Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  heiborlem8 Structured version   Visualization version   GIF version

Theorem heiborlem8 33747
Description: Lemma for heibor 33750. The previous lemmas establish that the sequence 𝑀 is Cauchy, so using completeness we now consider the convergent point 𝑌. By assumption, 𝑈 is an open cover, so 𝑌 is an element of some 𝑍𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀𝑛) of the sequence is contained in 𝑍. And finally we arrive at a contradiction, because {𝑍} is a finite subcover of 𝑈 that covers ball(𝑀𝑛), yet ball(𝑀𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑𝜓 where 𝜑 is all the accumulated hypotheses and 𝜓 is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
Hypotheses
Ref Expression
heibor.1 𝐽 = (MetOpen‘𝐷)
heibor.3 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
heibor.4 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
heibor.5 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
heibor.6 (𝜑𝐷 ∈ (CMet‘𝑋))
heibor.7 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
heibor.8 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
heibor.9 (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
heibor.10 (𝜑𝐶𝐺0)
heibor.11 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
heibor.12 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)
heibor.13 (𝜑𝑈𝐽)
heibor.14 𝑌 ∈ V
heibor.15 (𝜑𝑌𝑍)
heibor.16 (𝜑𝑍𝑈)
heibor.17 (𝜑 → (1st𝑀)(⇝𝑡𝐽)𝑌)
Assertion
Ref Expression
heiborlem8 (𝜑𝜓)
Distinct variable groups:   𝑥,𝑛,𝑦,𝑢,𝐹   𝑥,𝐺   𝜑,𝑥   𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧,𝐷   𝑚,𝑀,𝑢,𝑥,𝑦,𝑧   𝑇,𝑚,𝑛,𝑥,𝑦,𝑧   𝐵,𝑛,𝑢,𝑣,𝑦   𝑚,𝐽,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑈,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝜓,𝑦,𝑧   𝑆,𝑚,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝑚,𝑋,𝑛,𝑢,𝑣,𝑥,𝑦,𝑧   𝐶,𝑚,𝑛,𝑢,𝑣,𝑦   𝑛,𝐾,𝑥,𝑦,𝑧   𝑥,𝑌   𝑣,𝑍,𝑥   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝜓(𝑥,𝑣,𝑢,𝑚,𝑛)   𝐵(𝑧,𝑚)   𝐶(𝑥,𝑧)   𝑇(𝑣,𝑢)   𝑈(𝑚)   𝐹(𝑧,𝑣,𝑚)   𝐺(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝐾(𝑣,𝑢,𝑚)   𝑀(𝑣,𝑛)   𝑌(𝑦,𝑧,𝑣,𝑢,𝑚,𝑛)   𝑍(𝑦,𝑧,𝑢,𝑚,𝑛)

Proof of Theorem heiborlem8
Dummy variables 𝑡 𝑘 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 heibor.6 . . . 4 (𝜑𝐷 ∈ (CMet‘𝑋))
2 cmetmet 23130 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 22186 . . . 4 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
41, 2, 33syl 18 . . 3 (𝜑𝐷 ∈ (∞Met‘𝑋))
5 heibor.13 . . . 4 (𝜑𝑈𝐽)
6 heibor.16 . . . 4 (𝜑𝑍𝑈)
75, 6sseldd 3637 . . 3 (𝜑𝑍𝐽)
8 heibor.15 . . 3 (𝜑𝑌𝑍)
9 heibor.1 . . . 4 𝐽 = (MetOpen‘𝐷)
109mopni2 22345 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑍𝐽𝑌𝑍) → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
114, 7, 8, 10syl3anc 1366 . 2 (𝜑 → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
12 rphalfcl 11896 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
13 breq2 4689 . . . . . . . 8 (𝑟 = (𝑥 / 2) → ((2nd ‘(𝑀𝑘)) < 𝑟 ↔ (2nd ‘(𝑀𝑘)) < (𝑥 / 2)))
1413rexbidv 3081 . . . . . . 7 (𝑟 = (𝑥 / 2) → (∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟 ↔ ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2)))
15 heibor.3 . . . . . . . 8 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣}
16 heibor.4 . . . . . . . 8 𝐺 = {⟨𝑦, 𝑛⟩ ∣ (𝑛 ∈ ℕ0𝑦 ∈ (𝐹𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)}
17 heibor.5 . . . . . . . 8 𝐵 = (𝑧𝑋, 𝑚 ∈ ℕ0 ↦ (𝑧(ball‘𝐷)(1 / (2↑𝑚))))
18 heibor.7 . . . . . . . 8 (𝜑𝐹:ℕ0⟶(𝒫 𝑋 ∩ Fin))
19 heibor.8 . . . . . . . 8 (𝜑 → ∀𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ (𝐹𝑛)(𝑦𝐵𝑛))
20 heibor.9 . . . . . . . 8 (𝜑 → ∀𝑥𝐺 ((𝑇𝑥)𝐺((2nd𝑥) + 1) ∧ ((𝐵𝑥) ∩ ((𝑇𝑥)𝐵((2nd𝑥) + 1))) ∈ 𝐾))
21 heibor.10 . . . . . . . 8 (𝜑𝐶𝐺0)
22 heibor.11 . . . . . . . 8 𝑆 = seq0(𝑇, (𝑚 ∈ ℕ0 ↦ if(𝑚 = 0, 𝐶, (𝑚 − 1))))
23 heibor.12 . . . . . . . 8 𝑀 = (𝑛 ∈ ℕ ↦ ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩)
249, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem7 33746 . . . . . . 7 𝑟 ∈ ℝ+𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟
2514, 24vtoclri 3314 . . . . . 6 ((𝑥 / 2) ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2612, 25syl 17 . . . . 5 (𝑥 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2726adantl 481 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
28 nnnn0 11337 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 33743 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑆𝑘)𝐺𝑘)
30 fvex 6239 . . . . . . . . . 10 (𝑆𝑘) ∈ V
31 vex 3234 . . . . . . . . . 10 𝑘 ∈ V
329, 15, 16, 30, 31heiborlem2 33741 . . . . . . . . 9 ((𝑆𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆𝑘) ∈ (𝐹𝑘) ∧ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
3332simp3bi 1098 . . . . . . . 8 ((𝑆𝑘)𝐺𝑘 → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3429, 33syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3528, 34sylan2 490 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3635ad2ant2r 798 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
374ad2antrr 762 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐷 ∈ (∞Met‘𝑋))
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 33744 . . . . . . . . . . . . 13 (𝜑𝑀:ℕ⟶(𝑋 × ℝ+))
3938ffvelrnda 6399 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
4039ad2ant2r 798 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
41 xp1st 7242 . . . . . . . . . . 11 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
4240, 41syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
43 2nn 11223 . . . . . . . . . . . . . . 15 2 ∈ ℕ
44 nnexpcl 12913 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
4543, 28, 44sylancr 696 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ)
4645nnrpd 11908 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+)
4746rpreccld 11920 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ+)
4847ad2antrl 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ+)
4948rpxrd 11911 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ*)
50 xp2nd 7243 . . . . . . . . . . . 12 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5140, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5251rpxrd 11911 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ*)
53 1le3 11282 . . . . . . . . . . . . . 14 1 ≤ 3
54 elrp 11872 . . . . . . . . . . . . . . 15 ((2↑𝑘) ∈ ℝ+ ↔ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)))
55 1re 10077 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
56 3re 11132 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
57 lediv1 10926 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘))) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5855, 56, 57mp3an12 1454 . . . . . . . . . . . . . . 15 (((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5954, 58sylbi 207 . . . . . . . . . . . . . 14 ((2↑𝑘) ∈ ℝ+ → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
6053, 59mpbii 223 . . . . . . . . . . . . 13 ((2↑𝑘) ∈ ℝ+ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6146, 60syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6261ad2antrl 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
63 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑆𝑛) = (𝑆𝑘))
64 oveq2 6698 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
6564oveq2d 6706 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
6663, 65opeq12d 4441 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
67 opex 4962 . . . . . . . . . . . . . . 15 ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩ ∈ V
6866, 23, 67fvmpt 6321 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑀𝑘) = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
6968fveq2d 6233 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
70 ovex 6718 . . . . . . . . . . . . . 14 (3 / (2↑𝑘)) ∈ V
7130, 70op2nd 7219 . . . . . . . . . . . . 13 (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (3 / (2↑𝑘))
7269, 71syl6eq 2701 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7372ad2antrl 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7462, 73breqtrrd 4713 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀𝑘)))
75 ssbl 22275 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋) ∧ ((1 / (2↑𝑘)) ∈ ℝ* ∧ (2nd ‘(𝑀𝑘)) ∈ ℝ*) ∧ (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀𝑘))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))))
7637, 42, 49, 52, 74, 75syl221anc 1377 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))))
7728ad2antrl 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑘 ∈ ℕ0)
78 oveq1 6697 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝑀𝑘)) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))))
79 oveq2 6698 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
8079oveq2d 6706 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
8180oveq2d 6706 . . . . . . . . . . . 12 (𝑚 = 𝑘 → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
82 ovex 6718 . . . . . . . . . . . 12 ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
8378, 81, 17, 82ovmpt2 6838 . . . . . . . . . . 11 (((1st ‘(𝑀𝑘)) ∈ 𝑋𝑘 ∈ ℕ0) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8442, 77, 83syl2anc 694 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8568fveq2d 6233 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
8630, 70op1st 7218 . . . . . . . . . . . . 13 (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (𝑆𝑘)
8785, 86syl6eq 2701 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8887ad2antrl 764 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8988oveq1d 6705 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((𝑆𝑘)𝐵𝑘))
9084, 89eqtr3d 2687 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) = ((𝑆𝑘)𝐵𝑘))
91 1st2nd2 7249 . . . . . . . . . . . 12 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (𝑀𝑘) = ⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9240, 91syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑀𝑘) = ⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9392fveq2d 6233 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩))
94 df-ov 6693 . . . . . . . . . 10 ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9593, 94syl6reqr 2704 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘(𝑀𝑘)))
9676, 90, 953sstr3d 3680 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ ((ball‘𝐷)‘(𝑀𝑘)))
979mopntop 22292 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
9837, 97syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐽 ∈ Top)
99 blssm 22270 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝑀𝑘)) ∈ ℝ*) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
10037, 42, 52, 99syl3anc 1366 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
1019mopnuni 22293 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
10237, 101syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑋 = 𝐽)
103100, 95, 1023sstr3d 3680 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽)
104 eqid 2651 . . . . . . . . . . 11 𝐽 = 𝐽
105104sscls 20908 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10698, 103, 105syl2anc 694 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10795fveq2d 6233 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) = ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10812ad2antlr 763 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ+)
109108rpxrd 11911 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ*)
110 simprr 811 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
1119blsscls 22359 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋) ∧ ((2nd ‘(𝑀𝑘)) ∈ ℝ* ∧ (𝑥 / 2) ∈ ℝ* ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
11237, 42, 52, 109, 110, 111syl23anc 1373 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
113107, 112eqsstr3d 3673 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
114 rpre 11877 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
115114ad2antlr 763 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑥 ∈ ℝ)
116 heibor.17 . . . . . . . . . . . . . . 15 (𝜑 → (1st𝑀)(⇝𝑡𝐽)𝑌)
1179, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem6 33745 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑡 + 1))) ⊆ ((ball‘𝐷)‘(𝑀𝑡)))
1184, 38, 117, 9caublcls 23153 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
1191183expia 1286 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌) → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
120116, 119mpdan 703 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
121120imp 444 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
122121ad2ant2r 798 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
123113, 122sseldd 3637 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
124 blhalf 22257 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋) ∧ (𝑥 ∈ ℝ ∧ 𝑌 ∈ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
12537, 42, 115, 123, 124syl22anc 1367 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
126113, 125sstrd 3646 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ (𝑌(ball‘𝐷)𝑥))
127106, 126sstrd 3646 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ (𝑌(ball‘𝐷)𝑥))
12896, 127sstrd 3646 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥))
129 sstr2 3643 . . . . . . 7 (((𝑆𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
130128, 129syl 17 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
131 unisng 4484 . . . . . . . . . . . . 13 (𝑍𝑈 {𝑍} = 𝑍)
1326, 131syl 17 . . . . . . . . . . . 12 (𝜑 {𝑍} = 𝑍)
133132sseq2d 3666 . . . . . . . . . . 11 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ {𝑍} ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
134133biimpar 501 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍})
1356snssd 4372 . . . . . . . . . . . . 13 (𝜑 → {𝑍} ⊆ 𝑈)
136 snex 4938 . . . . . . . . . . . . . 14 {𝑍} ∈ V
137136elpw 4197 . . . . . . . . . . . . 13 ({𝑍} ∈ 𝒫 𝑈 ↔ {𝑍} ⊆ 𝑈)
138135, 137sylibr 224 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ 𝒫 𝑈)
139 snfi 8079 . . . . . . . . . . . . 13 {𝑍} ∈ Fin
140139a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ Fin)
141138, 140elind 3831 . . . . . . . . . . 11 (𝜑 → {𝑍} ∈ (𝒫 𝑈 ∩ Fin))
142 unieq 4476 . . . . . . . . . . . . 13 (𝑣 = {𝑍} → 𝑣 = {𝑍})
143142sseq2d 3666 . . . . . . . . . . . 12 (𝑣 = {𝑍} → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}))
144143rspcev 3340 . . . . . . . . . . 11 (({𝑍} ∈ (𝒫 𝑈 ∩ Fin) ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
145141, 144sylan 487 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
146134, 145syldan 486 . . . . . . . . 9 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
147 ovex 6718 . . . . . . . . . . 11 ((𝑆𝑘)𝐵𝑘) ∈ V
148 sseq1 3659 . . . . . . . . . . . . 13 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (𝑢 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
149148rexbidv 3081 . . . . . . . . . . . 12 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
150149notbid 307 . . . . . . . . . . 11 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
151147, 150, 15elab2 3386 . . . . . . . . . 10 (((𝑆𝑘)𝐵𝑘) ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
152151con2bii 346 . . . . . . . . 9 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
153146, 152sylib 208 . . . . . . . 8 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
154153ex 449 . . . . . . 7 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
155154ad2antrr 762 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
156130, 155syld 47 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
15736, 156mt2d 131 . . . 4 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
15827, 157rexlimddv 3064 . . 3 ((𝜑𝑥 ∈ ℝ+) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
159158nrexdv 3030 . 2 (𝜑 → ¬ ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
16011, 159pm2.21dd 186 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  {cab 2637  wral 2941  wrex 2942  Vcvv 3231  cin 3606  wss 3607  ifcif 4119  𝒫 cpw 4191  {csn 4210  cop 4216   cuni 4468   ciun 4552   class class class wbr 4685  {copab 4745  cmpt 4762   × cxp 5141  ccom 5147  wf 5922  cfv 5926  (class class class)co 6690  cmpt2 6692  1st c1st 7208  2nd c2nd 7209  Fincfn 7997  cr 9973  0cc0 9974  1c1 9975   + caddc 9977  *cxr 10111   < clt 10112  cle 10113  cmin 10304   / cdiv 10722  cn 11058  2c2 11108  3c3 11109  0cn0 11330  +crp 11870  seqcseq 12841  cexp 12900  ∞Metcxmt 19779  Metcme 19780  ballcbl 19781  MetOpencmopn 19784  Topctop 20746  clsccl 20870  𝑡clm 21078  CMetcms 23098
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-pre-sup 10052
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-er 7787  df-map 7901  df-pm 7902  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-sup 8389  df-inf 8390  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-div 10723  df-nn 11059  df-2 11117  df-3 11118  df-n0 11331  df-z 11416  df-uz 11726  df-q 11827  df-rp 11871  df-xneg 11984  df-xadd 11985  df-xmul 11986  df-fl 12633  df-seq 12842  df-exp 12901  df-topgen 16151  df-psmet 19786  df-xmet 19787  df-met 19788  df-bl 19789  df-mopn 19790  df-top 20747  df-topon 20764  df-bases 20798  df-cld 20871  df-ntr 20872  df-cls 20873  df-lm 21081  df-cmet 23101
This theorem is referenced by:  heiborlem9  33748
  Copyright terms: Public domain W3C validator