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Theorem ubthlem1 29812
Description: Lemma for ubth 29815. 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 24694, for some 𝑛 the set 𝐴𝑛 has an interior, meaning that there is a closed ball {𝑧𝑋 ∣ (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 4466 . . . . . . . . 9 (𝑇 = ∅ → ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
21ralrimivw 3147 . . . . . . . 8 (𝑇 = ∅ → ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
3 rabid2 3436 . . . . . . . 8 (𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
42, 3sylibr 233 . . . . . . 7 (𝑇 = ∅ → 𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
54eqcomd 2742 . . . . . 6 (𝑇 = ∅ → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} = 𝑋)
65eleq1d 2822 . . . . 5 (𝑇 = ∅ → ({𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7 iinrab 5029 . . . . . . 7 (𝑇 ≠ ∅ → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
87adantl 482 . . . . . 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 3944 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12 ubthlem.3 . . . . . . . . . . . . . . . . . . . 20 𝐷 = (IndMet‘𝑈)
13 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (IndMet‘𝑊) = (IndMet‘𝑊)
14 ubthlem.4 . . . . . . . . . . . . . . . . . . . 20 𝐽 = (MetOpen‘𝐷)
15 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16 eqid 2736 . . . . . . . . . . . . . . . . . . . 20 (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17 ubthlem.5 . . . . . . . . . . . . . . . . . . . . 21 𝑈 ∈ CBan
18 bnnv 29808 . . . . . . . . . . . . . . . . . . . . 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 29750 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22 ubth.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (BaseSet‘𝑈)
2322, 12cbncms 29807 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
2417, 23ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 𝐷 ∈ (CMet‘𝑋)
25 cmetmet 24650 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26 metxmet 23687 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
2724, 25, 26mp2b 10 . . . . . . . . . . . . . . . . . . . . 21 𝐷 ∈ (∞Met‘𝑋)
2814mopntopon 23792 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 𝐽 ∈ (TopOn‘𝑋)
30 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . 23 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3130, 13imsxmet 29634 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
3220, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
3315mopntopon 23792 . . . . . . . . . . . . . . . . . . . . 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 22620 . . . . . . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽)))
3721, 36sylib 217 . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . 16 ((𝜑𝑡𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
4039adantlr 713 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
4140ffvelcdmda 7035 . . . . . . . . . . . . . 14 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
4241biantrurd 533 . . . . . . . . . . . . 13 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → ((𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ ((𝑡𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
43 fveq2 6842 . . . . . . . . . . . . . . 15 (𝑦 = (𝑡𝑥) → (𝑁𝑦) = (𝑁‘(𝑡𝑥)))
4443breq1d 5115 . . . . . . . . . . . . . 14 (𝑦 = (𝑡𝑥) → ((𝑁𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4544elrab 3645 . . . . . . . . . . . . 13 ((𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ↔ ((𝑡𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4642, 45bitr4di 288 . . . . . . . . . . . 12 ((((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) ∧ 𝑥𝑋) → ((𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
4746pm5.32da 579 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → ((𝑥𝑋 ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘) ↔ (𝑥𝑋 ∧ (𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
48 2fveq3 6847 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑁‘(𝑡𝑧)) = (𝑁‘(𝑡𝑥)))
4948breq1d 5115 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
5049elrab 3645 . . . . . . . . . . . 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 6668 . . . . . . . . . . . 12 (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
53 elpreima 7008 . . . . . . . . . . . 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 310 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑥 ∈ {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
5655eqrdv 2734 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
57 imaeq2 6009 . . . . . . . . . . 11 (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} → (𝑡𝑥) = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
5857eleq1d 2822 . . . . . . . . . 10 (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} → ((𝑡𝑥) ∈ (Clsd‘𝐽) ↔ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽)))
5938simprd 496 . . . . . . . . . . 11 ((𝜑𝑡𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽))
6059adantlr 713 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽))
61 nnre 12160 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
6261ad2antlr 725 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ)
6362rexrd 11205 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ*)
64 eqid 2736 . . . . . . . . . . . . . 14 (0vec𝑊) = (0vec𝑊)
6530, 64nvzcl 29576 . . . . . . . . . . . . 13 (𝑊 ∈ NrmCVec → (0vec𝑊) ∈ (BaseSet‘𝑊))
6620, 65ax-mp 5 . . . . . . . . . . . 12 (0vec𝑊) ∈ (BaseSet‘𝑊)
67 ubth.2 . . . . . . . . . . . . . . . . . 18 𝑁 = (normCV𝑊)
6830, 64, 67, 13nvnd 29630 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
6920, 68mpan 688 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
70 xmetsym 23700 . . . . . . . . . . . . . . . . 17 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7132, 66, 70mp3an12 1451 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7269, 71eqtr4d 2779 . . . . . . . . . . . . . . 15 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = ((0vec𝑊)(IndMet‘𝑊)𝑦))
7372breq1d 5115 . . . . . . . . . . . . . 14 (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁𝑦) ≤ 𝑘 ↔ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
7473rabbiia 3411 . . . . . . . . . . . . 13 {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
7515, 74blcld 23861 . . . . . . . . . . . 12 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑘 ∈ ℝ*) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
7632, 66, 75mp3an12 1451 . . . . . . . . . . 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 3582 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
7956, 78eqeltrd 2838 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8079ralrimiva 3143 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
81 iincld 22390 . . . . . . 7 ((𝑇 ≠ ∅ ∧ ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽)) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
829, 80, 81syl2anr 597 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
838, 82eqeltrrd 2839 . . . . 5 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8414mopntop 23793 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
8527, 84ax-mp 5 . . . . . . 7 𝐽 ∈ Top
8629toponunii 22265 . . . . . . . 8 𝑋 = 𝐽
8786topcld 22386 . . . . . . 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 3030 . . . 4 ((𝜑𝑘 ∈ ℕ) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
91 ubthlem.9 . . . 4 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
9290, 91fmptd 7062 . . 3 (𝜑𝐴:ℕ⟶(Clsd‘𝐽))
9392frnd 6676 . . . . . 6 (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
9486cldss2 22381 . . . . . 6 (Clsd‘𝐽) ⊆ 𝒫 𝑋
9593, 94sstrdi 3956 . . . . 5 (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
96 sspwuni 5060 . . . . 5 (ran 𝐴 ⊆ 𝒫 𝑋 ran 𝐴𝑋)
9795, 96sylib 217 . . . 4 (𝜑 ran 𝐴𝑋)
98 ubthlem.8 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)
99 arch 12410 . . . . . . . . . 10 (𝑐 ∈ ℝ → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
10099adantl 482 . . . . . . . . 9 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → ∃𝑘 ∈ ℕ 𝑐 < 𝑘)
101 simpr 485 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) → 𝑐 ∈ ℝ)
102 ltle 11243 . . . . . . . . . . . . . . . . 17 ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘𝑐𝑘))
103101, 61, 102syl2an 596 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘𝑐𝑘))
104103impr 455 . . . . . . . . . . . . . . 15 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → 𝑐𝑘)
105104adantr 481 . . . . . . . . . . . . . 14 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑐𝑘)
10639ffvelcdmda 7035 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑡𝑇) ∧ 𝑥𝑋) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
107106an32s 650 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
10830, 67nvcl 29603 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ NrmCVec ∧ (𝑡𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
10920, 107, 108sylancr 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
110109adantlr 713 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
111110adantlr 713 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
112 simpllr 774 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑐 ∈ ℝ)
113 simplrl 775 . . . . . . . . . . . . . . . 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 11249 . . . . . . . . . . . . . . 15 (((𝑁‘(𝑡𝑥)) ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
116111, 112, 114, 115syl3anc 1371 . . . . . . . . . . . . . 14 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
117105, 116mpan2d 692 . . . . . . . . . . . . 13 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → ((𝑁‘(𝑡𝑥)) ≤ 𝑐 → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
118117ralimdva 3164 . . . . . . . . . . . 12 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
119118expr 457 . . . . . . . . . . 11 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑘 ∈ ℕ) → (𝑐 < 𝑘 → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
12022fvexi 6856 . . . . . . . . . . . . . . . . . 18 𝑋 ∈ V
121120rabex 5289 . . . . . . . . . . . . . . . . 17 {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V
12291fvmpt2 6959 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V) → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
123121, 122mpan2 689 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
124123eleq2d 2823 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘}))
12549ralbidv 3174 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
126125elrab 3645 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
127124, 126bitrdi 286 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴𝑘) ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
128 simpr 485 . . . . . . . . . . . . . . . 16 ((𝜑𝑥𝑋) → 𝑥𝑋)
129128biantrurd 533 . . . . . . . . . . . . . . 15 ((𝜑𝑥𝑋) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘 ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘)))
130129bicomd 222 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → ((𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘) ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
131127, 130sylan9bbr 511 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
13292ffnd 6669 . . . . . . . . . . . . . . 15 (𝜑𝐴 Fn ℕ)
133132adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐴 Fn ℕ)
134 fnfvelrn 7031 . . . . . . . . . . . . . . . 16 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ran 𝐴)
135 elssuni 4898 . . . . . . . . . . . . . . . 16 ((𝐴𝑘) ∈ ran 𝐴 → (𝐴𝑘) ⊆ ran 𝐴)
136134, 135syl 17 . . . . . . . . . . . . . . 15 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ⊆ ran 𝐴)
137136sseld 3943 . . . . . . . . . . . . . 14 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) → 𝑥 ran 𝐴))
138133, 137sylan 580 . . . . . . . . . . . . 13 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (𝑥 ∈ (𝐴𝑘) → 𝑥 ran 𝐴))
139131, 138sylbird 259 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘𝑥 ran 𝐴))
140139adantlr 713 . . . . . . . . . . 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 3152 . . . . . . . . 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 3152 . . . . . . 7 ((𝜑𝑥𝑋) → (∃𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴))
145144ralimdva 3164 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑥𝑋 𝑥 ran 𝐴))
14698, 145mpd 15 . . . . 5 (𝜑 → ∀𝑥𝑋 𝑥 ran 𝐴)
147 dfss3 3932 . . . . 5 (𝑋 ran 𝐴 ↔ ∀𝑥𝑋 𝑥 ran 𝐴)
148146, 147sylibr 233 . . . 4 (𝜑𝑋 ran 𝐴)
14997, 148eqssd 3961 . . 3 (𝜑 ran 𝐴 = 𝑋)
150 eqid 2736 . . . . . 6 (0vec𝑈) = (0vec𝑈)
15122, 150nvzcl 29576 . . . . 5 (𝑈 ∈ NrmCVec → (0vec𝑈) ∈ 𝑋)
152 ne0i 4294 . . . . 5 ((0vec𝑈) ∈ 𝑋𝑋 ≠ ∅)
15319, 151, 152mp2b 10 . . . 4 𝑋 ≠ ∅
15414bcth2 24694 . . . 4 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
15524, 153, 154mpanl12 700 . . 3 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ ran 𝐴 = 𝑋) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
15692, 149, 155syl2anc 584 . 2 (𝜑 → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
157 ffvelcdm 7032 . . . . . . . . . . 11 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ (Clsd‘𝐽))
15894, 157sselid 3942 . . . . . . . . . 10 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 𝑋)
159158elpwid 4569 . . . . . . . . 9 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16092, 159sylan 580 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16186ntrss3 22411 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
16285, 160, 161sylancr 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
163162sseld 3943 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → 𝑦𝑋))
16486ntropn 22400 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16585, 160, 164sylancr 587 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16614mopni2 23849 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
16727, 166mp3an1 1448 . . . . . . . . 9 ((((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
168165, 167sylan 580 . . . . . . . 8 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
169 elssuni 4898 . . . . . . . . . . . 12 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝐽)
170169, 86sseqtrrdi 3995 . . . . . . . . . . 11 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
171165, 170syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
172171sselda 3944 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → 𝑦𝑋)
17386ntrss2 22408 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
17485, 160, 173sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
175 sstr2 3951 . . . . . . . . . . . . 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 724 . . . . . . . . . . 11 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → (𝑦(ball‘𝐷)𝑥) ⊆ (𝐴𝑛)))
178 simpr 485 . . . . . . . . . . . . . 14 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → 𝑦𝑋)
179178, 27jctil 520 . . . . . . . . . . . . 13 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → (𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋))
180 rphalfcl 12942 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
181180rpxrd 12958 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
182 rpxr 12924 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
183 rphalflt 12944 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
184181, 182, 1833jca 1128 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
185 eqid 2736 . . . . . . . . . . . . . 14 {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
18614, 185blsscls2 23860 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥)) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
187179, 184, 186syl2an 596 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
188 sstr2 3951 . . . . . . . . . . . 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 482 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → (𝑥 / 2) ∈ ℝ+)
191 breq2 5109 . . . . . . . . . . . . . . . 16 (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
192191rabbidv 3415 . . . . . . . . . . . . . . 15 (𝑟 = (𝑥 / 2) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
193192sseq1d 3975 . . . . . . . . . . . . . 14 (𝑟 = (𝑥 / 2) → ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)))
194193rspcev 3581 . . . . . . . . . . . . 13 (((𝑥 / 2) ∈ ℝ+ ∧ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))
195194ex 413 . . . . . . . . . . . 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 3152 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) → (∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
199172, 198syldan 591 . . . . . . . 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 413 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
202163, 201jcad 513 . . . . 5 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → (𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))))
203202eximdv 1920 . . . 4 ((𝜑𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))))
204 n0 4306 . . . 4 (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)))
205 df-rex 3074 . . . 4 (∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
206203, 204, 2053imtr4g 295 . . 3 ((𝜑𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ → ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
207206reximdva 3165 . 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 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560   cuni 4865   ciin 4955   class class class wbr 5105  cmpt 5188  ccnv 5632  ran crn 5634  cima 5636   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cr 11050  *cxr 11188   < clt 11189  cle 11190   / cdiv 11812  cn 12153  2c2 12208  +crp 12915  ∞Metcxmet 20781  Metcmet 20782  ballcbl 20783  MetOpencmopn 20786  Topctop 22242  TopOnctopon 22259  Clsdccld 22367  intcnt 22368   Cn ccn 22575  CMetccmet 24618  NrmCVeccnv 29526  BaseSetcba 29528  0veccn0v 29530  normCVcnmcv 29532  IndMetcims 29533   BLnOp cblo 29684  CBanccbn 29804
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-dc 10382  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129  ax-addf 11130  ax-mulf 11131
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-sup 9378  df-inf 9379  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-xneg 13033  df-xadd 13034  df-xmul 13035  df-ico 13270  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-rest 17304  df-topgen 17325  df-psmet 20788  df-xmet 20789  df-met 20790  df-bl 20791  df-mopn 20792  df-fbas 20793  df-fg 20794  df-top 22243  df-topon 22260  df-bases 22296  df-cld 22370  df-ntr 22371  df-cls 22372  df-nei 22449  df-cn 22578  df-cnp 22579  df-lm 22580  df-fil 23197  df-fm 23289  df-flim 23290  df-flf 23291  df-cfil 24619  df-cau 24620  df-cmet 24621  df-grpo 29435  df-gid 29436  df-ginv 29437  df-gdiv 29438  df-ablo 29487  df-vc 29501  df-nv 29534  df-va 29537  df-ba 29538  df-sm 29539  df-0v 29540  df-vs 29541  df-nmcv 29542  df-ims 29543  df-lno 29686  df-nmoo 29687  df-blo 29688  df-0o 29689  df-cbn 29805
This theorem is referenced by:  ubthlem3  29814
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