MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ubthlem1 Structured version   Visualization version   GIF version

Theorem ubthlem1 28574
Description: Lemma for ubth 28577. 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 23860, 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 4449 . . . . . . . . 9 (𝑇 = ∅ → ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
21ralrimivw 3180 . . . . . . . 8 (𝑇 = ∅ → ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
3 rabid2 3379 . . . . . . . 8 (𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ ∀𝑧𝑋𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘)
42, 3sylibr 235 . . . . . . 7 (𝑇 = ∅ → 𝑋 = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
54eqcomd 2824 . . . . . 6 (𝑇 = ∅ → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} = 𝑋)
65eleq1d 2894 . . . . 5 (𝑇 = ∅ → ({𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽) ↔ 𝑋 ∈ (Clsd‘𝐽)))
7 iinrab 4982 . . . . . . 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 3964 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑡𝑇) → 𝑡 ∈ (𝑈 BLnOp 𝑊))
12 ubthlem.3 . . . . . . . . . . . . . . . . . . . 20 𝐷 = (IndMet‘𝑈)
13 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (IndMet‘𝑊) = (IndMet‘𝑊)
14 ubthlem.4 . . . . . . . . . . . . . . . . . . . 20 𝐽 = (MetOpen‘𝐷)
15 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (MetOpen‘(IndMet‘𝑊)) = (MetOpen‘(IndMet‘𝑊))
16 eqid 2818 . . . . . . . . . . . . . . . . . . . 20 (𝑈 BLnOp 𝑊) = (𝑈 BLnOp 𝑊)
17 ubthlem.5 . . . . . . . . . . . . . . . . . . . . 21 𝑈 ∈ CBan
18 bnnv 28570 . . . . . . . . . . . . . . . . . . . . 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 28512 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝑈 BLnOp 𝑊) → 𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))))
22 ubth.1 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑋 = (BaseSet‘𝑈)
2322, 12cbncms 28569 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑈 ∈ CBan → 𝐷 ∈ (CMet‘𝑋))
2417, 23ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 𝐷 ∈ (CMet‘𝑋)
25 cmetmet 23816 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
26 metxmet 22871 . . . . . . . . . . . . . . . . . . . . . 22 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
2724, 25, 26mp2b 10 . . . . . . . . . . . . . . . . . . . . 21 𝐷 ∈ (∞Met‘𝑋)
2814mopntopon 22976 . . . . . . . . . . . . . . . . . . . . 21 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2927, 28ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 𝐽 ∈ (TopOn‘𝑋)
30 eqid 2818 . . . . . . . . . . . . . . . . . . . . . . 23 (BaseSet‘𝑊) = (BaseSet‘𝑊)
3130, 13imsxmet 28396 . . . . . . . . . . . . . . . . . . . . . 22 (𝑊 ∈ NrmCVec → (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)))
3220, 31ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 (IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊))
3315mopntopon 22976 . . . . . . . . . . . . . . . . . . . . 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 21805 . . . . . . . . . . . . . . . . . . . 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 688 . . . . . . . . . . . . . . . . . . 19 (𝑡 ∈ (𝐽 Cn (MetOpen‘(IndMet‘𝑊))) ↔ (𝑡:𝑋⟶(BaseSet‘𝑊) ∧ ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽)))
3721, 36sylib 219 . . . . . . . . . . . . . . . . . 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 711 . . . . . . . . . . . . . . 15 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑡:𝑋⟶(BaseSet‘𝑊))
4140ffvelrnda 6843 . . . . . . . . . . . . . 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 6663 . . . . . . . . . . . . . . 15 (𝑦 = (𝑡𝑥) → (𝑁𝑦) = (𝑁‘(𝑡𝑥)))
4443breq1d 5067 . . . . . . . . . . . . . 14 (𝑦 = (𝑡𝑥) → ((𝑁𝑦) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4544elrab 3677 . . . . . . . . . . . . 13 ((𝑡𝑥) ∈ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ↔ ((𝑡𝑥) ∈ (BaseSet‘𝑊) ∧ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
4642, 45syl6bbr 290 . . . . . . . . . . . 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 6668 . . . . . . . . . . . . . 14 (𝑧 = 𝑥 → (𝑁‘(𝑡𝑧)) = (𝑁‘(𝑡𝑥)))
4948breq1d 5067 . . . . . . . . . . . . 13 (𝑧 = 𝑥 → ((𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ (𝑁‘(𝑡𝑥)) ≤ 𝑘))
5049elrab 3677 . . . . . . . . . . . 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 6507 . . . . . . . . . . . 12 (𝑡:𝑋⟶(BaseSet‘𝑊) → 𝑡 Fn 𝑋)
53 elpreima 6820 . . . . . . . . . . . 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 312 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑥 ∈ {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ 𝑥 ∈ (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘})))
5655eqrdv 2816 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
57 imaeq2 5918 . . . . . . . . . . 11 (𝑥 = {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} → (𝑡𝑥) = (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}))
5857eleq1d 2894 . . . . . . . . . 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 711 . . . . . . . . . 10 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → ∀𝑥 ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊)))(𝑡𝑥) ∈ (Clsd‘𝐽))
61 nnre 11633 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → 𝑘 ∈ ℝ)
6261ad2antlr 723 . . . . . . . . . . . 12 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ)
6362rexrd 10679 . . . . . . . . . . 11 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → 𝑘 ∈ ℝ*)
64 eqid 2818 . . . . . . . . . . . . . 14 (0vec𝑊) = (0vec𝑊)
6530, 64nvzcl 28338 . . . . . . . . . . . . 13 (𝑊 ∈ NrmCVec → (0vec𝑊) ∈ (BaseSet‘𝑊))
6620, 65ax-mp 5 . . . . . . . . . . . 12 (0vec𝑊) ∈ (BaseSet‘𝑊)
67 ubth.2 . . . . . . . . . . . . . . . . . 18 𝑁 = (normCV𝑊)
6830, 64, 67, 13nvnd 28392 . . . . . . . . . . . . . . . . 17 ((𝑊 ∈ NrmCVec ∧ 𝑦 ∈ (BaseSet‘𝑊)) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
6920, 68mpan 686 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
70 xmetsym 22884 . . . . . . . . . . . . . . . . 17 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑦 ∈ (BaseSet‘𝑊)) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7132, 66, 70mp3an12 1442 . . . . . . . . . . . . . . . 16 (𝑦 ∈ (BaseSet‘𝑊) → ((0vec𝑊)(IndMet‘𝑊)𝑦) = (𝑦(IndMet‘𝑊)(0vec𝑊)))
7269, 71eqtr4d 2856 . . . . . . . . . . . . . . 15 (𝑦 ∈ (BaseSet‘𝑊) → (𝑁𝑦) = ((0vec𝑊)(IndMet‘𝑊)𝑦))
7372breq1d 5067 . . . . . . . . . . . . . 14 (𝑦 ∈ (BaseSet‘𝑊) → ((𝑁𝑦) ≤ 𝑘 ↔ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘))
7473rabbiia 3470 . . . . . . . . . . . . 13 {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} = {𝑦 ∈ (BaseSet‘𝑊) ∣ ((0vec𝑊)(IndMet‘𝑊)𝑦) ≤ 𝑘}
7515, 74blcld 23042 . . . . . . . . . . . 12 (((IndMet‘𝑊) ∈ (∞Met‘(BaseSet‘𝑊)) ∧ (0vec𝑊) ∈ (BaseSet‘𝑊) ∧ 𝑘 ∈ ℝ*) → {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘} ∈ (Clsd‘(MetOpen‘(IndMet‘𝑊))))
7632, 66, 75mp3an12 1442 . . . . . . . . . . 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 3622 . . . . . . . . 9 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → (𝑡 “ {𝑦 ∈ (BaseSet‘𝑊) ∣ (𝑁𝑦) ≤ 𝑘}) ∈ (Clsd‘𝐽))
7956, 78eqeltrd 2910 . . . . . . . 8 (((𝜑𝑘 ∈ ℕ) ∧ 𝑡𝑇) → {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8079ralrimiva 3179 . . . . . . 7 ((𝜑𝑘 ∈ ℕ) → ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
81 iincld 21575 . . . . . . 7 ((𝑇 ≠ ∅ ∧ ∀𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽)) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
829, 80, 81syl2anr 596 . . . . . 6 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → 𝑡𝑇 {𝑧𝑋 ∣ (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
838, 82eqeltrrd 2911 . . . . 5 (((𝜑𝑘 ∈ ℕ) ∧ 𝑇 ≠ ∅) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
8414mopntop 22977 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
8527, 84ax-mp 5 . . . . . . 7 𝐽 ∈ Top
8629toponunii 21452 . . . . . . . 8 𝑋 = 𝐽
8786topcld 21571 . . . . . . 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 3099 . . . 4 ((𝜑𝑘 ∈ ℕ) → {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ (Clsd‘𝐽))
91 ubthlem.9 . . . 4 𝐴 = (𝑘 ∈ ℕ ↦ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
9290, 91fmptd 6870 . . 3 (𝜑𝐴:ℕ⟶(Clsd‘𝐽))
9392frnd 6514 . . . . . 6 (𝜑 → ran 𝐴 ⊆ (Clsd‘𝐽))
9486cldss2 21566 . . . . . 6 (Clsd‘𝐽) ⊆ 𝒫 𝑋
9593, 94sstrdi 3976 . . . . 5 (𝜑 → ran 𝐴 ⊆ 𝒫 𝑋)
96 sspwuni 5013 . . . . 5 (ran 𝐴 ⊆ 𝒫 𝑋 ran 𝐴𝑋)
9795, 96sylib 219 . . . 4 (𝜑 ran 𝐴𝑋)
98 ubthlem.8 . . . . . 6 (𝜑 → ∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐)
99 arch 11882 . . . . . . . . . 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 10717 . . . . . . . . . . . . . . . . 17 ((𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (𝑐 < 𝑘𝑐𝑘))
103101, 61, 102syl2an 595 . . . . . . . . . . . . . . . 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)
10639ffvelrnda 6843 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑡𝑇) ∧ 𝑥𝑋) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
107106an32s 648 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑡𝑥) ∈ (BaseSet‘𝑊))
10830, 67nvcl 28365 . . . . . . . . . . . . . . . . . 18 ((𝑊 ∈ NrmCVec ∧ (𝑡𝑥) ∈ (BaseSet‘𝑊)) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
10920, 107, 108sylancr 587 . . . . . . . . . . . . . . . . 17 (((𝜑𝑥𝑋) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
110109adantlr 711 . . . . . . . . . . . . . . . 16 ((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
111110adantlr 711 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (𝑁‘(𝑡𝑥)) ∈ ℝ)
112 simpllr 772 . . . . . . . . . . . . . . 15 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → 𝑐 ∈ ℝ)
113 simplrl 773 . . . . . . . . . . . . . . . 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 10722 . . . . . . . . . . . . . . 15 (((𝑁‘(𝑡𝑥)) ∈ ℝ ∧ 𝑐 ∈ ℝ ∧ 𝑘 ∈ ℝ) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
116111, 112, 114, 115syl3anc 1363 . . . . . . . . . . . . . 14 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → (((𝑁‘(𝑡𝑥)) ≤ 𝑐𝑐𝑘) → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
117105, 116mpan2d 690 . . . . . . . . . . . . 13 (((((𝜑𝑥𝑋) ∧ 𝑐 ∈ ℝ) ∧ (𝑘 ∈ ℕ ∧ 𝑐 < 𝑘)) ∧ 𝑡𝑇) → ((𝑁‘(𝑡𝑥)) ≤ 𝑐 → (𝑁‘(𝑡𝑥)) ≤ 𝑘))
118117ralimdva 3174 . . . . . . . . . . . 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 6677 . . . . . . . . . . . . . . . . . 18 𝑋 ∈ V
121120rabex 5226 . . . . . . . . . . . . . . . . 17 {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V
12291fvmpt2 6771 . . . . . . . . . . . . . . . . 17 ((𝑘 ∈ ℕ ∧ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ∈ V) → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
123121, 122mpan2 687 . . . . . . . . . . . . . . . 16 (𝑘 ∈ ℕ → (𝐴𝑘) = {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘})
124123eleq2d 2895 . . . . . . . . . . . . . . 15 (𝑘 ∈ ℕ → (𝑥 ∈ (𝐴𝑘) ↔ 𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘}))
12549ralbidv 3194 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑥 → (∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘 ↔ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
126125elrab 3677 . . . . . . . . . . . . . . 15 (𝑥 ∈ {𝑧𝑋 ∣ ∀𝑡𝑇 (𝑁‘(𝑡𝑧)) ≤ 𝑘} ↔ (𝑥𝑋 ∧ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘))
127124, 126syl6bb 288 . . . . . . . . . . . . . 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 224 . . . . . . . . . . . . . 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 6508 . . . . . . . . . . . . . . 15 (𝜑𝐴 Fn ℕ)
133132adantr 481 . . . . . . . . . . . . . 14 ((𝜑𝑥𝑋) → 𝐴 Fn ℕ)
134 fnfvelrn 6840 . . . . . . . . . . . . . . . 16 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ∈ ran 𝐴)
135 elssuni 4859 . . . . . . . . . . . . . . . 16 ((𝐴𝑘) ∈ ran 𝐴 → (𝐴𝑘) ⊆ ran 𝐴)
136134, 135syl 17 . . . . . . . . . . . . . . 15 ((𝐴 Fn ℕ ∧ 𝑘 ∈ ℕ) → (𝐴𝑘) ⊆ ran 𝐴)
137136sseld 3963 . . . . . . . . . . . . . 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 261 . . . . . . . . . . . 12 (((𝜑𝑥𝑋) ∧ 𝑘 ∈ ℕ) → (∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑘𝑥 ran 𝐴))
140139adantlr 711 . . . . . . . . . . 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 3281 . . . . . . . . 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 3281 . . . . . . 7 ((𝜑𝑥𝑋) → (∃𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐𝑥 ran 𝐴))
145144ralimdva 3174 . . . . . 6 (𝜑 → (∀𝑥𝑋𝑐 ∈ ℝ ∀𝑡𝑇 (𝑁‘(𝑡𝑥)) ≤ 𝑐 → ∀𝑥𝑋 𝑥 ran 𝐴))
14698, 145mpd 15 . . . . 5 (𝜑 → ∀𝑥𝑋 𝑥 ran 𝐴)
147 dfss3 3953 . . . . 5 (𝑋 ran 𝐴 ↔ ∀𝑥𝑋 𝑥 ran 𝐴)
148146, 147sylibr 235 . . . 4 (𝜑𝑋 ran 𝐴)
14997, 148eqssd 3981 . . 3 (𝜑 ran 𝐴 = 𝑋)
150 eqid 2818 . . . . . 6 (0vec𝑈) = (0vec𝑈)
15122, 150nvzcl 28338 . . . . 5 (𝑈 ∈ NrmCVec → (0vec𝑈) ∈ 𝑋)
152 ne0i 4297 . . . . 5 ((0vec𝑈) ∈ 𝑋𝑋 ≠ ∅)
15319, 151, 152mp2b 10 . . . 4 𝑋 ≠ ∅
15414bcth2 23860 . . . 4 (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑋 ≠ ∅) ∧ (𝐴:ℕ⟶(Clsd‘𝐽) ∧ ran 𝐴 = 𝑋)) → ∃𝑛 ∈ ℕ ((int‘𝐽)‘(𝐴𝑛)) ≠ ∅)
15524, 153, 154mpanl12 698 . . 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 ffvelrn 6841 . . . . . . . . . . 11 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ (Clsd‘𝐽))
15894, 157sseldi 3962 . . . . . . . . . 10 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ∈ 𝒫 𝑋)
159158elpwid 4549 . . . . . . . . 9 ((𝐴:ℕ⟶(Clsd‘𝐽) ∧ 𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16092, 159sylan 580 . . . . . . . 8 ((𝜑𝑛 ∈ ℕ) → (𝐴𝑛) ⊆ 𝑋)
16186ntrss3 21596 . . . . . . . 8 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
16285, 160, 161sylancr 587 . . . . . . 7 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
163162sseld 3963 . . . . . 6 ((𝜑𝑛 ∈ ℕ) → (𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → 𝑦𝑋))
16486ntropn 21585 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16585, 160, 164sylancr 587 . . . . . . . . 9 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽)
16614mopni2 23030 . . . . . . . . . 10 ((𝐷 ∈ (∞Met‘𝑋) ∧ ((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → ∃𝑥 ∈ ℝ+ (𝑦(ball‘𝐷)𝑥) ⊆ ((int‘𝐽)‘(𝐴𝑛)))
16727, 166mp3an1 1439 . . . . . . . . 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 4859 . . . . . . . . . . . 12 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝐽)
170169, 86sseqtrrdi 4015 . . . . . . . . . . 11 (((int‘𝐽)‘(𝐴𝑛)) ∈ 𝐽 → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
171165, 170syl 17 . . . . . . . . . 10 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ 𝑋)
172171sselda 3964 . . . . . . . . 9 (((𝜑𝑛 ∈ ℕ) ∧ 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛))) → 𝑦𝑋)
17386ntrss2 21593 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ (𝐴𝑛) ⊆ 𝑋) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
17485, 160, 173sylancr 587 . . . . . . . . . . . . 13 ((𝜑𝑛 ∈ ℕ) → ((int‘𝐽)‘(𝐴𝑛)) ⊆ (𝐴𝑛))
175 sstr2 3971 . . . . . . . . . . . . 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 722 . . . . . . . . . . 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 12404 . . . . . . . . . . . . . . 15 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
181180rpxrd 12420 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ*)
182 rpxr 12386 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+𝑥 ∈ ℝ*)
183 rphalflt 12406 . . . . . . . . . . . . . 14 (𝑥 ∈ ℝ+ → (𝑥 / 2) < 𝑥)
184181, 182, 1833jca 1120 . . . . . . . . . . . . 13 (𝑥 ∈ ℝ+ → ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥))
185 eqid 2818 . . . . . . . . . . . . . 14 {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)}
18614, 185blsscls2 23041 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ ((𝑥 / 2) ∈ ℝ*𝑥 ∈ ℝ* ∧ (𝑥 / 2) < 𝑥)) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
187179, 184, 186syl2an 595 . . . . . . . . . . . 12 ((((𝜑𝑛 ∈ ℕ) ∧ 𝑦𝑋) ∧ 𝑥 ∈ ℝ+) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝑦(ball‘𝐷)𝑥))
188 sstr2 3971 . . . . . . . . . . . 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 5061 . . . . . . . . . . . . . . . 16 (𝑟 = (𝑥 / 2) → ((𝑦𝐷𝑧) ≤ 𝑟 ↔ (𝑦𝐷𝑧) ≤ (𝑥 / 2)))
192191rabbidv 3478 . . . . . . . . . . . . . . 15 (𝑟 = (𝑥 / 2) → {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} = {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)})
193192sseq1d 3995 . . . . . . . . . . . . . 14 (𝑟 = (𝑥 / 2) → ({𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ (𝑥 / 2)} ⊆ (𝐴𝑛)))
194193rspcev 3620 . . . . . . . . . . . . 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 3281 . . . . . . . . 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 1909 . . . 4 ((𝜑𝑛 ∈ ℕ) → (∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)) → ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛))))
204 n0 4307 . . . 4 (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ ↔ ∃𝑦 𝑦 ∈ ((int‘𝐽)‘(𝐴𝑛)))
205 df-rex 3141 . . . 4 (∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛) ↔ ∃𝑦(𝑦𝑋 ∧ ∃𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
206203, 204, 2053imtr4g 297 . . 3 ((𝜑𝑛 ∈ ℕ) → (((int‘𝐽)‘(𝐴𝑛)) ≠ ∅ → ∃𝑦𝑋𝑟 ∈ ℝ+ {𝑧𝑋 ∣ (𝑦𝐷𝑧) ≤ 𝑟} ⊆ (𝐴𝑛)))
207206reximdva 3271 . 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 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  wss 3933  c0 4288  𝒫 cpw 4535   cuni 4830   ciin 4911   class class class wbr 5057  cmpt 5137  ccnv 5547  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  cr 10524  *cxr 10662   < clt 10663  cle 10664   / cdiv 11285  cn 11626  2c2 11680  +crp 12377  ∞Metcxmet 20458  Metcmet 20459  ballcbl 20460  MetOpencmopn 20463  Topctop 21429  TopOnctopon 21446  Clsdccld 21552  intcnt 21553   Cn ccn 21760  CMetccmet 23784  NrmCVeccnv 28288  BaseSetcba 28290  0veccn0v 28292  normCVcnmcv 28294  IndMetcims 28295   BLnOp cblo 28446  CBanccbn 28566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-dc 9856  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603  ax-addf 10604  ax-mulf 10605
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cn 21763  df-cnp 21764  df-lm 21765  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-cfil 23785  df-cau 23786  df-cmet 23787  df-grpo 28197  df-gid 28198  df-ginv 28199  df-gdiv 28200  df-ablo 28249  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-vs 28303  df-nmcv 28304  df-ims 28305  df-lno 28448  df-nmoo 28449  df-blo 28450  df-0o 28451  df-cbn 28567
This theorem is referenced by:  ubthlem3  28576
  Copyright terms: Public domain W3C validator