Step |
Hyp |
Ref |
Expression |
1 |
|
heibor.1 |
⊢ 𝐽 = ( MetOpen ‘ 𝐷 ) |
2 |
|
heibor.3 |
⊢ 𝐾 = { 𝑢 ∣ ¬ ∃ 𝑣 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑢 ⊆ ∪ 𝑣 } |
3 |
|
heibor.4 |
⊢ 𝐺 = { 〈 𝑦 , 𝑛 〉 ∣ ( 𝑛 ∈ ℕ0 ∧ 𝑦 ∈ ( 𝐹 ‘ 𝑛 ) ∧ ( 𝑦 𝐵 𝑛 ) ∈ 𝐾 ) } |
4 |
|
heibor.5 |
⊢ 𝐵 = ( 𝑧 ∈ 𝑋 , 𝑚 ∈ ℕ0 ↦ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / ( 2 ↑ 𝑚 ) ) ) ) |
5 |
|
heibor.6 |
⊢ ( 𝜑 → 𝐷 ∈ ( CMet ‘ 𝑋 ) ) |
6 |
|
heibor.7 |
⊢ ( 𝜑 → 𝐹 : ℕ0 ⟶ ( 𝒫 𝑋 ∩ Fin ) ) |
7 |
|
heibor.8 |
⊢ ( 𝜑 → ∀ 𝑛 ∈ ℕ0 𝑋 = ∪ 𝑦 ∈ ( 𝐹 ‘ 𝑛 ) ( 𝑦 𝐵 𝑛 ) ) |
8 |
|
heibor.9 |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐺 ( ( 𝑇 ‘ 𝑥 ) 𝐺 ( ( 2nd ‘ 𝑥 ) + 1 ) ∧ ( ( 𝐵 ‘ 𝑥 ) ∩ ( ( 𝑇 ‘ 𝑥 ) 𝐵 ( ( 2nd ‘ 𝑥 ) + 1 ) ) ) ∈ 𝐾 ) ) |
9 |
|
heibor.10 |
⊢ ( 𝜑 → 𝐶 𝐺 0 ) |
10 |
|
heibor.11 |
⊢ 𝑆 = seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) ) |
11 |
|
heibor.12 |
⊢ 𝑀 = ( 𝑛 ∈ ℕ ↦ 〈 ( 𝑆 ‘ 𝑛 ) , ( 3 / ( 2 ↑ 𝑛 ) ) 〉 ) |
12 |
|
3re |
⊢ 3 ∈ ℝ |
13 |
|
3pos |
⊢ 0 < 3 |
14 |
12 13
|
elrpii |
⊢ 3 ∈ ℝ+ |
15 |
|
rpdivcl |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 3 ∈ ℝ+ ) → ( 𝑟 / 3 ) ∈ ℝ+ ) |
16 |
14 15
|
mpan2 |
⊢ ( 𝑟 ∈ ℝ+ → ( 𝑟 / 3 ) ∈ ℝ+ ) |
17 |
|
2re |
⊢ 2 ∈ ℝ |
18 |
|
1lt2 |
⊢ 1 < 2 |
19 |
|
expnlbnd |
⊢ ( ( ( 𝑟 / 3 ) ∈ ℝ+ ∧ 2 ∈ ℝ ∧ 1 < 2 ) → ∃ 𝑘 ∈ ℕ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) |
20 |
17 18 19
|
mp3an23 |
⊢ ( ( 𝑟 / 3 ) ∈ ℝ+ → ∃ 𝑘 ∈ ℕ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) |
21 |
16 20
|
syl |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑘 ∈ ℕ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) |
22 |
|
2nn |
⊢ 2 ∈ ℕ |
23 |
|
nnnn0 |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℕ0 ) |
24 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝑘 ∈ ℕ0 ) → ( 2 ↑ 𝑘 ) ∈ ℕ ) |
25 |
22 23 24
|
sylancr |
⊢ ( 𝑘 ∈ ℕ → ( 2 ↑ 𝑘 ) ∈ ℕ ) |
26 |
25
|
nnrpd |
⊢ ( 𝑘 ∈ ℕ → ( 2 ↑ 𝑘 ) ∈ ℝ+ ) |
27 |
|
rpcn |
⊢ ( ( 2 ↑ 𝑘 ) ∈ ℝ+ → ( 2 ↑ 𝑘 ) ∈ ℂ ) |
28 |
|
rpne0 |
⊢ ( ( 2 ↑ 𝑘 ) ∈ ℝ+ → ( 2 ↑ 𝑘 ) ≠ 0 ) |
29 |
|
3cn |
⊢ 3 ∈ ℂ |
30 |
|
divrec |
⊢ ( ( 3 ∈ ℂ ∧ ( 2 ↑ 𝑘 ) ∈ ℂ ∧ ( 2 ↑ 𝑘 ) ≠ 0 ) → ( 3 / ( 2 ↑ 𝑘 ) ) = ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) ) |
31 |
29 30
|
mp3an1 |
⊢ ( ( ( 2 ↑ 𝑘 ) ∈ ℂ ∧ ( 2 ↑ 𝑘 ) ≠ 0 ) → ( 3 / ( 2 ↑ 𝑘 ) ) = ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) ) |
32 |
27 28 31
|
syl2anc |
⊢ ( ( 2 ↑ 𝑘 ) ∈ ℝ+ → ( 3 / ( 2 ↑ 𝑘 ) ) = ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) ) |
33 |
26 32
|
syl |
⊢ ( 𝑘 ∈ ℕ → ( 3 / ( 2 ↑ 𝑘 ) ) = ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) ) |
34 |
33
|
adantl |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ ) → ( 3 / ( 2 ↑ 𝑘 ) ) = ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) ) |
35 |
34
|
breq1d |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ ) → ( ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ↔ ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) < 𝑟 ) ) |
36 |
25
|
nnrecred |
⊢ ( 𝑘 ∈ ℕ → ( 1 / ( 2 ↑ 𝑘 ) ) ∈ ℝ ) |
37 |
|
rpre |
⊢ ( 𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ ) |
38 |
12 13
|
pm3.2i |
⊢ ( 3 ∈ ℝ ∧ 0 < 3 ) |
39 |
|
ltmuldiv2 |
⊢ ( ( ( 1 / ( 2 ↑ 𝑘 ) ) ∈ ℝ ∧ 𝑟 ∈ ℝ ∧ ( 3 ∈ ℝ ∧ 0 < 3 ) ) → ( ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) < 𝑟 ↔ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) ) |
40 |
38 39
|
mp3an3 |
⊢ ( ( ( 1 / ( 2 ↑ 𝑘 ) ) ∈ ℝ ∧ 𝑟 ∈ ℝ ) → ( ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) < 𝑟 ↔ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) ) |
41 |
36 37 40
|
syl2anr |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ ) → ( ( 3 · ( 1 / ( 2 ↑ 𝑘 ) ) ) < 𝑟 ↔ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) ) |
42 |
35 41
|
bitrd |
⊢ ( ( 𝑟 ∈ ℝ+ ∧ 𝑘 ∈ ℕ ) → ( ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ↔ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) ) |
43 |
42
|
rexbidva |
⊢ ( 𝑟 ∈ ℝ+ → ( ∃ 𝑘 ∈ ℕ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ↔ ∃ 𝑘 ∈ ℕ ( 1 / ( 2 ↑ 𝑘 ) ) < ( 𝑟 / 3 ) ) ) |
44 |
21 43
|
mpbird |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑘 ∈ ℕ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ) |
45 |
|
fveq2 |
⊢ ( 𝑛 = 𝑘 → ( 𝑆 ‘ 𝑛 ) = ( 𝑆 ‘ 𝑘 ) ) |
46 |
|
oveq2 |
⊢ ( 𝑛 = 𝑘 → ( 2 ↑ 𝑛 ) = ( 2 ↑ 𝑘 ) ) |
47 |
46
|
oveq2d |
⊢ ( 𝑛 = 𝑘 → ( 3 / ( 2 ↑ 𝑛 ) ) = ( 3 / ( 2 ↑ 𝑘 ) ) ) |
48 |
45 47
|
opeq12d |
⊢ ( 𝑛 = 𝑘 → 〈 ( 𝑆 ‘ 𝑛 ) , ( 3 / ( 2 ↑ 𝑛 ) ) 〉 = 〈 ( 𝑆 ‘ 𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) 〉 ) |
49 |
|
opex |
⊢ 〈 ( 𝑆 ‘ 𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) 〉 ∈ V |
50 |
48 11 49
|
fvmpt |
⊢ ( 𝑘 ∈ ℕ → ( 𝑀 ‘ 𝑘 ) = 〈 ( 𝑆 ‘ 𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) 〉 ) |
51 |
50
|
fveq2d |
⊢ ( 𝑘 ∈ ℕ → ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) = ( 2nd ‘ 〈 ( 𝑆 ‘ 𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) 〉 ) ) |
52 |
|
fvex |
⊢ ( 𝑆 ‘ 𝑘 ) ∈ V |
53 |
|
ovex |
⊢ ( 3 / ( 2 ↑ 𝑘 ) ) ∈ V |
54 |
52 53
|
op2nd |
⊢ ( 2nd ‘ 〈 ( 𝑆 ‘ 𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) 〉 ) = ( 3 / ( 2 ↑ 𝑘 ) ) |
55 |
51 54
|
eqtrdi |
⊢ ( 𝑘 ∈ ℕ → ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) = ( 3 / ( 2 ↑ 𝑘 ) ) ) |
56 |
55
|
breq1d |
⊢ ( 𝑘 ∈ ℕ → ( ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) < 𝑟 ↔ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ) ) |
57 |
56
|
rexbiia |
⊢ ( ∃ 𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) < 𝑟 ↔ ∃ 𝑘 ∈ ℕ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ) |
58 |
44 57
|
sylibr |
⊢ ( 𝑟 ∈ ℝ+ → ∃ 𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) < 𝑟 ) |
59 |
58
|
rgen |
⊢ ∀ 𝑟 ∈ ℝ+ ∃ 𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀 ‘ 𝑘 ) ) < 𝑟 |