Theorem ubthlem1 28650

Description: Lemma for ubth 28653. 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 23936, 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 4456	. . . . . . . . 9 ⊢ (𝑇 = ∅ → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
2	1	ralrimivw 3186	. . . . . . . 8 ⊢ (𝑇 = ∅ → ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
3		rabid2 3384	. . . . . . . 8 ⊢ (𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
4	2, 3	sylibr 236	. . . . . . 7 ⊢ (𝑇 = ∅ → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
5	4	eqcomd 2830	. . . . . 6 ⊢ (𝑇 = ∅ → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = 𝑋)
6	5	eleq1d 2900	. . . . 5 ⊢ (𝑇 = ∅ → ({𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7		iinrab 4994	. . . . . . 7 ⊢ (𝑇 ≠ ∅ → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
8	7	adantl 484	. . . . . 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 3970	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12		ubthlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = (IndMet‘𝑈)
13		eqid 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
14		ubthlem.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 = (MetOpen‘𝐷)
15		eqid 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16		eqid 2824	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17		ubthlem.5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑈 ∈ CBan
18		bnnv 28646	. . . . . . . . . . . . . . . . . . . . 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 28588	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22		ubth.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (BaseSet‘𝑈)
23	22, 12	cbncms 28645	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
24	17, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 ∈ (CMet‘𝑋)
25		cmetmet 23892	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26		metxmet 22947	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	24, 25, 26	mp2b 10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 ∈ (∞Met‘𝑋)
28	14	mopntopon 23052	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 ∈ (TopOn‘𝑋)
30		eqid 2824	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
31	30, 13	imsxmet 28472	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
32	20, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
33	15	mopntopon 23052	. . . . . . . . . . . . . . . . . . . . 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 21880	. . . . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
37	21, 36	sylib 220	. . . . . . . . . . . . . . . . . 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 497	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
40	39	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
41	40	ffvelrnda 6854	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
42	41	biantrurd 535	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
43		fveq2 6673	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑡‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑡‘𝑥)))
44	43	breq1d 5079	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑡‘𝑥) → ((𝑁‘𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
45	44	elrab 3683	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
46	42, 45	syl6bbr 291	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
47	46	pm5.32da 581	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ((𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
48		2fveq3 6678	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑥)))
49	48	breq1d 5079	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
50	49	elrab 3683	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
51	50	a1i 11	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
52		ffn 6517	. . . . . . . . . . . 12 ⊢ (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
53		elpreima 6831	. . . . . . . . . . . 12 ⊢ (𝑡 Fn 𝑋 → (𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
54	40, 52, 53	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})))
55	47, 51, 54	3bitr4d 313	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
56	55	eqrdv 2822	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
57		imaeq2 5928	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → (◡𝑡 “ 𝑥) = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
58	57	eleq1d 2900	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → ((◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽) ↔ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
59	38	simprd 498	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
60	59	adantlr 713	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
61		nnre 11648	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
62	61	ad2antlr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ)
63	62	rexrd 10694	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ*)
64		eqid 2824	. . . . . . . . . . . . . 14 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
65	30, 64	nvzcl 28414	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ NrmCVec → (0vec‘𝑊) ∈ (BaseSet‘𝑊))
66	20, 65	ax-mp 5	. . . . . . . . . . . 12 ⊢ (0vec‘𝑊) ∈ (BaseSet‘𝑊)
67		ubth.2	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑁 = (normCV‘𝑊)
68	30, 64, 67, 13	nvnd 28468	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
69	20, 68	mpan 688	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
70		xmetsym 22960	. . . . . . . . . . . . . . . . 17 ⊢ (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((0vec‘𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
71	32, 66, 70	mp3an12 1447	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((0vec‘𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
72	69, 71	eqtr4d 2862	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = ((0vec‘𝑊)(IndMet‘𝑊)𝑦))
73	72	breq1d 5079	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁‘𝑦) ≤ 𝑘 ↔ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
74	73	rabbiia 3475	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
75	15, 74	blcld 23118	. . . . . . . . . . . 12 ⊢ (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec‘𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑘 ∈ ℝ*) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
76	32, 66, 75	mp3an12 1447	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℝ* → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
77	63, 76	syl 17	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
78	58, 60, 77	rspcdva 3628	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
79	56, 78	eqeltrd 2916	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
80	79	ralrimiva 3185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
81		iincld 21650	. . . . . . 7 ⊢ ((𝑇 ≠ ∅ ∧ ∀𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽)) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
82	9, 80, 81	syl2anr 598	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
83	8, 82	eqeltrrd 2917	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
84	14	mopntop 23053	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
85	27, 84	ax-mp 5	. . . . . . 7 ⊢ 𝐽 ∈ Top
86	29	toponunii 21527	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
87	86	topcld 21646	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
88	85, 87	ax-mp 5	. . . . . 6 ⊢ 𝑋 ∈ (Clsd‘𝐽)
89	88	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝑋 ∈ (Clsd‘𝐽))
90	6, 83, 89	pm2.61ne 3105	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
91		ubthlem.9	. . . 4 ⊢ 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
92	90, 91	fmptd 6881	. . 3 ⊢ (𝜑 → 𝐴:ℕ⟶(Clsd‘𝐽))
93	92	frnd 6524	. . . . . 6 ⊢ (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
94	86	cldss2 21641	. . . . . 6 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
95	93, 94	sstrdi 3982	. . . . 5 ⊢ (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
96		sspwuni 5025	. . . . 5 ⊢ (ran 𝐴 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝐴 ⊆ 𝑋)
97	95, 96	sylib 220	. . . 4 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑋)
98		ubthlem.8	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐)
99		arch 11897	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
100	99	adantl 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
101		simpr 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
102		ltle 10732	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
103	101, 61, 102	syl2an 597	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
104	103	impr 457	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → 𝑐 ≤ 𝑘)
105	104	adantr 483	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ≤ 𝑘)
106	39	ffvelrnda 6854	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
107	106	an32s 650	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
108	30, 67	nvcl 28441	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
109	20, 107, 108	sylancr 589	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
110	109	adantlr 713	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
111	110	adantlr 713	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
112		simpllr 774	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ∈ ℝ)
113		simplrl 775	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℕ)
114	113, 61	syl 17	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ)
115		letr 10737	. . . . . . . . . . . . . . 15 ⊢ (((𝑁‘(𝑡‘𝑥)) ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ∧ 𝑐 ≤ 𝑘) → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
116	111, 112, 114, 115	syl3anc 1367	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → (((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 ∧ 𝑐 ≤ 𝑘) → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
117	105, 116	mpan2d 692	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
118	117	ralimdva 3180	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
119	118	expr 459	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
120	22	fvexi 6687	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 ∈ V
121	120	rabex 5238	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ V
122	91	fvmpt2 6782	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑘 ∈ ℕ ∧ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ V) → (𝐴‘𝑘) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
123	121, 122	mpan2 689	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
124	123	eleq2d 2901	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ 𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}))
125	49	ralbidv 3200	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
126	125	elrab 3683	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
127	124, 126	syl6bb 289	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
128		simpr 487	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
129	128	biantrurd 535	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
130	129	bicomd 225	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
131	127, 130	sylan9bbr 513	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
132	92	ffnd 6518	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn ℕ)
133	132	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 Fn ℕ)
134		fnfvelrn 6851	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ran 𝐴)
135		elssuni 4871	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴‘𝑘) ∈ ran 𝐴 → (𝐴‘𝑘) ⊆ ∪ ran 𝐴)
136	134, 135	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ⊆ ∪ ran 𝐴)
137	136	sseld 3969	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
138	133, 137	sylan 582	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
139	131, 138	sylbird 262	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 → 𝑥 ∈ ∪ ran 𝐴))
140	139	adantlr 713	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 → 𝑥 ∈ ∪ ran 𝐴))
141	119, 140	syl6d 75	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴)))
142	141	rexlimdva 3287	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → (∃𝑘 ∈ ℕ 𝑐 < 𝑘 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴)))
143	100, 142	mpd 15	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴))
144	143	rexlimdva 3287	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴))
145	144	ralimdva 3180	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴))
146	98, 145	mpd 15	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴)
147		dfss3 3959	. . . . 5 ⊢ (𝑋 ⊆ ∪ ran 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴)
148	146, 147	sylibr 236	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ∪ ran 𝐴)
149	97, 148	eqssd 3987	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑋)
150		eqid 2824	. . . . . 6 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
151	22, 150	nvzcl 28414	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (0vec‘𝑈) ∈ 𝑋)
152		ne0i 4303	. . . . 5 ⊢ ((0vec‘𝑈) ∈ 𝑋 → 𝑋 ≠ ∅)
153	19, 151, 152	mp2b 10	. . . 4 ⊢ 𝑋 ≠ ∅
154	14	bcth2 23936	. . . 4 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
155	24, 153, 154	mpanl12 700	. . 3 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
156	92, 149, 155	syl2anc 586	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
157		ffvelrn 6852	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ (Clsd‘𝐽))
158	94, 157	sseldi 3968	. . . . . . . . . 10 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 𝑋)
159	158	elpwid 4553	. . . . . . . . 9 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
160	92, 159	sylan 582	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
161	86	ntrss3 21671	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
162	85, 160, 161	sylancr 589	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
163	162	sseld 3969	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → 𝑦 ∈ 𝑋))
164	86	ntropn 21660	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
165	85, 160, 164	sylancr 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
166	14	mopni2 23106	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
167	27, 166	mp3an1 1444	. . . . . . . . 9 ⊢ ((((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
168	165, 167	sylan 582	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
169		elssuni 4871	. . . . . . . . . . . 12 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ ∪ 𝐽)
170	169, 86	sseqtrrdi 4021	. . . . . . . . . . 11 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
171	165, 170	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
172	171	sselda 3970	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → 𝑦 ∈ 𝑋)
173	86	ntrss2 21668	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
174	85, 160, 173	sylancr 589	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
175		sstr2 3977	. . . . . . . . . . . . 13 ⊢ ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
176	174, 175	syl5com 31	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
177	176	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
178		simpr 487	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
179	178, 27	jctil 522	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋))
180		rphalfcl 12419	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
181	180	rpxrd 12435	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
182		rpxr 12401	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
183		rphalflt 12421	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
184	181, 182, 183	3jca 1124	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
185		eqid 2824	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
186	14, 185	blsscls2 23117	. . . . . . . . . . . . 13 ⊢ (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ ((𝑥 / 2) ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥)) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
187	179, 184, 186	syl2an 597	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
188		sstr2 3977	. . . . . . . . . . . 12 ⊢ ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
189	187, 188	syl 17	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
190	180	adantl 484	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
191		breq2 5073	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
192	191	rabbidv 3483	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑥 / 2) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
193	192	sseq1d 4001	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑥 / 2) → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛) ↔ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
194	193	rspcev 3626	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 2) ∈ ℝ+ ∧ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))
195	194	ex 415	. . . . . . . . . . . 12 ⊢ ((𝑥 / 2) ∈ ℝ+ → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
196	190, 195	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)))
197	177, 189, 196	3syld 60	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
198	197	rexlimdva 3287	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
199	172, 198	syldan 593	. . . . . . . 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)))
200	168, 199	mpd 15	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))
201	200	ex 415	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
202	163, 201	jcad 515	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
203	202	eximdv 1917	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
204		n0 4313	. . . 4 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)))
205		df-rex 3147	. . . 4 ⊢ (∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛) ↔ ∃𝑦(𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
206	203, 204, 205	3imtr4g 298	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ → ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
207	206	reximdva 3277	. 2 ⊢ (𝜑 → (∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
208	156, 207	mpd 15	1 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1536  ∃wex 1779   ∈ wcel 2113   ≠ wne 3019  ∀wral 3141  ∃wrex 3142  {crab 3145  Vcvv 3497   ⊆ wss 3939  ∅c0 4294  𝒫 cpw 4542  ∪ cuni 4841  ∩ ciin 4923   class class class wbr 5069   ↦ cmpt 5149  ^◡ccnv 5557  ran crn 5559   “ cima 5561   Fn wfn 6353  ⟶wf 6354  ‘cfv 6358  (class class class)co 7159  ℝcr 10539  ℝ^*cxr 10677   < clt 10678   ≤ cle 10679   / cdiv 11300  ℕcn 11641  2c2 11695  ℝ⁺crp 12392  ∞Metcxmet 20533  Metcmet 20534  ballcbl 20535  MetOpencmopn 20538  Topctop 21504  TopOnctopon 21521  Clsdccld 21627  intcnt 21628   Cn ccn 21835  CMetccmet 23860  NrmCVeccnv 28364  BaseSetcba 28366  0_veccn0v 28368  norm_CVcnmcv 28370  IndMetcims 28371   BLnOp cblo 28522  CBanccbn 28642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-rep 5193  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464  ax-inf2 9107  ax-dc 9871  ax-cnex 10596  ax-resscn 10597  ax-1cn 10598  ax-icn 10599  ax-addcl 10600  ax-addrcl 10601  ax-mulcl 10602  ax-mulrcl 10603  ax-mulcom 10604  ax-addass 10605  ax-mulass 10606  ax-distr 10607  ax-i2m1 10608  ax-1ne0 10609  ax-1rid 10610  ax-rnegex 10611  ax-rrecex 10612  ax-cnre 10613  ax-pre-lttri 10614  ax-pre-lttrn 10615  ax-pre-ltadd 10616  ax-pre-mulgt0 10617  ax-pre-sup 10618  ax-addf 10619  ax-mulf 10620

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ne 3020  df-nel 3127  df-ral 3146  df-rex 3147  df-reu 3148  df-rmo 3149  df-rab 3150  df-v 3499  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-tp 4575  df-op 4577  df-uni 4842  df-int 4880  df-iun 4924  df-iin 4925  df-br 5070  df-opab 5132  df-mpt 5150  df-tr 5176  df-id 5463  df-eprel 5468  df-po 5477  df-so 5478  df-fr 5517  df-we 5519  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-ima 5571  df-pred 6151  df-ord 6197  df-on 6198  df-lim 6199  df-suc 6200  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-riota 7117  df-ov 7162  df-oprab 7163  df-mpo 7164  df-om 7584  df-1st 7692  df-2nd 7693  df-wrecs 7950  df-recs 8011  df-rdg 8049  df-1o 8105  df-er 8292  df-map 8411  df-pm 8412  df-en 8513  df-dom 8514  df-sdom 8515  df-sup 8909  df-inf 8910  df-pnf 10680  df-mnf 10681  df-xr 10682  df-ltxr 10683  df-le 10684  df-sub 10875  df-neg 10876  df-div 11301  df-nn 11642  df-2 11703  df-3 11704  df-n0 11901  df-z 11985  df-uz 12247  df-q 12352  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-ico 12747  df-seq 13373  df-exp 13433  df-cj 14461  df-re 14462  df-im 14463  df-sqrt 14597  df-abs 14598  df-rest 16699  df-topgen 16720  df-psmet 20540  df-xmet 20541  df-met 20542  df-bl 20543  df-mopn 20544  df-fbas 20545  df-fg 20546  df-top 21505  df-topon 21522  df-bases 21557  df-cld 21630  df-ntr 21631  df-cls 21632  df-nei 21709  df-cn 21838  df-cnp 21839  df-lm 21840  df-fil 22457  df-fm 22549  df-flim 22550  df-flf 22551  df-cfil 23861  df-cau 23862  df-cmet 23863  df-grpo 28273  df-gid 28274  df-ginv 28275  df-gdiv 28276  df-ablo 28325  df-vc 28339  df-nv 28372  df-va 28375  df-ba 28376  df-sm 28377  df-0v 28378  df-vs 28379  df-nmcv 28380  df-ims 28381  df-lno 28524  df-nmoo 28525  df-blo 28526  df-0o 28527  df-cbn 28643

This theorem is referenced by:  ubthlem3  28652