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Theorem ubthlem1 28663
 Description: Lemma for ubth 28666. 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 23944, 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 = (normCV𝑊)
ubthlem.3 𝐷 = (IndMet‘𝑈)
ubthlem.4 𝐽 = (MetOpen‘𝐷)
ubthlem.5 𝑈 ∈ CBan
ubthlem.6 𝑊 ∈ NrmCVec
ubthlem.7 (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))
ubthlem.8 (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)
ubthlem.9 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
Assertion
Ref Expression
ubthlem1 (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))
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
StepHypRef Expression
1 rzal 4411 . . . . . . . . 9 (𝑇 = ∅ → ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
21ralrimivw 3150 . . . . . . . 8 (𝑇 = ∅ → ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
3 rabid2 3334 . . . . . . . 8 (𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
42, 3sylibr 237 . . . . . . 7 (𝑇 = ∅ → 𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
54eqcomd 2804 . . . . . 6 (𝑇 = ∅ → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} = 𝑋)
65eleq1d 2874 . . . . 5 (𝑇 = ∅ → ({𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7 iinrab 4955 . . . . . . 7 (𝑇 ≠ ∅ → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
87adantl 485 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
9 id 22 . . . . . . 7 (𝑇 ≠ ∅ → 𝑇 ≠ ∅)
10 ubthlem.7 . . . . . . . . . . . . . . . . . . 19 (𝜑𝑇 ⊆ (𝑈 BLnOp 𝑊))
1110sselda 3915 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12 ubthlem.3 . . . . . . . . . . . . . . . . . . . 20 𝐷 = (IndMet‘𝑈)
13 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (IndMet‘𝑊) = (IndMet‘𝑊)
14 ubthlem.4 . . . . . . . . . . . . . . . . . . . 20 𝐽 = (MetOpen‘𝐷)
15 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16 eqid 2798 . . . . . . . . . . . . . . . . . . . 20 (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17 ubthlem.5 . . . . . . . . . . . . . . . . . . . . 21 𝑈 ∈ CBan
18 bnnv 28659 . . . . . . . . . . . . . . . . . . . . 21 (𝑈 ∈ CBan → 𝑈 ∈ NrmCVec)
1917, 18ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 𝑈 ∈ NrmCVec
20 ubthlem.6 . . . . . . . . . . . . . . . . . . . 20 𝑊 ∈ NrmCVec
2112, 13, 14, 15, 16, 19, 20blocn2 28601 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22 ubth.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (BaseSet‘𝑈)
2322, 12cbncms 28658 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
2417, 23ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 𝐷 ∈ (CMet‘𝑋)
25 cmetmet 23900 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26 metxmet 22951 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
2724, 25, 26mp2b 10 . . . . . . . . . . . . . . . . . . . . 21 𝐷 ∈ (∞Met‘𝑋)
2814mopntopon 23056 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 𝐽 ∈ (TopOn‘𝑋)
30 eqid 2798 . . . . . . . . . . . . . . . . . . . . . . 23 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3130, 13imsxmet 28485 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
3220, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
3315mopntopon 23056 . . . . . . . . . . . . . . . . . . . . 21 ((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) → (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊)))
3432, 33ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 (MetOpen‘(IndMet‘𝑊)) ∈ (TopOn‘(BaseSet‘𝑊))
35 iscncl 21884 . . . . . . . . . . . . . . . . . . . 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)))(𝑡𝑥) ∈ (Clsd‘𝐽))))
3629, 34, 35mp2an 691 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽)))
3721, 36sylib 221 . . . . . . . . . . . . . . . . . 18 (𝑡 ∈ (𝑈 BLnOp 𝑊) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽)))
3811, 37syl 17 . . . . . . . . . . . . . . . . 17 ((𝜑𝑡𝑇) → (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽)))
3938simpld 498 . . . . . . . . . . . . . . . 16 ((𝜑𝑡𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
4039adantlr 714 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
4140ffvelrnda 6829 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
4241biantrurd 536 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → ((𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ ((𝑡𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
43 fveq2 6646 . . . . . . . . . . . . . . 15 (𝑦 = (𝑡𝑥) → (𝑁𝑦) = (𝑁‘(𝑡𝑥)))
4443breq1d 5041 . . . . . . . . . . . . . 14 (𝑦 = (𝑡𝑥) → ((𝑁𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4544elrab 3628 . . . . . . . . . . . . 13 ((𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ↔ ((𝑡𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4642, 45bitr4di 292 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → ((𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
4746pm5.32da 582 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → ((𝑥𝑋 ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘) ↔ (𝑥𝑋 ∧ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
48 2fveq3 6651 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑁‘(𝑡𝑧)) = (𝑁‘(𝑡𝑥)))
4948breq1d 5041 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
5049elrab 3628 . . . . . . . . . . . 12 (𝑥 ∈ {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ (𝑥𝑋 ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
5150a1i 11 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑥 ∈ {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ (𝑥𝑋 ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
52 ffn 6488 . . . . . . . . . . . 12 (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
53 elpreima 6806 . . . . . . . . . . . 12 (𝑡 Fn 𝑋 → (𝑥 ∈ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ↔ (𝑥𝑋 ∧ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
5440, 52, 533syl 18 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑥 ∈ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ↔ (𝑥𝑋 ∧ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
5547, 51, 543bitr4d 314 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑥 ∈ {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
5655eqrdv 2796 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
57 imaeq2 5893 . . . . . . . . . . 11 (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} → (𝑡𝑥) = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
5857eleq1d 2874 . . . . . . . . . 10 (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} → ((𝑡𝑥) ∈ (Clsd‘𝐽) ↔ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
5938simprd 499 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽))
6059adantlr 714 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽))
61 nnre 11635 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
6261ad2antlr 726 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ)
6362rexrd 10683 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ*)
64 eqid 2798 . . . . . . . . . . . . . 14 (0vec𝑊) = (0vec𝑊)
6530, 64nvzcl 28427 . . . . . . . . . . . . 13 (𝑊 ∈ NrmCVec → (0vec𝑊) ∈ (BaseSet‘𝑊))
6620, 65ax-mp 5 . . . . . . . . . . . 12 (0vec𝑊) ∈ (BaseSet‘𝑊)
67 ubth.2 . . . . . . . . . . . . . . . . . 18 𝑁 = (normCV𝑊)
6830, 64, 67, 13nvnd 28481 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
6920, 68mpan 689 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
70 xmetsym 22964 . . . . . . . . . . . . . . . . 17 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7132, 66, 70mp3an12 1448 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7269, 71eqtr4d 2836 . . . . . . . . . . . . . . 15 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = ((0vec𝑊)(IndMet‘𝑊)𝑦))
7372breq1d 5041 . . . . . . . . . . . . . 14 (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁𝑦) ≤ 𝑘 ↔ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
7473rabbiia 3419 . . . . . . . . . . . . 13 {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
7515, 74blcld 23122 . . . . . . . . . . . 12 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑘 ∈ ℝ*) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
7632, 66, 75mp3an12 1448 . . . . . . . . . . 11 (𝑘 ∈ ℝ* → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
7763, 76syl 17 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
7858, 60, 77rspcdva 3573 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
7956, 78eqeltrd 2890 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8079ralrimiva 3149 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
81 iincld 21654 . . . . . . 7 ((𝑇 ≠ ∅ ∧ ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽)) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
829, 80, 81syl2anr 599 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
838, 82eqeltrrd 2891 . . . . 5 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8414mopntop 23057 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
8527, 84ax-mp 5 . . . . . . 7 𝐽 ∈ Top
8629toponunii 21531 . . . . . . . 8 𝑋 = 𝐽
8786topcld 21650 . . . . . . 7 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
8885, 87ax-mp 5 . . . . . 6 𝑋 ∈ (Clsd‘𝐽)
8988a1i 11 . . . . 5 ((𝜑𝑘 ∈ ℕ) → 𝑋 ∈ (Clsd‘𝐽))
906, 83, 89pm2.61ne 3072 . . . 4 ((𝜑𝑘 ∈ ℕ) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
91 ubthlem.9 . . . 4 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
9290, 91fmptd 6856 . . 3 (𝜑𝐴:ℕ⟶(Clsd‘𝐽))
9392frnd 6495 . . . . . 6 (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
9486cldss2 21645 . . . . . 6 (Clsd‘𝐽) ⊆ 𝒫 𝑋
9593, 94sstrdi 3927 . . . . 5 (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
96 sspwuni 4986 . . . . 5 (ran 𝐴 ⊆ 𝒫 𝑋 ran 𝐴𝑋)
9795, 96sylib 221 . . . 4 (𝜑 ran 𝐴𝑋)
98 ubthlem.8 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)
99 arch 11885 . . . . . . . . . 10 (𝑐 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
10099adantl 485 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
101 simpr 488 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
102 ltle 10721 . . . . . . . . . . . . . . . . 17 ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘𝑐𝑘))
103101, 61, 102syl2an 598 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘𝑐𝑘))
104103impr 458 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → 𝑐𝑘)
105104adantr 484 . . . . . . . . . . . . . 14 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑐𝑘)
10639ffvelrnda 6829 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑡𝑇) ∧ 𝑥𝑋) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
107106an32s 651 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
10830, 67nvcl 28454 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ NrmCVec ∧ (𝑡𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
10920, 107, 108sylancr 590 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
110109adantlr 714 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
111110adantlr 714 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
112 simpllr 775 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑐 ∈ ℝ)
113 simplrl 776 . . . . . . . . . . . . . . . 16 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑘 ∈ ℕ)
114113, 61syl 17 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ)
115 letr 10726 . . . . . . . . . . . . . . 15 (((𝑁‘(𝑡𝑥)) ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
116111, 112, 114, 115syl3anc 1368 . . . . . . . . . . . . . 14 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
117105, 116mpan2d 693 . . . . . . . . . . . . 13 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → ((𝑁‘(𝑡𝑥)) ≤ 𝑐 → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
118117ralimdva 3144 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
119118expr 460 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
12022fvexi 6660 . . . . . . . . . . . . . . . . . 18 𝑋 ∈ V
121120rabex 5200 . . . . . . . . . . . . . . . . 17 {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V
12291fvmpt2 6757 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V) → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
123121, 122mpan2 690 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
124123eleq2d 2875 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘}))
12549ralbidv 3162 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
126125elrab 3628 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
127124, 126syl6bb 290 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴𝑘) ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
128 simpr 488 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
129128biantrurd 536 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
130129bicomd 226 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ((𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘) ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
131127, 130sylan9bbr 514 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
13292ffnd 6489 . . . . . . . . . . . . . . 15 (𝜑𝐴 Fn ℕ)
133132adantr 484 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐴 Fn ℕ)
134 fnfvelrn 6826 . . . . . . . . . . . . . . . 16 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ran 𝐴)
135 elssuni 4831 . . . . . . . . . . . . . . . 16 ((𝐴𝑘) ∈ ran 𝐴 → (𝐴𝑘) ⊆ ran 𝐴)
136134, 135syl 17 . . . . . . . . . . . . . . 15 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ⊆ ran 𝐴)
137136sseld 3914 . . . . . . . . . . . . . 14 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) → 𝑥 ran 𝐴))
138133, 137sylan 583 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) → 𝑥 ran 𝐴))
139131, 138sylbird 263 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘𝑥 ran 𝐴))
140139adantlr 714 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘𝑥 ran 𝐴))
141119, 140syl6d 75 . . . . . . . . . 10 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴)))
142141rexlimdva 3243 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → (∃𝑘 ∈ ℕ 𝑐 < 𝑘 → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴)))
143100, 142mpd 15 . . . . . . . 8 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴))
144143rexlimdva 3243 . . . . . . 7 ((𝜑𝑥𝑋) → (∃𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴))
145144ralimdva 3144 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑥𝑋 𝑥 ran 𝐴))
14698, 145mpd 15 . . . . 5 (𝜑 → ∀𝑥𝑋 𝑥 ran 𝐴)
147 dfss3 3903 . . . . 5 (𝑋 ran 𝐴 ↔ ∀𝑥𝑋 𝑥 ran 𝐴)
148146, 147sylibr 237 . . . 4 (𝜑𝑋 ran 𝐴)
14997, 148eqssd 3932 . . 3 (𝜑 ran 𝐴 = 𝑋)
150 eqid 2798 . . . . . 6 (0vec𝑈) = (0vec𝑈)
15122, 150nvzcl 28427 . . . . 5 (𝑈 ∈ NrmCVec → (0vec𝑈) ∈ 𝑋)
152 ne0i 4250 . . . . 5 ((0vec𝑈) ∈ 𝑋𝑋 ≠ ∅)
15319, 151, 152mp2b 10 . . . 4 𝑋 ≠ ∅
15414bcth2 23944 . . . 4 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
15524, 153, 154mpanl12 701 . . 3 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ ran 𝐴 = 𝑋) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
15692, 149, 155syl2anc 587 . 2 (𝜑 → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
157 ffvelrn 6827 . . . . . . . . . . 11 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ (Clsd‘𝐽))
15894, 157sseldi 3913 . . . . . . . . . 10 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 𝑋)
159158elpwid 4508 . . . . . . . . 9 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16092, 159sylan 583 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16186ntrss3 21675 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
16285, 160, 161sylancr 590 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
163162sseld 3914 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → 𝑦𝑋))
16486ntropn 21664 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16585, 160, 164sylancr 590 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16614mopni2 23110 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
16727, 166mp3an1 1445 . . . . . . . . 9 ((((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
168165, 167sylan 583 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
169 elssuni 4831 . . . . . . . . . . . 12 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝐽)
170169, 86sseqtrrdi 3966 . . . . . . . . . . 11 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
171165, 170syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
172171sselda 3915 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → 𝑦𝑋)
17386ntrss2 21672 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
17485, 160, 173sylancr 590 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
175 sstr2 3922 . . . . . . . . . . . . 13 ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → (((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛)))
176174, 175syl5com 31 . . . . . . . . . . . 12 ((𝜑𝑛 ∈ ℕ) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛)))
177176ad2antrr 725 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛)))
178 simpr 488 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → 𝑦𝑋)
179178, 27jctil 523 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋))
180 rphalfcl 12407 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
181180rpxrd 12423 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
182 rpxr 12389 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
183 rphalflt 12409 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
184181, 182, 1833jca 1125 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
185 eqid 2798 . . . . . . . . . . . . . 14 {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
18614, 185blsscls2 23121 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥)) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
187179, 184, 186syl2an 598 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
188 sstr2 3922 . . . . . . . . . . . 12 ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)))
189187, 188syl 17 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)))
190180adantl 485 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
191 breq2 5035 . . . . . . . . . . . . . . . 16 (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
192191rabbidv 3427 . . . . . . . . . . . . . . 15 (𝑟 = (𝑥 / 2) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
193192sseq1d 3946 . . . . . . . . . . . . . 14 (𝑟 = (𝑥 / 2) → ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)))
194193rspcev 3571 . . . . . . . . . . . . 13 (((𝑥 / 2) ∈ ℝ+ ∧ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))
195194ex 416 . . . . . . . . . . . 12 ((𝑥 / 2) ∈ ℝ+ → ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
196190, 195syl 17 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
197177, 189, 1963syld 60 . . . . . . . . . 10 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
198197rexlimdva 3243 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
199172, 198syldan 594 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
200168, 199mpd 15 . . . . . . 7 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))
201200ex 416 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
202163, 201jcad 516 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → (𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))))
203202eximdv 1918 . . . 4 ((𝜑𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))))
204 n0 4260 . . . 4 (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)))
205 df-rex 3112 . . . 4 (∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
206203, 204, 2053imtr4g 299 . . 3 ((𝜑𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ → ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
207206reximdva 3233 . 2 (𝜑 → (∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ → ∃𝑛 ∈ ℕ ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
208156, 207mpd 15 1 (𝜑 → ∃𝑛 ∈ ℕ ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  𝒫 cpw 4497  ∪ cuni 4801  ∩ ciin 4883   class class class wbr 5031   ↦ cmpt 5111  ◡ccnv 5519  ran crn 5521   “ cima 5523   Fn wfn 6320  ⟶wf 6321  ‘cfv 6325  (class class class)co 7136  ℝcr 10528  ℝ*cxr 10666   < clt 10667   ≤ cle 10668   / cdiv 11289  ℕcn 11628  2c2 11683  ℝ+crp 12380  ∞Metcxmet 20080  Metcmet 20081  ballcbl 20082  MetOpencmopn 20085  Topctop 21508  TopOnctopon 21525  Clsdccld 21631  intcnt 21632   Cn ccn 21839  CMetccmet 23868  NrmCVeccnv 28377  BaseSetcba 28379  0veccn0v 28381  normCVcnmcv 28383  IndMetcims 28384   BLnOp cblo 28535  CBanccbn 28655 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444  ax-inf2 9091  ax-dc 9860  ax-cnex 10585  ax-resscn 10586  ax-1cn 10587  ax-icn 10588  ax-addcl 10589  ax-addrcl 10590  ax-mulcl 10591  ax-mulrcl 10592  ax-mulcom 10593  ax-addass 10594  ax-mulass 10595  ax-distr 10596  ax-i2m1 10597  ax-1ne0 10598  ax-1rid 10599  ax-rnegex 10600  ax-rrecex 10601  ax-cnre 10602  ax-pre-lttri 10603  ax-pre-lttrn 10604  ax-pre-ltadd 10605  ax-pre-mulgt0 10606  ax-pre-sup 10607  ax-addf 10608  ax-mulf 10609 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-iin 4885  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6117  df-ord 6163  df-on 6164  df-lim 6165  df-suc 6166  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-riota 7094  df-ov 7139  df-oprab 7140  df-mpo 7141  df-om 7564  df-1st 7674  df-2nd 7675  df-wrecs 7933  df-recs 7994  df-rdg 8032  df-1o 8088  df-er 8275  df-map 8394  df-pm 8395  df-en 8496  df-dom 8497  df-sdom 8498  df-sup 8893  df-inf 8894  df-pnf 10669  df-mnf 10670  df-xr 10671  df-ltxr 10672  df-le 10673  df-sub 10864  df-neg 10865  df-div 11290  df-nn 11629  df-2 11691  df-3 11692  df-n0 11889  df-z 11973  df-uz 12235  df-q 12340  df-rp 12381  df-xneg 12498  df-xadd 12499  df-xmul 12500  df-ico 12735  df-seq 13368  df-exp 13429  df-cj 14453  df-re 14454  df-im 14455  df-sqrt 14589  df-abs 14590  df-rest 16691  df-topgen 16712  df-psmet 20087  df-xmet 20088  df-met 20089  df-bl 20090  df-mopn 20091  df-fbas 20092  df-fg 20093  df-top 21509  df-topon 21526  df-bases 21561  df-cld 21634  df-ntr 21635  df-cls 21636  df-nei 21713  df-cn 21842  df-cnp 21843  df-lm 21844  df-fil 22461  df-fm 22553  df-flim 22554  df-flf 22555  df-cfil 23869  df-cau 23870  df-cmet 23871  df-grpo 28286  df-gid 28287  df-ginv 28288  df-gdiv 28289  df-ablo 28338  df-vc 28352  df-nv 28385  df-va 28388  df-ba 28389  df-sm 28390  df-0v 28391  df-vs 28392  df-nmcv 28393  df-ims 28394  df-lno 28537  df-nmoo 28538  df-blo 28539  df-0o 28540  df-cbn 28656 This theorem is referenced by:  ubthlem3  28665
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