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Theorem heiborlem8 32787
 Description: Lemma for heibor 32790. 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 𝑍 ∈ 𝑈, and some ball centered at 𝑌 is contained in 𝑍. But the sequence contains arbitrarily small balls close to 𝑌, so some element ball(𝑀‘𝑛) 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(𝑀‘𝑛), yet ball(𝑀‘𝑛) ∈ 𝐾. For convenience, we write this contradiction as 𝜑 → 𝜓 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 22892 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 21949 . . . 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 3569 . . 3 (𝜑𝑍𝐽)
8 heibor.15 . . 3 (𝜑𝑌𝑍)
9 heibor.1 . . . 4 𝐽 = (MetOpen‘𝐷)
109mopni2 22108 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑍𝐽𝑌𝑍) → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
114, 7, 8, 10syl3anc 1318 . 2 (𝜑 → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
12 rphalfcl 11734 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
13 breq2 4587 . . . . . . . 8 (𝑟 = (𝑥 / 2) → ((2nd ‘(𝑀𝑘)) < 𝑟 ↔ (2nd ‘(𝑀𝑘)) < (𝑥 / 2)))
1413rexbidv 3034 . . . . . . 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 32786 . . . . . . 7 𝑟 ∈ ℝ+𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟
2514, 24vtoclri 3256 . . . . . 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 11176 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 32783 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑆𝑘)𝐺𝑘)
30 fvex 6113 . . . . . . . . . 10 (𝑆𝑘) ∈ V
31 vex 3176 . . . . . . . . . 10 𝑘 ∈ V
329, 15, 16, 30, 31heiborlem2 32781 . . . . . . . . 9 ((𝑆𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆𝑘) ∈ (𝐹𝑘) ∧ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
3332simp3bi 1071 . . . . . . . 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 779 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
374ad2antrr 758 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐷 ∈ (∞Met‘𝑋))
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 32784 . . . . . . . . . . . . 13 (𝜑𝑀:ℕ⟶(𝑋 × ℝ+))
3938ffvelrnda 6267 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
4039ad2ant2r 779 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
41 xp1st 7089 . . . . . . . . . . 11 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
4240, 41syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
43 2nn 11062 . . . . . . . . . . . . . . 15 2 ∈ ℕ
44 nnexpcl 12735 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
4543, 28, 44sylancr 694 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ)
4645nnrpd 11746 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+)
4746rpreccld 11758 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ+)
4847ad2antrl 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ+)
4948rpxrd 11749 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ*)
50 xp2nd 7090 . . . . . . . . . . . 12 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5140, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5251rpxrd 11749 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ*)
53 1le3 11121 . . . . . . . . . . . . . 14 1 ≤ 3
54 elrp 11710 . . . . . . . . . . . . . . 15 ((2↑𝑘) ∈ ℝ+ ↔ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)))
55 1re 9918 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
56 3re 10971 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
57 lediv1 10767 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘))) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5855, 56, 57mp3an12 1406 . . . . . . . . . . . . . . 15 (((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5954, 58sylbi 206 . . . . . . . . . . . . . 14 ((2↑𝑘) ∈ ℝ+ → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
6053, 59mpbii 222 . . . . . . . . . . . . 13 ((2↑𝑘) ∈ ℝ+ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6146, 60syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6261ad2antrl 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
63 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑆𝑛) = (𝑆𝑘))
64 oveq2 6557 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
6564oveq2d 6565 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
6663, 65opeq12d 4348 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
67 opex 4859 . . . . . . . . . . . . . . 15 ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩ ∈ V
6866, 23, 67fvmpt 6191 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑀𝑘) = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
6968fveq2d 6107 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
70 ovex 6577 . . . . . . . . . . . . . 14 (3 / (2↑𝑘)) ∈ V
7130, 70op2nd 7068 . . . . . . . . . . . . 13 (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (3 / (2↑𝑘))
7269, 71syl6eq 2660 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7372ad2antrl 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7462, 73breqtrrd 4611 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀𝑘)))
75 ssbl 22038 . . . . . . . . . 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 1329 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))))
7728ad2antrl 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑘 ∈ ℕ0)
78 oveq1 6556 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝑀𝑘)) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))))
79 oveq2 6557 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
8079oveq2d 6565 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
8180oveq2d 6565 . . . . . . . . . . . 12 (𝑚 = 𝑘 → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
82 ovex 6577 . . . . . . . . . . . 12 ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
8378, 81, 17, 82ovmpt2 6694 . . . . . . . . . . 11 (((1st ‘(𝑀𝑘)) ∈ 𝑋𝑘 ∈ ℕ0) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8442, 77, 83syl2anc 691 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8568fveq2d 6107 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
8630, 70op1st 7067 . . . . . . . . . . . . 13 (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (𝑆𝑘)
8785, 86syl6eq 2660 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8887ad2antrl 760 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8988oveq1d 6564 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((𝑆𝑘)𝐵𝑘))
9084, 89eqtr3d 2646 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) = ((𝑆𝑘)𝐵𝑘))
91 1st2nd2 7096 . . . . . . . . . . . 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 6107 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩))
94 df-ov 6552 . . . . . . . . . 10 ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9593, 94syl6reqr 2663 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘(𝑀𝑘)))
9676, 90, 953sstr3d 3610 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ ((ball‘𝐷)‘(𝑀𝑘)))
979mopntop 22055 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
9837, 97syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐽 ∈ Top)
99 blssm 22033 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝑀𝑘)) ∈ ℝ*) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
10037, 42, 52, 99syl3anc 1318 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
1019mopnuni 22056 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
10237, 101syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑋 = 𝐽)
103100, 95, 1023sstr3d 3610 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽)
104 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
105104sscls 20670 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10698, 103, 105syl2anc 691 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10795fveq2d 6107 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) = ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10812ad2antlr 759 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ+)
109108rpxrd 11749 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ*)
110 simprr 792 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
1119blsscls 22122 . . . . . . . . . . . 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 1325 . . . . . . . . . . 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 3603 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
114 rpre 11715 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
115114ad2antlr 759 . . . . . . . . . . 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 32785 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑡 + 1))) ⊆ ((ball‘𝐷)‘(𝑀𝑡)))
1184, 38, 117, 9caublcls 22915 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
1191183expia 1259 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌) → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
120116, 119mpdan 699 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
121120imp 444 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
122121ad2ant2r 779 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
123113, 122sseldd 3569 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
124 blhalf 22020 . . . . . . . . . . 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 1319 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
126113, 125sstrd 3578 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ (𝑌(ball‘𝐷)𝑥))
127106, 126sstrd 3578 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ (𝑌(ball‘𝐷)𝑥))
12896, 127sstrd 3578 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥))
129 sstr2 3575 . . . . . . 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 4388 . . . . . . . . . . . . 13 (𝑍𝑈 {𝑍} = 𝑍)
1326, 131syl 17 . . . . . . . . . . . 12 (𝜑 {𝑍} = 𝑍)
133132sseq2d 3596 . . . . . . . . . . 11 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ {𝑍} ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
134133biimpar 501 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍})
1356snssd 4281 . . . . . . . . . . . . 13 (𝜑 → {𝑍} ⊆ 𝑈)
136 snex 4835 . . . . . . . . . . . . . 14 {𝑍} ∈ V
137136elpw 4114 . . . . . . . . . . . . 13 ({𝑍} ∈ 𝒫 𝑈 ↔ {𝑍} ⊆ 𝑈)
138135, 137sylibr 223 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ 𝒫 𝑈)
139 snfi 7923 . . . . . . . . . . . . 13 {𝑍} ∈ Fin
140139a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ Fin)
141138, 140elind 3760 . . . . . . . . . . 11 (𝜑 → {𝑍} ∈ (𝒫 𝑈 ∩ Fin))
142 unieq 4380 . . . . . . . . . . . . 13 (𝑣 = {𝑍} → 𝑣 = {𝑍})
143142sseq2d 3596 . . . . . . . . . . . 12 (𝑣 = {𝑍} → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}))
144143rspcev 3282 . . . . . . . . . . 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 6577 . . . . . . . . . . 11 ((𝑆𝑘)𝐵𝑘) ∈ V
148 sseq1 3589 . . . . . . . . . . . . 13 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (𝑢 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
149148rexbidv 3034 . . . . . . . . . . . 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 3323 . . . . . . . . . 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 207 . . . . . . . 8 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
154153ex 449 . . . . . . 7 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
155154ad2antrr 758 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
156130, 155syld 46 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
15736, 156mt2d 130 . . . 4 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
15827, 157rexlimddv 3017 . . 3 ((𝜑𝑥 ∈ ℝ+) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
159158nrexdv 2984 . 2 (𝜑 → ¬ ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
16011, 159pm2.21dd 185 1 (𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ifcif 4036  𝒫 cpw 4108  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455   class class class wbr 4583  {copab 4642   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  1st c1st 7057  2nd c2nd 7058  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818  ℝ*cxr 9952   < clt 9953   ≤ cle 9954   − cmin 10145   / cdiv 10563  ℕcn 10897  2c2 10947  3c3 10948  ℕ0cn0 11169  ℝ+crp 11708  seqcseq 12663  ↑cexp 12722  ∞Metcxmt 19552  Metcme 19553  ballcbl 19554  MetOpencmopn 19557  Topctop 20517  clsccl 20632  ⇝𝑡clm 20840  CMetcms 22860 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-q 11665  df-rp 11709  df-xneg 11822  df-xadd 11823  df-xmul 11824  df-fl 12455  df-seq 12664  df-exp 12723  df-topgen 15927  df-psmet 19559  df-xmet 19560  df-met 19561  df-bl 19562  df-mopn 19563  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-lm 20843  df-cmet 22863 This theorem is referenced by:  heiborlem9  32788
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