Theorem ubthlem1 27110

Description: Lemma for ubth 27113. The function 𝐴 exhibits a countable collection of sets that are closed, being the inverse image under 𝑡 of the closed ball of radius 𝑘, and by assumption they cover 𝑋. Thus, by the Baire Category theorem bcth2 22935, for some 𝑛 the set 𝐴‘𝑛 has an interior, meaning that there is a closed ball {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} in the set. (Contributed by Mario Carneiro, 11-Jan-2014.) (New usage is discouraged.)

Hypotheses

Ref	Expression
ubth.1	⊢ 𝑋 = (BaseSet‘𝑈)
ubth.2	⊢ 𝑁 = (normCV‘𝑊)
ubthlem.3	⊢ 𝐷 = (IndMet‘𝑈)
ubthlem.4	⊢ 𝐽 = (MetOpen‘𝐷)
ubthlem.5	⊢ 𝑈 ∈ CBan
ubthlem.6	⊢ 𝑊 ∈ NrmCVec
ubthlem.7	⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊))
ubthlem.8	⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐)
ubthlem.9	⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})

Assertion

Ref	Expression
ubthlem1	⊢ (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))

Distinct variable groups:   𝑘,𝑐,𝑛,𝑟,𝑥,𝑦,𝑧,𝐴   𝑡,𝑐,𝐷,𝑘,𝑛,𝑟,𝑥,𝑧   𝑘,𝐽,𝑛   𝑦,𝑡,𝐽,𝑥   𝑁,𝑐,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦,𝑧   𝜑,𝑐,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦   𝑇,𝑐,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦,𝑧   𝑈,𝑐,𝑛,𝑟,𝑡,𝑥,𝑦,𝑧   𝑊,𝑐,𝑛,𝑟,𝑡,𝑥,𝑦   𝑋,𝑐,𝑘,𝑛,𝑟,𝑡,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑡)   𝐷(𝑦)   𝑈(𝑘)   𝐽(𝑧,𝑟,𝑐)   𝑊(𝑧,𝑘)

Proof of Theorem ubthlem1

Step	Hyp	Ref	Expression
1		rzal 4025	. . . . . . . . 9 ⊢ (𝑇 = ∅ → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
2	1	ralrimivw 2950	. . . . . . . 8 ⊢ (𝑇 = ∅ → ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
3		rabid2 3096	. . . . . . . 8 ⊢ (𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
4	2, 3	sylibr 223	. . . . . . 7 ⊢ (𝑇 = ∅ → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
5	4	eqcomd 2616	. . . . . 6 ⊢ (𝑇 = ∅ → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = 𝑋)
6	5	eleq1d 2672	. . . . 5 ⊢ (𝑇 = ∅ → ({𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7		iinrab 4518	. . . . . . 7 ⊢ (𝑇 ≠ ∅ → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
8	7	adantl 481	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
9		id 22	. . . . . . 7 ⊢ (𝑇 ≠ ∅ → 𝑇 ≠ ∅)
10		ubthlem.7	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑇 ⊆ (𝑈 BLnOp 𝑊))
11	10	sselda 3568	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12		ubthlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = (IndMet‘𝑈)
13		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
14		ubthlem.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 = (MetOpen‘𝐷)
15		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16		eqid 2610	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17		ubthlem.5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑈 ∈ CBan
18		bnnv 27106	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
19	17, 18	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑈 ∈ NrmCVec
20		ubthlem.6	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝑊 ∈ NrmCVec
21	12, 13, 14, 15, 16, 19, 20	blocn2 27047	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22		ubth.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (BaseSet‘𝑈)
23	22, 12	cbncms 27105	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
24	17, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 ∈ (CMet‘𝑋)
25		cmetmet 22892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26		metxmet 21949	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	24, 25, 26	mp2b 10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 ∈ (∞Met‘𝑋)
28	14	mopntopon 22054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 ∈ (TopOn‘𝑋)
30		eqid 2610	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
31	30, 13	imsxmet 26931	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
32	20, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
33	15	mopntopon 22054	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) → (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊)))
34	32, 33	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))
35		iscncl 20883	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))) → (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))))
36	29, 34, 35	mp2an 704	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
37	21, 36	sylib 207	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
38	11, 37	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
39	38	simpld 474	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
40	39	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
41	40	ffvelrnda 6267	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
42	41	biantrurd 528	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
43		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑡‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑡‘𝑥)))
44	43	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑡‘𝑥) → ((𝑁‘𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
45	44	elrab 3331	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
46	42, 45	syl6bbr 277	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
47	46	pm5.32da 671	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ((𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
48		fveq2 6103	. . . . . . . . . . . . . . 15 ⊢ (𝑧 = 𝑥 → (𝑡‘𝑧) = (𝑡‘𝑥))
49	48	fveq2d 6107	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑥)))
50	49	breq1d 4593	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
51	50	elrab 3331	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
52	51	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
53		ffn 5958	. . . . . . . . . . . 12 ⊢ (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
54		elpreima 6245	. . . . . . . . . . . 12 ⊢ (𝑡 Fn 𝑋 → (𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
55	40, 53, 54	3syl 18	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
56	47, 52, 55	3bitr4d 299	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
57	56	eqrdv 2608	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
58		nnre 10904	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
59	58	ad2antlr 759	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ)
60	59	rexrd 9968	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ*)
61		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
62	30, 61	nvzcl 26873	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊))
63	20, 62	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0vec‘𝑊) ∈ (BaseSet‘𝑊)
64		ubth.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = (normCV‘𝑊)
65	30, 61, 64, 13	nvnd 26927	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
66	20, 65	mpan 702	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
67		xmetsym 21962	. . . . . . . . . . . . . . . . 17 ⊢ (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((0vec‘𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
68	32, 63, 67	mp3an12 1406	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((0vec‘𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
69	66, 68	eqtr4d 2647	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = ((0vec‘𝑊)(IndMet‘𝑊)𝑦))
70	69	breq1d 4593	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁‘𝑦) ≤ 𝑘 ↔ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
71	70	rabbiia 3161	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
72	15, 71	blcld 22120	. . . . . . . . . . . 12 ⊢ (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑘 ∈ ℝ*) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
73	32, 63, 72	mp3an12 1406	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ* → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
74	60, 73	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
75	38	simprd 478	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
76	75	adantlr 747	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
77		imaeq2 5381	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → (◡𝑡 “ 𝑥) = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
78	77	eleq1d 2672	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → ((◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽) ↔ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
79	78	rspcv 3278	. . . . . . . . . 10 ⊢ ({𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))) → (∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽) → (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
80	74, 76, 79	sylc 63	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
81	57, 80	eqeltrd 2688	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
82	81	ralrimiva 2949	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
83		iincld 20653	. . . . . . 7 ⊢ ((𝑇 ≠ ∅ ∧ ∀𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽)) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
84	9, 82, 83	syl2anr 494	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
85	8, 84	eqeltrrd 2689	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
86	14	mopntop 22055	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
87	27, 86	ax-mp 5	. . . . . . 7 ⊢ 𝐽 ∈ Top
88	29	toponunii 20547	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
89	88	topcld 20649	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
90	87, 89	ax-mp 5	. . . . . 6 ⊢ 𝑋 ∈ (Clsd‘𝐽)
91	90	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑋 ∈ (Clsd‘𝐽))
92	6, 85, 91	pm2.61ne 2867	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
93		ubthlem.9	. . . 4 ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
94	92, 93	fmptd 6292	. . 3 ⊢ (𝜑 → 𝐴:ℕ⟶(Clsd‘𝐽))
95		frn 5966	. . . . . . 7 ⊢ (𝐴:ℕ⟶(Clsd‘𝐽) → ran 𝐴 ⊆ (Clsd‘𝐽))
96	94, 95	syl 17	. . . . . 6 ⊢ (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
97	88	cldss2 20644	. . . . . 6 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
98	96, 97	syl6ss 3580	. . . . 5 ⊢ (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
99		sspwuni 4547	. . . . 5 ⊢ (ran 𝐴 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝐴 ⊆ 𝑋)
100	98, 99	sylib 207	. . . 4 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑋)
101		ubthlem.8	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐)
102		arch 11166	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
103	102	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
104		simpr 476	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
105		ltle 10005	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
106	104, 58, 105	syl2an 493	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
107	106	impr 647	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → 𝑐 ≤ 𝑘)
108	107	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ≤ 𝑘)
109	39	ffvelrnda 6267	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
110	109	an32s 842	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
111	30, 64	nvcl 26900	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
112	20, 110, 111	sylancr 694	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
113	112	adantlr 747	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
114	113	adantlr 747	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
115		simpllr 795	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ∈ ℝ)
116		simplrl 796	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℕ)
117	116, 58	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ)
118		letr 10010	. . . . . . . . . . . . . . 15 ⊢ (((𝑁‘(𝑡‘𝑥)) ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ∧ 𝑐 ≤ 𝑘) → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
119	114, 115, 117, 118	syl3anc 1318	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → (((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ∧ 𝑐 ≤ 𝑘) → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
120	108, 119	mpan2d 706	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
121	120	ralimdva 2945	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
122	121	expr 641	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
123		fvex 6113	. . . . . . . . . . . . . . . . . . 19 ⊢ (BaseSet‘𝑈) ∈ V
124	22, 123	eqeltri 2684	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 ∈ V
125	124	rabex 4740	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ V
126	93	fvmpt2 6200	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ ∧ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ V) → (𝐴‘𝑘) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
127	125, 126	mpan2 703	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
128	127	eleq2d 2673	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ 𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}))
129	50	ralbidv 2969	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
130	129	elrab 3331	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
131	128, 130	syl6bb 275	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
132		simpr 476	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
133	132	biantrurd 528	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
134	133	bicomd 212	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
135	131, 134	sylan9bbr 733	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
136		ffn 5958	. . . . . . . . . . . . . . . 16 ⊢ (𝐴:ℕ⟶(Clsd‘𝐽) → 𝐴 Fn ℕ)
137	94, 136	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn ℕ)
138	137	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 Fn ℕ)
139		fnfvelrn 6264	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ran 𝐴)
140		elssuni 4403	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴‘𝑘) ∈ ran 𝐴 → (𝐴‘𝑘) ⊆ ∪ ran 𝐴)
141	139, 140	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ⊆ ∪ ran 𝐴)
142	141	sseld 3567	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
143	138, 142	sylan 487	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
144	135, 143	sylbird 249	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 → 𝑥 ∈ ∪ ran 𝐴))
145	144	adantlr 747	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 → 𝑥 ∈ ∪ ran 𝐴))
146	122, 145	syl6d 73	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴)))
147	146	rexlimdva 3013	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → (∃𝑘 ∈ ℕ 𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴)))
148	103, 147	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴))
149	148	rexlimdva 3013	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴))
150	149	ralimdva 2945	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴))
151	101, 150	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴)
152		dfss3 3558	. . . . 5 ⊢ (𝑋 ⊆ ∪ ran 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴)
153	151, 152	sylibr 223	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ∪ ran 𝐴)
154	100, 153	eqssd 3585	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑋)
155		eqid 2610	. . . . . 6 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
156	22, 155	nvzcl 26873	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (0vec‘𝑈) ∈ 𝑋)
157		ne0i 3880	. . . . 5 ⊢ ((0vec‘𝑈) ∈ 𝑋 → 𝑋 ≠ ∅)
158	19, 156, 157	mp2b 10	. . . 4 ⊢ 𝑋 ≠ ∅
159	14	bcth2 22935	. . . 4 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
160	24, 158, 159	mpanl12 714	. . 3 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
161	94, 154, 160	syl2anc 691	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
162		ffvelrn 6265	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ (Clsd‘𝐽))
163	97, 162	sseldi 3566	. . . . . . . . . 10 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 𝑋)
164	163	elpwid 4118	. . . . . . . . 9 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
165	94, 164	sylan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
166	88	ntrss3 20674	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
167	87, 165, 166	sylancr 694	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
168	167	sseld 3567	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → 𝑦 ∈ 𝑋))
169	88	ntropn 20663	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
170	87, 165, 169	sylancr 694	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
171	14	mopni2 22108	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
172	27, 171	mp3an1 1403	. . . . . . . . 9 ⊢ ((((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
173	170, 172	sylan 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
174		elssuni 4403	. . . . . . . . . . . 12 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ ∪ 𝐽)
175	174, 88	syl6sseqr 3615	. . . . . . . . . . 11 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
176	170, 175	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
177	176	sselda 3568	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → 𝑦 ∈ 𝑋)
178	88	ntrss2 20671	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
179	87, 165, 178	sylancr 694	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
180		sstr2 3575	. . . . . . . . . . . . 13 ⊢ ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
181	179, 180	syl5com 31	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
182	181	ad2antrr 758	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
183		simpr 476	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
184	183, 27	jctil 558	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋))
185		rphalfcl 11734	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
186	185	rpxrd 11749	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
187		rpxr 11716	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
188		rphalflt 11736	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
189	186, 187, 188	3jca 1235	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
190		eqid 2610	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
191	14, 190	blsscls2 22119	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑥 / 2) ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥)) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
192	184, 189, 191	syl2an 493	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
193		sstr2 3575	. . . . . . . . . . . 12 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
194	192, 193	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
195	185	adantl 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
196		breq2 4587	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
197	196	rabbidv 3164	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑥 / 2) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
198	197	sseq1d 3595	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑥 / 2) → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛) ↔ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
199	198	rspcev 3282	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 2) ∈ ℝ+ ∧ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))
200	199	ex 449	. . . . . . . . . . . 12 ⊢ ((𝑥 / 2) ∈ ℝ+ → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
201	195, 200	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
202	182, 194, 201	3syld 58	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
203	202	rexlimdva 3013	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
204	177, 203	syldan 486	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
205	173, 204	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))
206	205	ex 449	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
207	168, 206	jcad 554	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
208	207	eximdv 1833	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
209		n0 3890	. . . 4 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)))
210		df-rex 2902	. . . 4 ⊢ (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
211	208, 209, 210	3imtr4g 284	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ → ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
212	211	reximdva 3000	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
213	161, 212	mpd 15	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  ∪ cuni 4372  ∩ ciin 4456   class class class wbr 4583   ↦ cmpt 4643  ^◡ccnv 5037  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  ℝ^*cxr 9952   < clt 9953   ≤ cle 9954   / cdiv 10563  ℕcn 10897  2c2 10947  ℝ⁺crp 11708  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  Topctop 20517  TopOnctopon 20518  Clsdccld 20630  intcnt 20631   Cn ccn 20838  CMetcms 22860  NrmCVeccnv 26823  BaseSetcba 26825  0_veccn0v 26827  norm_CVcnmcv 26829  IndMetcims 26830   BLnOp cblo 26981  CBanccbn 27102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-dc 9151  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893  ax-addf 9894  ax-mulf 9895

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-ico 12052  df-seq 12664  df-exp 12723  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-rest 15906  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cn 20841  df-cnp 20842  df-lm 20843  df-fil 21460  df-fm 21552  df-flim 21553  df-flf 21554  df-cfil 22861  df-cau 22862  df-cmet 22863  df-grpo 26731  df-gid 26732  df-ginv 26733  df-gdiv 26734  df-ablo 26783  df-vc 26798  df-nv 26831  df-va 26834  df-ba 26835  df-sm 26836  df-0v 26837  df-vs 26838  df-nmcv 26839  df-ims 26840  df-lno 26983  df-nmoo 26984  df-blo 26985  df-0o 26986  df-cbn 27103

This theorem is referenced by:  ubthlem3  27112