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Theorem heiborlem8 32583
Description: Lemma for heibor 32586. 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 22810 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 21890 . . . 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 3568 . . 3 (𝜑𝑍𝐽)
8 heibor.15 . . 3 (𝜑𝑌𝑍)
9 heibor.1 . . . 4 𝐽 = (MetOpen‘𝐷)
109mopni2 22049 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑍𝐽𝑌𝑍) → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
114, 7, 8, 10syl3anc 1317 . 2 (𝜑 → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
12 rphalfcl 11690 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
13 breq2 4581 . . . . . . . 8 (𝑟 = (𝑥 / 2) → ((2nd ‘(𝑀𝑘)) < 𝑟 ↔ (2nd ‘(𝑀𝑘)) < (𝑥 / 2)))
1413rexbidv 3033 . . . . . . 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 32582 . . . . . . 7 𝑟 ∈ ℝ+𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟
2514, 24vtoclri 3255 . . . . . 6 ((𝑥 / 2) ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2612, 25syl 17 . . . . 5 (𝑥 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2726adantl 480 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
28 nnnn0 11146 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 32579 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑆𝑘)𝐺𝑘)
30 fvex 6098 . . . . . . . . . 10 (𝑆𝑘) ∈ V
31 vex 3175 . . . . . . . . . 10 𝑘 ∈ V
329, 15, 16, 30, 31heiborlem2 32577 . . . . . . . . 9 ((𝑆𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆𝑘) ∈ (𝐹𝑘) ∧ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
3332simp3bi 1070 . . . . . . . 8 ((𝑆𝑘)𝐺𝑘 → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3429, 33syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3528, 34sylan2 489 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3635ad2ant2r 778 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
374ad2antrr 757 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐷 ∈ (∞Met‘𝑋))
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 32580 . . . . . . . . . . . . 13 (𝜑𝑀:ℕ⟶(𝑋 × ℝ+))
3938ffvelrnda 6252 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
4039ad2ant2r 778 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
41 xp1st 7066 . . . . . . . . . . 11 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
4240, 41syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
43 2nn 11032 . . . . . . . . . . . . . . 15 2 ∈ ℕ
44 nnexpcl 12690 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
4543, 28, 44sylancr 693 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ)
4645nnrpd 11702 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+)
4746rpreccld 11714 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ+)
4847ad2antrl 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ+)
4948rpxrd 11705 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ*)
50 xp2nd 7067 . . . . . . . . . . . 12 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5140, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5251rpxrd 11705 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ*)
53 1le3 11091 . . . . . . . . . . . . . 14 1 ≤ 3
54 elrp 11666 . . . . . . . . . . . . . . 15 ((2↑𝑘) ∈ ℝ+ ↔ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)))
55 1re 9895 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
56 3re 10941 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
57 lediv1 10737 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘))) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5855, 56, 57mp3an12 1405 . . . . . . . . . . . . . . 15 (((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5954, 58sylbi 205 . . . . . . . . . . . . . 14 ((2↑𝑘) ∈ ℝ+ → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
6053, 59mpbii 221 . . . . . . . . . . . . 13 ((2↑𝑘) ∈ ℝ+ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6146, 60syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6261ad2antrl 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
63 fveq2 6088 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑆𝑛) = (𝑆𝑘))
64 oveq2 6535 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
6564oveq2d 6543 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
6663, 65opeq12d 4342 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
67 opex 4853 . . . . . . . . . . . . . . 15 ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩ ∈ V
6866, 23, 67fvmpt 6176 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑀𝑘) = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
6968fveq2d 6092 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
70 ovex 6555 . . . . . . . . . . . . . 14 (3 / (2↑𝑘)) ∈ V
7130, 70op2nd 7045 . . . . . . . . . . . . 13 (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (3 / (2↑𝑘))
7269, 71syl6eq 2659 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7372ad2antrl 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7462, 73breqtrrd 4605 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀𝑘)))
75 ssbl 21979 . . . . . . . . . 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 1328 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))))
7728ad2antrl 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑘 ∈ ℕ0)
78 oveq1 6534 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝑀𝑘)) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))))
79 oveq2 6535 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
8079oveq2d 6543 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
8180oveq2d 6543 . . . . . . . . . . . 12 (𝑚 = 𝑘 → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
82 ovex 6555 . . . . . . . . . . . 12 ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
8378, 81, 17, 82ovmpt2 6672 . . . . . . . . . . 11 (((1st ‘(𝑀𝑘)) ∈ 𝑋𝑘 ∈ ℕ0) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8442, 77, 83syl2anc 690 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8568fveq2d 6092 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
8630, 70op1st 7044 . . . . . . . . . . . . 13 (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (𝑆𝑘)
8785, 86syl6eq 2659 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8887ad2antrl 759 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8988oveq1d 6542 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((𝑆𝑘)𝐵𝑘))
9084, 89eqtr3d 2645 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) = ((𝑆𝑘)𝐵𝑘))
91 1st2nd2 7073 . . . . . . . . . . . 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 6092 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩))
94 df-ov 6530 . . . . . . . . . 10 ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9593, 94syl6reqr 2662 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘(𝑀𝑘)))
9676, 90, 953sstr3d 3609 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ ((ball‘𝐷)‘(𝑀𝑘)))
979mopntop 21996 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
9837, 97syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐽 ∈ Top)
99 blssm 21974 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝑀𝑘)) ∈ ℝ*) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
10037, 42, 52, 99syl3anc 1317 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
1019mopnuni 21997 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
10237, 101syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑋 = 𝐽)
103100, 95, 1023sstr3d 3609 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽)
104 eqid 2609 . . . . . . . . . . 11 𝐽 = 𝐽
105104sscls 20612 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10698, 103, 105syl2anc 690 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10795fveq2d 6092 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) = ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10812ad2antlr 758 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ+)
109108rpxrd 11705 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ*)
110 simprr 791 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
1119blsscls 22063 . . . . . . . . . . . 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 1324 . . . . . . . . . . 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 3602 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
114 rpre 11671 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
115114ad2antlr 758 . . . . . . . . . . 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 32581 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑡 + 1))) ⊆ ((ball‘𝐷)‘(𝑀𝑡)))
1184, 38, 117, 9caublcls 22832 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
1191183expia 1258 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌) → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
120116, 119mpdan 698 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
121120imp 443 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
122121ad2ant2r 778 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
123113, 122sseldd 3568 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
124 blhalf 21961 . . . . . . . . . . 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 1318 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
126113, 125sstrd 3577 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ (𝑌(ball‘𝐷)𝑥))
127106, 126sstrd 3577 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ (𝑌(ball‘𝐷)𝑥))
12896, 127sstrd 3577 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥))
129 sstr2 3574 . . . . . . 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 4382 . . . . . . . . . . . . 13 (𝑍𝑈 {𝑍} = 𝑍)
1326, 131syl 17 . . . . . . . . . . . 12 (𝜑 {𝑍} = 𝑍)
133132sseq2d 3595 . . . . . . . . . . 11 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ {𝑍} ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
134133biimpar 500 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍})
1356snssd 4280 . . . . . . . . . . . . 13 (𝜑 → {𝑍} ⊆ 𝑈)
136 snex 4830 . . . . . . . . . . . . . 14 {𝑍} ∈ V
137136elpw 4113 . . . . . . . . . . . . 13 ({𝑍} ∈ 𝒫 𝑈 ↔ {𝑍} ⊆ 𝑈)
138135, 137sylibr 222 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ 𝒫 𝑈)
139 snfi 7900 . . . . . . . . . . . . 13 {𝑍} ∈ Fin
140139a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ Fin)
141138, 140elind 3759 . . . . . . . . . . 11 (𝜑 → {𝑍} ∈ (𝒫 𝑈 ∩ Fin))
142 unieq 4374 . . . . . . . . . . . . 13 (𝑣 = {𝑍} → 𝑣 = {𝑍})
143142sseq2d 3595 . . . . . . . . . . . 12 (𝑣 = {𝑍} → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}))
144143rspcev 3281 . . . . . . . . . . 11 (({𝑍} ∈ (𝒫 𝑈 ∩ Fin) ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
145141, 144sylan 486 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
146134, 145syldan 485 . . . . . . . . 9 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
147 ovex 6555 . . . . . . . . . . 11 ((𝑆𝑘)𝐵𝑘) ∈ V
148 sseq1 3588 . . . . . . . . . . . . 13 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (𝑢 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
149148rexbidv 3033 . . . . . . . . . . . 12 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
150149notbid 306 . . . . . . . . . . 11 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
151147, 150, 15elab2 3322 . . . . . . . . . 10 (((𝑆𝑘)𝐵𝑘) ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
152151con2bii 345 . . . . . . . . 9 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
153146, 152sylib 206 . . . . . . . 8 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
154153ex 448 . . . . . . 7 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
155154ad2antrr 757 . . . . . 6 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
156130, 155syld 45 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑌(ball‘𝐷)𝑥) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
15736, 156mt2d 129 . . . 4 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
15827, 157rexlimddv 3016 . . 3 ((𝜑𝑥 ∈ ℝ+) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
159158nrexdv 2983 . 2 (𝜑 → ¬ ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
16011, 159pm2.21dd 184 1 (𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  {cab 2595  wral 2895  wrex 2896  Vcvv 3172  cin 3538  wss 3539  ifcif 4035  𝒫 cpw 4107  {csn 4124  cop 4130   cuni 4366   ciun 4449   class class class wbr 4577  {copab 4636  cmpt 4637   × cxp 5026  ccom 5032  wf 5786  cfv 5790  (class class class)co 6527  cmpt2 6529  1st c1st 7034  2nd c2nd 7035  Fincfn 7818  cr 9791  0cc0 9792  1c1 9793   + caddc 9795  *cxr 9929   < clt 9930  cle 9931  cmin 10117   / cdiv 10533  cn 10867  2c2 10917  3c3 10918  0cn0 11139  +crp 11664  seqcseq 12618  cexp 12677  ∞Metcxmt 19498  Metcme 19499  ballcbl 19500  MetOpencmopn 19503  Topctop 20459  clsccl 20574  𝑡clm 20782  CMetcms 22778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824  ax-cnex 9848  ax-resscn 9849  ax-1cn 9850  ax-icn 9851  ax-addcl 9852  ax-addrcl 9853  ax-mulcl 9854  ax-mulrcl 9855  ax-mulcom 9856  ax-addass 9857  ax-mulass 9858  ax-distr 9859  ax-i2m1 9860  ax-1ne0 9861  ax-1rid 9862  ax-rnegex 9863  ax-rrecex 9864  ax-cnre 9865  ax-pre-lttri 9866  ax-pre-lttrn 9867  ax-pre-ltadd 9868  ax-pre-mulgt0 9869  ax-pre-sup 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6935  df-1st 7036  df-2nd 7037  df-wrecs 7271  df-recs 7332  df-rdg 7370  df-1o 7424  df-er 7606  df-map 7723  df-pm 7724  df-en 7819  df-dom 7820  df-sdom 7821  df-fin 7822  df-sup 8208  df-inf 8209  df-pnf 9932  df-mnf 9933  df-xr 9934  df-ltxr 9935  df-le 9936  df-sub 10119  df-neg 10120  df-div 10534  df-nn 10868  df-2 10926  df-3 10927  df-n0 11140  df-z 11211  df-uz 11520  df-q 11621  df-rp 11665  df-xneg 11778  df-xadd 11779  df-xmul 11780  df-fl 12410  df-seq 12619  df-exp 12678  df-topgen 15873  df-psmet 19505  df-xmet 19506  df-met 19507  df-bl 19508  df-mopn 19509  df-top 20463  df-bases 20464  df-topon 20465  df-cld 20575  df-ntr 20576  df-cls 20577  df-lm 20785  df-cmet 22781
This theorem is referenced by:  heiborlem9  32584
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