Theorem ubthlem1 30899

Description: Lemma for ubth 30902. 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 25378, 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 4515	. . . . . . . . 9 ⊢ (𝑇 = ∅ → ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
2	1	ralrimivw 3148	. . . . . . . 8 ⊢ (𝑇 = ∅ → ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
3		rabid2 3468	. . . . . . . 8 ⊢ (𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ ∀𝑧 ∈ 𝑋 ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘)
4	2, 3	sylibr 234	. . . . . . 7 ⊢ (𝑇 = ∅ → 𝑋 = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
5	4	eqcomd 2741	. . . . . 6 ⊢ (𝑇 = ∅ → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = 𝑋)
6	5	eleq1d 2824	. . . . 5 ⊢ (𝑇 = ∅ → ({𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7		iinrab 5074	. . . . . . 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 3995	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12		ubthlem.3	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐷 = (IndMet‘𝑈)
13		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (IndMet‘𝑊) = (IndMet‘𝑊)
14		ubthlem.4	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 = (MetOpen‘𝐷)
15		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16		eqid 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17		ubthlem.5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑈 ∈ CBan
18		bnnv 30895	. . . . . . . . . . . . . . . . . . . . 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 30837	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22		ubth.1	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ 𝑋 = (BaseSet‘𝑈)
23	22, 12	cbncms 30894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
24	17, 23	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐷 ∈ (CMet‘𝑋)
25		cmetmet 25334	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26		metxmet 24360	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
27	24, 25, 26	mp2b 10	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝐷 ∈ (∞Met‘𝑋)
28	14	mopntopon 24465	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐽 ∈ (TopOn‘𝑋)
30		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (BaseSet‘𝑊) = (BaseSet‘𝑊)
31	30, 13	imsxmet 30721	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
32	20, 31	ax-mp 5	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
33	15	mopntopon 24465	. . . . . . . . . . . . . . . . . . . . 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 23293	. . . . . . . . . . . . . . . . . . . 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 692	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽)))
37	21, 36	sylib 218	. . . . . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
40	39	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
41	40	ffvelcdmda 7104	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
42	41	biantrurd 532	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
43		fveq2 6907	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = (𝑡‘𝑥) → (𝑁‘𝑦) = (𝑁‘(𝑡‘𝑥)))
44	43	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑦 = (𝑡‘𝑥) → ((𝑁‘𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
45	44	elrab 3695	. . . . . . . . . . . . 13 ⊢ ((𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} ↔ ((𝑡‘𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
46	42, 45	bitr4di 289	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
47	46	pm5.32da 579	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ((𝑥 ∈ 𝑋 ∧ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ (𝑥 ∈ 𝑋 ∧ (𝑡‘𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
48		2fveq3 6912	. . . . . . . . . . . . . 14 ⊢ (𝑧 = 𝑥 → (𝑁‘(𝑡‘𝑧)) = (𝑁‘(𝑡‘𝑥)))
49	48	breq1d 5158	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝑥 → ((𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
50	49	elrab 3695	. . . . . . . . . . . 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 6737	. . . . . . . . . . . 12 ⊢ (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
53		elpreima 7078	. . . . . . . . . . . 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 311	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘})))
56	55	eqrdv 2733	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
57		imaeq2 6076	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → (◡𝑡 “ 𝑥) = (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}))
58	57	eleq1d 2824	. . . . . . . . . 10 ⊢ (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} → ((◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽) ↔ (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
59	38	simprd 495	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
60	59	adantlr 715	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(◡𝑡 “ 𝑥) ∈ (Clsd‘𝐽))
61		nnre 12271	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
62	61	ad2antlr 727	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ)
63	62	rexrd 11309	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → 𝑘 ∈ ℝ*)
64		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (0vec‘𝑊) = (0vec‘𝑊)
65	30, 64	nvzcl 30663	. . . . . . . . . . . . 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 30717	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
69	20, 68	mpan 690	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
70		xmetsym 24373	. . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((0vec‘𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec‘𝑊)))
72	69, 71	eqtr4d 2778	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → (𝑁‘𝑦) = ((0vec‘𝑊)(IndMet‘𝑊)𝑦))
73	72	breq1d 5158	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁‘𝑦) ≤ 𝑘 ↔ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
74	73	rabbiia 3437	. . . . . . . . . . . . 13 ⊢ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec‘𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
75	15, 74	blcld 24534	. . . . . . . . . . . 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 1450	. . . . . . . . . . 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 3623	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → (◡𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁‘𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
79	56, 78	eqeltrd 2839	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑡 ∈ 𝑇) → {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
80	79	ralrimiva 3144	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ∀𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
81		iincld 23063	. . . . . . 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 597	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → ∩ 𝑡 ∈ 𝑇 {𝑧 ∈ 𝑋 ∣ (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
83	8, 82	eqeltrrd 2840	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
84	14	mopntop 24466	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
85	27, 84	ax-mp 5	. . . . . . 7 ⊢ 𝐽 ∈ Top
86	29	toponunii 22938	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
87	86	topcld 23059	. . . . . . 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 3025	. . . 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 7134	. . 3 ⊢ (𝜑 → 𝐴:ℕ⟶(Clsd‘𝐽))
93	92	frnd 6745	. . . . . 6 ⊢ (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
94	86	cldss2 23054	. . . . . 6 ⊢ (Clsd‘𝐽) ⊆ 𝒫 𝑋
95	93, 94	sstrdi 4008	. . . . 5 ⊢ (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
96		sspwuni 5105	. . . . 5 ⊢ (ran 𝐴 ⊆ 𝒫 𝑋 ↔ ∪ ran 𝐴 ⊆ 𝑋)
97	95, 96	sylib 218	. . . 4 ⊢ (𝜑 → ∪ ran 𝐴 ⊆ 𝑋)
98		ubthlem.8	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑋 ∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐)
99		arch 12521	. . . . . . . . . 10 ⊢ (𝑐 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
100	99	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
101		simpr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
102		ltle 11347	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
103	101, 61, 102	syl2an 596	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → 𝑐 ≤ 𝑘))
104	103	impr 454	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → 𝑐 ≤ 𝑘)
105	104	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ≤ 𝑘)
106	39	ffvelcdmda 7104	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑇) ∧ 𝑥 ∈ 𝑋) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
107	106	an32s 652	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑡‘𝑥) ∈ (BaseSet‘𝑊))
108	30, 67	nvcl 30690	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑊 ∈ NrmCVec ∧ (𝑡‘𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
109	20, 107, 108	sylancr 587	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
110	109	adantlr 715	. . . . . . . . . . . . . . . 16 ⊢ ((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
111	110	adantlr 715	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → (𝑁‘(𝑡‘𝑥)) ∈ ℝ)
112		simpllr 776	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → 𝑐 ∈ ℝ)
113		simplrl 777	. . . . . . . . . . . . . . . 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 11353	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 694	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡 ∈ 𝑇) → ((𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
118	117	ralimdva 3165	. . . . . . . . . . . 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 456	. . . . . . . . . . 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 6921	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑋 ∈ V
121	120	rabex 5345	. . . . . . . . . . . . . . . . 17 ⊢ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ∈ V
122	91	fvmpt2 7027	. . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ ℕ → (𝐴‘𝑘) = {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘})
124	123	eleq2d 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ 𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘}))
125	49	ralbidv 3176	. . . . . . . . . . . . . . . 16 ⊢ (𝑧 = 𝑥 → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘 ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
126	125	elrab 3695	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ {𝑧 ∈ 𝑋 ∣ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑧)) ≤ 𝑘} ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
127	124, 126	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴‘𝑘) ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
128		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝑥 ∈ 𝑋)
129	128	biantrurd 532	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 ↔ (𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘)))
130	129	bicomd 223	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑥 ∈ 𝑋 ∧ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
131	127, 130	sylan9bbr 510	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) ↔ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘))
132	92	ffnd 6738	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 Fn ℕ)
133	132	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐴 Fn ℕ)
134		fnfvelrn 7100	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴‘𝑘) ∈ ran 𝐴)
135		elssuni 4942	. . . . . . . . . . . . . . . 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 3994	. . . . . . . . . . . . . 14 ⊢ ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
138	133, 137	sylan 580	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴‘𝑘) → 𝑥 ∈ ∪ ran 𝐴))
139	131, 138	sylbird 260	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑘 → 𝑥 ∈ ∪ ran 𝐴))
140	139	adantlr 715	. . . . . . . . . . 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 3153	. . . . . . . . 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 3153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (∃𝑐 ∈ ℝ ∀𝑡 ∈ 𝑇 (𝑁‘(𝑡‘𝑥)) ≤ 𝑐 → 𝑥 ∈ ∪ ran 𝐴))
145	144	ralimdva 3165	. . . . . 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 3984	. . . . 5 ⊢ (𝑋 ⊆ ∪ ran 𝐴 ↔ ∀𝑥 ∈ 𝑋 𝑥 ∈ ∪ ran 𝐴)
148	146, 147	sylibr 234	. . . 4 ⊢ (𝜑 → 𝑋 ⊆ ∪ ran 𝐴)
149	97, 148	eqssd 4013	. . 3 ⊢ (𝜑 → ∪ ran 𝐴 = 𝑋)
150		eqid 2735	. . . . . 6 ⊢ (0vec‘𝑈) = (0vec‘𝑈)
151	22, 150	nvzcl 30663	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → (0vec‘𝑈) ∈ 𝑋)
152		ne0i 4347	. . . . 5 ⊢ ((0vec‘𝑈) ∈ 𝑋 → 𝑋 ≠ ∅)
153	19, 151, 152	mp2b 10	. . . 4 ⊢ 𝑋 ≠ ∅
154	14	bcth2 25378	. . . 4 ⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
155	24, 153, 154	mpanl12 702	. . 3 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ ∪ ran 𝐴 = 𝑋) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
156	92, 149, 155	syl2anc 584	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅)
157		ffvelcdm 7101	. . . . . . . . . . 11 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ (Clsd‘𝐽))
158	94, 157	sselid 3993	. . . . . . . . . 10 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ∈ 𝒫 𝑋)
159	158	elpwid 4614	. . . . . . . . 9 ⊢ ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
160	92, 159	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐴‘𝑛) ⊆ 𝑋)
161	86	ntrss3 23084	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
162	85, 160, 161	sylancr 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ 𝑋)
163	162	sseld 3994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → 𝑦 ∈ 𝑋))
164	86	ntropn 23073	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
165	85, 160, 164	sylancr 587	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽)
166	14	mopni2 24522	. . . . . . . . . 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 1447	. . . . . . . . 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 580	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)))
169		elssuni 4942	. . . . . . . . . . . 12 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ ∪ 𝐽)
170	169, 86	sseqtrrdi 4047	. . . . . . . . . . 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 3995	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛))) → 𝑦 ∈ 𝑋)
173	86	ntrss2 23081	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ (𝐴‘𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
174	85, 160, 173	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴‘𝑛)) ⊆ (𝐴‘𝑛))
175		sstr2 4002	. . . . . . . . . . . . 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 726	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴‘𝑛)))
178		simpr 484	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ 𝑋)
179	178, 27	jctil 519	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋))
180		rphalfcl 13060	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
181	180	rpxrd 13076	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
182		rpxr 13042	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → 𝑥 ∈ ℝ*)
183		rphalflt 13062	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
184	181, 182, 183	3jca 1127	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ* ∧ 𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
185		eqid 2735	. . . . . . . . . . . . . 14 ⊢ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
186	14, 185	blsscls2 24533	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
188		sstr2 4002	. . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑛 ∈ ℕ) ∧ 𝑦 ∈ 𝑋) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
191		breq2 5152	. . . . . . . . . . . . . . . 16 ⊢ (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
192	191	rabbidv 3441	. . . . . . . . . . . . . . 15 ⊢ (𝑟 = (𝑥 / 2) → {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
193	192	sseq1d 4027	. . . . . . . . . . . . . 14 ⊢ (𝑟 = (𝑥 / 2) → ({𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛) ↔ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)))
194	193	rspcev 3622	. . . . . . . . . . . . 13 ⊢ (((𝑥 / 2) ∈ ℝ+ ∧ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))
195	194	ex 412	. . . . . . . . . . . 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 3153	. . . . . . . . 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 591	. . . . . . . 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 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
202	163, 201	jcad 512	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → (𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
203	202	eximdv 1915	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)) → ∃𝑦(𝑦 ∈ 𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛))))
204		n0 4359	. . . 4 ⊢ (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴‘𝑛)))
205		df-rex 3069	. . . 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 296	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴‘𝑛)) ≠ ∅ → ∃𝑦 ∈ 𝑋 ∃𝑟 ∈ ℝ+ {𝑧 ∈ 𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴‘𝑛)))
207	206	reximdva 3166	. 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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  {crab 3433  Vcvv 3478   ⊆ wss 3963  ∅c0 4339  𝒫 cpw 4605  ∪ cuni 4912  ∩ ciin 4997   class class class wbr 5148   ↦ cmpt 5231  ^◡ccnv 5688  ran crn 5690   “ cima 5692   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  ℝ^*cxr 11292   < clt 11293   ≤ cle 11294   / cdiv 11918  ℕcn 12264  2c2 12319  ℝ⁺crp 13032  ∞Metcxmet 21367  Metcmet 21368  ballcbl 21369  MetOpencmopn 21372  Topctop 22915  TopOnctopon 22932  Clsdccld 23040  intcnt 23041   Cn ccn 23248  CMetccmet 25302  NrmCVeccnv 30613  BaseSetcba 30615  0_veccn0v 30617  norm_CVcnmcv 30619  IndMetcims 30620   BLnOp cblo 30771  CBanccbn 30891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679  ax-dc 10484  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231  ax-addf 11232  ax-mulf 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ico 13390  df-seq 14040  df-exp 14100  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-rest 17469  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-top 22916  df-topon 22933  df-bases 22969  df-cld 23043  df-ntr 23044  df-cls 23045  df-nei 23122  df-cn 23251  df-cnp 23252  df-lm 23253  df-fil 23870  df-fm 23962  df-flim 23963  df-flf 23964  df-cfil 25303  df-cau 25304  df-cmet 25305  df-grpo 30522  df-gid 30523  df-ginv 30524  df-gdiv 30525  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-vs 30628  df-nmcv 30629  df-ims 30630  df-lno 30773  df-nmoo 30774  df-blo 30775  df-0o 30776  df-cbn 30892

This theorem is referenced by:  ubthlem3  30901