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Theorem heiborlem8 33971
Description: Lemma for heibor 33974. 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 23362 . . . 4 (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
3 metxmet 22417 . . . 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 3761 . . 3 (𝜑𝑍𝐽)
8 heibor.15 . . 3 (𝜑𝑌𝑍)
9 heibor.1 . . . 4 𝐽 = (MetOpen‘𝐷)
109mopni2 22576 . . 3 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑍𝐽𝑌𝑍) → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
114, 7, 8, 10syl3anc 1490 . 2 (𝜑 → ∃𝑥 ∈ ℝ+ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
12 rphalfcl 12055 . . . . . 6 (𝑥 ∈ ℝ+ → (𝑥 / 2) ∈ ℝ+)
13 breq2 4812 . . . . . . . 8 (𝑟 = (𝑥 / 2) → ((2nd ‘(𝑀𝑘)) < 𝑟 ↔ (2nd ‘(𝑀𝑘)) < (𝑥 / 2)))
1413rexbidv 3198 . . . . . . 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 33970 . . . . . . 7 𝑟 ∈ ℝ+𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < 𝑟
2514, 24vtoclri 3434 . . . . . 6 ((𝑥 / 2) ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2612, 25syl 17 . . . . 5 (𝑥 ∈ ℝ+ → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
2726adantl 473 . . . 4 ((𝜑𝑥 ∈ ℝ+) → ∃𝑘 ∈ ℕ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
28 nnnn0 11545 . . . . . . 7 (𝑘 ∈ ℕ → 𝑘 ∈ ℕ0)
299, 15, 16, 17, 1, 18, 19, 20, 21, 22heiborlem4 33967 . . . . . . . 8 ((𝜑𝑘 ∈ ℕ0) → (𝑆𝑘)𝐺𝑘)
30 fvex 6387 . . . . . . . . . 10 (𝑆𝑘) ∈ V
31 vex 3352 . . . . . . . . . 10 𝑘 ∈ V
329, 15, 16, 30, 31heiborlem2 33965 . . . . . . . . 9 ((𝑆𝑘)𝐺𝑘 ↔ (𝑘 ∈ ℕ0 ∧ (𝑆𝑘) ∈ (𝐹𝑘) ∧ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
3332simp3bi 1177 . . . . . . . 8 ((𝑆𝑘)𝐺𝑘 → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3429, 33syl 17 . . . . . . 7 ((𝜑𝑘 ∈ ℕ0) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3528, 34sylan2 586 . . . . . 6 ((𝜑𝑘 ∈ ℕ) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
3635ad2ant2r 753 . . . . 5 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
374ad2antrr 717 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐷 ∈ (∞Met‘𝑋))
389, 15, 16, 17, 1, 18, 19, 20, 21, 22, 23heiborlem5 33968 . . . . . . . . . . . . 13 (𝜑𝑀:ℕ⟶(𝑋 × ℝ+))
3938ffvelrnda 6548 . . . . . . . . . . . 12 ((𝜑𝑘 ∈ ℕ) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
4039ad2ant2r 753 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑀𝑘) ∈ (𝑋 × ℝ+))
41 xp1st 7397 . . . . . . . . . . 11 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
4240, 41syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) ∈ 𝑋)
43 2nn 11344 . . . . . . . . . . . . . . 15 2 ∈ ℕ
44 nnexpcl 13079 . . . . . . . . . . . . . . 15 ((2 ∈ ℕ ∧ 𝑘 ∈ ℕ0) → (2↑𝑘) ∈ ℕ)
4543, 28, 44sylancr 581 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℕ)
4645nnrpd 12067 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2↑𝑘) ∈ ℝ+)
4746rpreccld 12079 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ∈ ℝ+)
4847ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ+)
4948rpxrd 12070 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ∈ ℝ*)
50 xp2nd 7398 . . . . . . . . . . . 12 ((𝑀𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5140, 50syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ+)
5251rpxrd 12070 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) ∈ ℝ*)
53 1le3 11489 . . . . . . . . . . . . . 14 1 ≤ 3
54 elrp 12029 . . . . . . . . . . . . . . 15 ((2↑𝑘) ∈ ℝ+ ↔ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)))
55 1re 10292 . . . . . . . . . . . . . . . 16 1 ∈ ℝ
56 3re 11351 . . . . . . . . . . . . . . . 16 3 ∈ ℝ
57 lediv1 11141 . . . . . . . . . . . . . . . 16 ((1 ∈ ℝ ∧ 3 ∈ ℝ ∧ ((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘))) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5855, 56, 57mp3an12 1575 . . . . . . . . . . . . . . 15 (((2↑𝑘) ∈ ℝ ∧ 0 < (2↑𝑘)) → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
5954, 58sylbi 208 . . . . . . . . . . . . . 14 ((2↑𝑘) ∈ ℝ+ → (1 ≤ 3 ↔ (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘))))
6053, 59mpbii 224 . . . . . . . . . . . . 13 ((2↑𝑘) ∈ ℝ+ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6146, 60syl 17 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
6261ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (3 / (2↑𝑘)))
63 fveq2 6374 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (𝑆𝑛) = (𝑆𝑘))
64 oveq2 6849 . . . . . . . . . . . . . . . . 17 (𝑛 = 𝑘 → (2↑𝑛) = (2↑𝑘))
6564oveq2d 6857 . . . . . . . . . . . . . . . 16 (𝑛 = 𝑘 → (3 / (2↑𝑛)) = (3 / (2↑𝑘)))
6663, 65opeq12d 4566 . . . . . . . . . . . . . . 15 (𝑛 = 𝑘 → ⟨(𝑆𝑛), (3 / (2↑𝑛))⟩ = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
67 opex 5087 . . . . . . . . . . . . . . 15 ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩ ∈ V
6866, 23, 67fvmpt 6470 . . . . . . . . . . . . . 14 (𝑘 ∈ ℕ → (𝑀𝑘) = ⟨(𝑆𝑘), (3 / (2↑𝑘))⟩)
6968fveq2d 6378 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
70 ovex 6873 . . . . . . . . . . . . . 14 (3 / (2↑𝑘)) ∈ V
7130, 70op2nd 7374 . . . . . . . . . . . . 13 (2nd ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (3 / (2↑𝑘))
7269, 71syl6eq 2814 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7372ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) = (3 / (2↑𝑘)))
7462, 73breqtrrd 4836 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1 / (2↑𝑘)) ≤ (2nd ‘(𝑀𝑘)))
75 ssbl 22506 . . . . . . . . . 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 1500 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))))
7728ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑘 ∈ ℕ0)
78 oveq1 6848 . . . . . . . . . . . 12 (𝑧 = (1st ‘(𝑀𝑘)) → (𝑧(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))))
79 oveq2 6849 . . . . . . . . . . . . . 14 (𝑚 = 𝑘 → (2↑𝑚) = (2↑𝑘))
8079oveq2d 6857 . . . . . . . . . . . . 13 (𝑚 = 𝑘 → (1 / (2↑𝑚)) = (1 / (2↑𝑘)))
8180oveq2d 6857 . . . . . . . . . . . 12 (𝑚 = 𝑘 → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑚))) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
82 ovex 6873 . . . . . . . . . . . 12 ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) ∈ V
8378, 81, 17, 82ovmpt2 6993 . . . . . . . . . . 11 (((1st ‘(𝑀𝑘)) ∈ 𝑋𝑘 ∈ ℕ0) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8442, 77, 83syl2anc 579 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))))
8568fveq2d 6378 . . . . . . . . . . . . 13 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩))
8630, 70op1st 7373 . . . . . . . . . . . . 13 (1st ‘⟨(𝑆𝑘), (3 / (2↑𝑘))⟩) = (𝑆𝑘)
8785, 86syl6eq 2814 . . . . . . . . . . . 12 (𝑘 ∈ ℕ → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8887ad2antrl 719 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (1st ‘(𝑀𝑘)) = (𝑆𝑘))
8988oveq1d 6856 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))𝐵𝑘) = ((𝑆𝑘)𝐵𝑘))
9084, 89eqtr3d 2800 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(1 / (2↑𝑘))) = ((𝑆𝑘)𝐵𝑘))
91 1st2nd2 7404 . . . . . . . . . . . 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 6378 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩))
94 df-ov 6844 . . . . . . . . . 10 ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝑀𝑘)), (2nd ‘(𝑀𝑘))⟩)
9593, 94syl6reqr 2817 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) = ((ball‘𝐷)‘(𝑀𝑘)))
9676, 90, 953sstr3d 3806 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ ((ball‘𝐷)‘(𝑀𝑘)))
979mopntop 22523 . . . . . . . . . . 11 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
9837, 97syl 17 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝐽 ∈ Top)
99 blssm 22501 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑀𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝑀𝑘)) ∈ ℝ*) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
10037, 42, 52, 99syl3anc 1490 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘))) ⊆ 𝑋)
1019mopnuni 22524 . . . . . . . . . . . 12 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = 𝐽)
10237, 101syl 17 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑋 = 𝐽)
103100, 95, 1023sstr3d 3806 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽)
104 eqid 2764 . . . . . . . . . . 11 𝐽 = 𝐽
105104sscls 21139 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ ((ball‘𝐷)‘(𝑀𝑘)) ⊆ 𝐽) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10698, 103, 105syl2anc 579 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10795fveq2d 6378 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((1st ‘(𝑀𝑘))(ball‘𝐷)(2nd ‘(𝑀𝑘)))) = ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
10812ad2antlr 718 . . . . . . . . . . . . 13 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ+)
109108rpxrd 12070 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (𝑥 / 2) ∈ ℝ*)
110 simprr 789 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → (2nd ‘(𝑀𝑘)) < (𝑥 / 2))
1119blsscls 22590 . . . . . . . . . . . 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 1496 . . . . . . . . . . 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 3799 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
114 rpre 12035 . . . . . . . . . . . 12 (𝑥 ∈ ℝ+𝑥 ∈ ℝ)
115114ad2antlr 718 . . . . . . . . . . 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 33969 . . . . . . . . . . . . . . . . 17 (𝜑 → ∀𝑡 ∈ ℕ ((ball‘𝐷)‘(𝑀‘(𝑡 + 1))) ⊆ ((ball‘𝐷)‘(𝑀𝑡)))
1184, 38, 117, 9caublcls 23385 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
1191183expia 1150 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (1st𝑀)(⇝𝑡𝐽)𝑌) → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
120116, 119mpdan 678 . . . . . . . . . . . . . 14 (𝜑 → (𝑘 ∈ ℕ → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘)))))
121120imp 395 . . . . . . . . . . . . 13 ((𝜑𝑘 ∈ ℕ) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
122121ad2ant2r 753 . . . . . . . . . . . 12 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))))
123113, 122sseldd 3761 . . . . . . . . . . 11 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → 𝑌 ∈ ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)))
124 blhalf 22488 . . . . . . . . . . 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 867 . . . . . . . . . 10 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((1st ‘(𝑀𝑘))(ball‘𝐷)(𝑥 / 2)) ⊆ (𝑌(ball‘𝐷)𝑥))
126113, 125sstrd 3770 . . . . . . . . 9 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((cls‘𝐽)‘((ball‘𝐷)‘(𝑀𝑘))) ⊆ (𝑌(ball‘𝐷)𝑥))
127106, 126sstrd 3770 . . . . . . . 8 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((ball‘𝐷)‘(𝑀𝑘)) ⊆ (𝑌(ball‘𝐷)𝑥))
12896, 127sstrd 3770 . . . . . . 7 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ((𝑆𝑘)𝐵𝑘) ⊆ (𝑌(ball‘𝐷)𝑥))
129 sstr2 3767 . . . . . . 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 4608 . . . . . . . . . . . . 13 (𝑍𝑈 {𝑍} = 𝑍)
1326, 131syl 17 . . . . . . . . . . . 12 (𝜑 {𝑍} = 𝑍)
133132sseq2d 3792 . . . . . . . . . . 11 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ {𝑍} ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍))
134133biimpar 469 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍})
1356snssd 4493 . . . . . . . . . . . . 13 (𝜑 → {𝑍} ⊆ 𝑈)
136 snex 5063 . . . . . . . . . . . . . 14 {𝑍} ∈ V
137136elpw 4320 . . . . . . . . . . . . 13 ({𝑍} ∈ 𝒫 𝑈 ↔ {𝑍} ⊆ 𝑈)
138135, 137sylibr 225 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ 𝒫 𝑈)
139 snfi 8244 . . . . . . . . . . . . 13 {𝑍} ∈ Fin
140139a1i 11 . . . . . . . . . . . 12 (𝜑 → {𝑍} ∈ Fin)
141138, 140elind 3959 . . . . . . . . . . 11 (𝜑 → {𝑍} ∈ (𝒫 𝑈 ∩ Fin))
142 unieq 4601 . . . . . . . . . . . . 13 (𝑣 = {𝑍} → 𝑣 = {𝑍})
143142sseq2d 3792 . . . . . . . . . . . 12 (𝑣 = {𝑍} → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}))
144143rspcev 3460 . . . . . . . . . . 11 (({𝑍} ∈ (𝒫 𝑈 ∩ Fin) ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
145141, 144sylan 575 . . . . . . . . . 10 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ {𝑍}) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
146134, 145syldan 585 . . . . . . . . 9 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
147 ovex 6873 . . . . . . . . . . 11 ((𝑆𝑘)𝐵𝑘) ∈ V
148 sseq1 3785 . . . . . . . . . . . . 13 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (𝑢 𝑣 ↔ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
149148rexbidv 3198 . . . . . . . . . . . 12 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
150149notbid 309 . . . . . . . . . . 11 (𝑢 = ((𝑆𝑘)𝐵𝑘) → (¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 𝑣 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣))
151147, 150, 15elab2 3508 . . . . . . . . . 10 (((𝑆𝑘)𝐵𝑘) ∈ 𝐾 ↔ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣)
152151con2bii 348 . . . . . . . . 9 (∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)((𝑆𝑘)𝐵𝑘) ⊆ 𝑣 ↔ ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
153146, 152sylib 209 . . . . . . . 8 ((𝜑 ∧ ((𝑆𝑘)𝐵𝑘) ⊆ 𝑍) → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾)
154153ex 401 . . . . . . 7 (𝜑 → (((𝑆𝑘)𝐵𝑘) ⊆ 𝑍 → ¬ ((𝑆𝑘)𝐵𝑘) ∈ 𝐾))
155154ad2antrr 717 . . . . . 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 133 . . . 4 (((𝜑𝑥 ∈ ℝ+) ∧ (𝑘 ∈ ℕ ∧ (2nd ‘(𝑀𝑘)) < (𝑥 / 2))) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
15827, 157rexlimddv 3181 . . 3 ((𝜑𝑥 ∈ ℝ+) → ¬ (𝑌(ball‘𝐷)𝑥) ⊆ 𝑍)
159158nrexdv 3146 . 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 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  {cab 2750  wral 3054  wrex 3055  Vcvv 3349  cin 3730  wss 3731  ifcif 4242  𝒫 cpw 4314  {csn 4333  cop 4339   cuni 4593   ciun 4675   class class class wbr 4808  {copab 4870  cmpt 4887   × cxp 5274  ccom 5280  wf 6063  cfv 6067  (class class class)co 6841  cmpt2 6843  1st c1st 7363  2nd c2nd 7364  Fincfn 8159  cr 10187  0cc0 10188  1c1 10189   + caddc 10191  *cxr 10326   < clt 10327  cle 10328  cmin 10519   / cdiv 10937  cn 11273  2c2 11326  3c3 11327  0cn0 11537  +crp 12027  seqcseq 13007  cexp 13066  ∞Metcxmet 20003  Metcmet 20004  ballcbl 20005  MetOpencmopn 20008  Topctop 20976  clsccl 21101  𝑡clm 21309  CMetccmet 23330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742  ax-rep 4929  ax-sep 4940  ax-nul 4948  ax-pow 5000  ax-pr 5061  ax-un 7146  ax-cnex 10244  ax-resscn 10245  ax-1cn 10246  ax-icn 10247  ax-addcl 10248  ax-addrcl 10249  ax-mulcl 10250  ax-mulrcl 10251  ax-mulcom 10252  ax-addass 10253  ax-mulass 10254  ax-distr 10255  ax-i2m1 10256  ax-1ne0 10257  ax-1rid 10258  ax-rnegex 10259  ax-rrecex 10260  ax-cnre 10261  ax-pre-lttri 10262  ax-pre-lttrn 10263  ax-pre-ltadd 10264  ax-pre-mulgt0 10265  ax-pre-sup 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2564  df-eu 2581  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-nel 3040  df-ral 3059  df-rex 3060  df-reu 3061  df-rmo 3062  df-rab 3063  df-v 3351  df-sbc 3596  df-csb 3691  df-dif 3734  df-un 3736  df-in 3738  df-ss 3745  df-pss 3747  df-nul 4079  df-if 4243  df-pw 4316  df-sn 4334  df-pr 4336  df-tp 4338  df-op 4340  df-uni 4594  df-int 4633  df-iun 4677  df-iin 4678  df-br 4809  df-opab 4871  df-mpt 4888  df-tr 4911  df-id 5184  df-eprel 5189  df-po 5197  df-so 5198  df-fr 5235  df-we 5237  df-xp 5282  df-rel 5283  df-cnv 5284  df-co 5285  df-dm 5286  df-rn 5287  df-res 5288  df-ima 5289  df-pred 5864  df-ord 5910  df-on 5911  df-lim 5912  df-suc 5913  df-iota 6030  df-fun 6069  df-fn 6070  df-f 6071  df-f1 6072  df-fo 6073  df-f1o 6074  df-fv 6075  df-riota 6802  df-ov 6844  df-oprab 6845  df-mpt2 6846  df-om 7263  df-1st 7365  df-2nd 7366  df-wrecs 7609  df-recs 7671  df-rdg 7709  df-1o 7763  df-er 7946  df-map 8061  df-pm 8062  df-en 8160  df-dom 8161  df-sdom 8162  df-fin 8163  df-sup 8554  df-inf 8555  df-pnf 10329  df-mnf 10330  df-xr 10331  df-ltxr 10332  df-le 10333  df-sub 10521  df-neg 10522  df-div 10938  df-nn 11274  df-2 11334  df-3 11335  df-n0 11538  df-z 11624  df-uz 11886  df-q 11989  df-rp 12028  df-xneg 12145  df-xadd 12146  df-xmul 12147  df-fl 12800  df-seq 13008  df-exp 13067  df-topgen 16371  df-psmet 20010  df-xmet 20011  df-met 20012  df-bl 20013  df-mopn 20014  df-top 20977  df-topon 20994  df-bases 21029  df-cld 21102  df-ntr 21103  df-cls 21104  df-lm 21312  df-cmet 23333
This theorem is referenced by:  heiborlem9  33972
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