3	. . . . 5 ⊢ 𝑇 = (𝐽 ↾_t 𝑋)
5		simpr 485	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑇 ∈ Comp)
6	4, 5	eqeltrrid 2844	. . . 4 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝐽 ↾_t 𝑋) ∈ Comp)
7	1	cnfldtopon 23946	. . . . . 6 ⊢ 𝐽 ∈ (TopOn‘ℂ)
8	7	toponunii 22065	. . . . 5 ⊢ ℂ = ∪ 𝐽
9	8	hauscmp 22558	. . . 4 ⊢ ((𝐽 ∈ Haus ∧ 𝑋 ⊆ ℂ ∧ (𝐽 ↾_t 𝑋) ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
10	2, 3, 6, 9	mp3an2i 1465	. . 3 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ (Clsd‘𝐽))
11	1	cnfldtop 23947	. . . . . . . . . . 11 ⊢ 𝐽 ∈ Top
12	8	restuni 22313	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ⊆ ℂ) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
13	11, 3, 12	sylancr 587	. . . . . . . . . 10 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ (𝐽 ↾_t 𝑋))
14	4	unieqi 4852	. . . . . . . . . 10 ⊢ ∪ 𝑇 = ∪ (𝐽 ↾_t 𝑋)
15	13, 14	eqtr4di 2796	. . . . . . . . 9 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 = ∪ 𝑇)
16	15	eleq2d 2824	. . . . . . . 8 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑇))
17	16	biimpar 478	. . . . . . 7 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → 𝑥 ∈ 𝑋)
18		cnex 10952	. . . . . . . . . . . 12 ⊢ ℂ ∈ V
19		ssexg 5247	. . . . . . . . . . . 12 ⊢ ((𝑋 ⊆ ℂ ∧ ℂ ∈ V) → 𝑋 ∈ V)
20	3, 18, 19	sylancl 586	. . . . . . . . . . 11 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → 𝑋 ∈ V)
21	20	adantr 481	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑋 ∈ V)
22		cnxmet 23936	. . . . . . . . . . 11 ⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
23		0cnd 10968	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ∈ ℂ)
24	3	sselda 3921	. . . . . . . . . . . . . 14 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ℂ)
25	24	abscld 15148	. . . . . . . . . . . . 13 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) ∈ ℝ)
26		peano2re 11148	. . . . . . . . . . . . 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 11025	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ^*)
29	1	cnfldtopn 23945	. . . . . . . . . . . 12 ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
30	29	blopn 23656	. . . . . . . . . . 11 ⊢ (((abs ∘ − ) ∈ (∞Met‘ℂ) ∧ 0 ∈ ℂ ∧ ((abs‘𝑥) + 1) ∈ ℝ^*) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
31	22, 23, 28, 30	mp3an2i 1465	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∈ 𝐽)
32		elrestr 17139	. . . . . . . . . 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 1465	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ (𝐽 ↾_t 𝑋))
34	33, 4	eleqtrrdi 2850	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) ∈ 𝑇)
35		0cn 10967	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℂ
36		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ (abs ∘ − ) = (abs ∘ − )
37	36	cnmetdval 23934	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
38	35, 37	mpan 687	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (0(abs ∘ − )𝑥) = (abs‘(0 − 𝑥)))
39		df-neg 11208	. . . . . . . . . . . . . . 15 ⊢ -𝑥 = (0 − 𝑥)
40	39	fveq2i 6777	. . . . . . . . . . . . . 14 ⊢ (abs‘-𝑥) = (abs‘(0 − 𝑥))
41		absneg 14989	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℂ → (abs‘-𝑥) = (abs‘𝑥))
42	40, 41	eqtr3id 2792	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℂ → (abs‘(0 − 𝑥)) = (abs‘𝑥))
43	38, 42	eqtrd 2778	. . . . . . . . . . . 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 11905	. . . . . . . . . . 11 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (abs‘𝑥) < ((abs‘𝑥) + 1))
46	44, 45	eqbrtrd 5096	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → (0(abs ∘ − )𝑥) < ((abs‘𝑥) + 1))
47		elbl 23541	. . . . . . . . . . 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 1465	. . . . . . . . . 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 710	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
50		simpr 485	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
51	49, 50	elind 4128	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
52	24	absge0d 15156	. . . . . . . . . 10 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → 0 ≤ (abs‘𝑥))
53	25, 52	ge0p1rpd 12802	. . . . . . . . 9 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ((abs‘𝑥) + 1) ∈ ℝ⁺)
54		eqid 2738	. . . . . . . . 9 ⊢ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)
55		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑟 = ((abs‘𝑥) + 1) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)))
56	55	ineq1d 4145	. . . . . . . . . 10 ⊢ (𝑟 = ((abs‘𝑥) + 1) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋))
57	56	rspceeqv 3575	. . . . . . . . 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 586	. . . . . . . 8 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ 𝑋) → ∃𝑟 ∈ ℝ⁺ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋))
59		eleq2 2827	. . . . . . . . . 10 ⊢ (𝑢 = ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋) → (𝑥 ∈ 𝑢 ↔ 𝑥 ∈ ((0(ball‘(abs ∘ − ))((abs‘𝑥) + 1)) ∩ 𝑋)))
60		eqeq1 2742	. . . . . . . . . . 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 3226	. . . . . . . . . 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 631	. . . . . . . . 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 3561	. . . . . . . 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 591	. . . . . 6 ⊢ (((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) ∧ 𝑥 ∈ ∪ 𝑇) → ∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
66	65	ralrimiva 3103	. . . . 5 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → ∀𝑥 ∈ ∪ 𝑇∃𝑢 ∈ 𝑇 (𝑥 ∈ 𝑢 ∧ ∃𝑟 ∈ ℝ⁺ 𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋)))
67		eqid 2738	. . . . . 6 ⊢ ∪ 𝑇 = ∪ 𝑇
68		oveq2 7283	. . . . . . . 8 ⊢ (𝑟 = (𝑓‘𝑢) → (0(ball‘(abs ∘ − ))𝑟) = (0(ball‘(abs ∘ − ))(𝑓‘𝑢)))
69	68	ineq1d 4145	. . . . . . 7 ⊢ (𝑟 = (𝑓‘𝑢) → ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋))
70	69	eqeq2d 2749	. . . . . 6 ⊢ (𝑟 = (𝑓‘𝑢) → (𝑢 = ((0(ball‘(abs ∘ − ))𝑟) ∩ 𝑋) ↔ 𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋)))
71	67, 70	cmpcovf 22542	. . . . 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 584	. . . 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 729	. . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 2824	. . . . . . . . . . . 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 4843	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑠 ↔ ∃𝑧 ∈ 𝑠 𝑥 ∈ 𝑧)
78	76, 77	bitrdi 287	. . . . . . . . . . 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 4871	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 ∈ 𝑠 → 𝑧 ⊆ ∪ 𝑠)
80	79	ad2antrl 725	. . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . 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 3962	. . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 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 3922	. . . . . . . . . . . . . 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 15148	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . 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 6962	. . . . . . . . . . . . . . 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 12772	. . . . . . . . . . . . . 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 6774	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑢 = 𝑧 → (𝑓‘𝑢) = (𝑓‘𝑧))
97	96	oveq2d 7291	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑢 = 𝑧 → (0(ball‘(abs ∘ − ))(𝑓‘𝑢)) = (0(ball‘(abs ∘ − ))(𝑓‘𝑧)))
98	97	ineq1d 4145	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑢 = 𝑧 → ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋))
99	95, 98	eqeq12d 2754	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑢 = 𝑧 → (𝑢 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑢)) ∩ 𝑋) ↔ 𝑧 = ((0(ball‘(abs ∘ − ))(𝑓‘𝑧)) ∩ 𝑋)))
100		simprr 770	. . . . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . . . . . . 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 2841	. . . . . . . . . . . . . . . . . 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 4132	. . . . . . . . . . . . . . . . 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 10968	. . . . . . . . . . . . . . . . . 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 12773	. . . . . . . . . . . . . . . . . 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 23541	. . . . . . . . . . . . . . . . . 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 1465	. . . . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . . 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 5098	. . . . . . . . . . . . . 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 5084	. . . . . . . . . . . . . . 15 ⊢ (𝑢 = 𝑧 → ((𝑓‘𝑢) ≤ 𝑟 ↔ (𝑓‘𝑧) ≤ 𝑟))
113		simplrr 775	. . . . . . . . . . . . . . 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 3562	. . . . . . . . . . . . . 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 11135	. . . . . . . . . . . . 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 11123	. . . . . . . . . . . 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 3214	. . . . . . . . . . 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 239	. . . . . . . . . 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 3102	. . . . . . . . 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 773	. . . . . . . . . . 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 4133	. . . . . . . . . 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 6959	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ⁺)
123	122	rpred 12772	. . . . . . . . . . . 12 ⊢ ((𝑓:𝑠⟶ℝ⁺ ∧ 𝑢 ∈ 𝑠) → (𝑓‘𝑢) ∈ ℝ)
124	123	ralrimiva 3103	. . . . . . . . . . 11 ⊢ (𝑓:𝑠⟶ℝ⁺ → ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ)
125	124	ad2antrl 725	. . . . . . . . . 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 11921	. . . . . . . . . 10 ⊢ ((𝑠 ∈ Fin ∧ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ 𝑠 (𝑓‘𝑢) ≤ 𝑟)
127	121, 125, 126	syl2anc 584	. . . . . . . . 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 3204	. . . . . . . 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 413	. . . . . . 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 1936	. . . . . 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 454	. . . . 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 3213	. . . 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 512	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ 𝑇 ∈ Comp) → (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟))
135		eqid 2738	. . . . . 6 ⊢ (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧))) = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (𝑦 + (i · 𝑧)))
136		eqid 2738	. . . . . 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 24117	. . . . 5 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ (𝑟 ∈ ℝ ∧ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
138	137	rexlimdvaa 3214	. . . 4 ⊢ (𝑋 ∈ (Clsd‘𝐽) → (∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟 → 𝑇 ∈ Comp))
139	138	imp 407	. . 3 ⊢ ((𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟) → 𝑇 ∈ Comp)
140	139	adantl 482	. 2 ⊢ ((𝑋 ⊆ ℂ ∧ (𝑋 ∈ (Clsd‘𝐽) ∧ ∃𝑟 ∈ ℝ ∀𝑥 ∈ 𝑋 (abs‘𝑥) ≤ 𝑟)) → 𝑇 ∈ Comp)
141	134, 140	impbida 798	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 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ∩ cin 3886 ⊆ wss 3887 𝒫 cpw 4533 ∪ cuni 4839 class class class wbr 5074 × cxp 5587 “ cima 5592 ∘ ccom 5593 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 ∈ cmpo 7277 Fincfn 8733 ℂcc 10869 ℝcr 10870 0cc0 10871 1c1 10872 ici 10873 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 − cmin 11205 -cneg 11206 ℝ⁺crp 12730 [,]cicc 13082 abscabs 14945 ↾_t crest 17131 TopOpenctopn 17132 ∞Metcxmet 20582 ballcbl 20584 ℂ_fldccnfld 20597 Topctop 22042 Clsdccld 22167 Hauscha 22459 Compccmp 22537

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-cls 22172 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041

This theorem is referenced by: cnllycmp 24119 cncmet 24486 ftalem3 26224

W3C validator