Metamath Proof Explorer


Theorem heiborlem7

Description: Lemma for heibor . Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014)

Ref Expression
Hypotheses heibor.1 𝐽 = ( MetOpen ‘ 𝐷 )
heibor.3 𝐾 = { 𝑢 ∣ ¬ ∃ 𝑣 ∈ ( 𝒫 𝑈 ∩ Fin ) 𝑢 𝑣 }
heibor.4 𝐺 = { ⟨ 𝑦 , 𝑛 ⟩ ∣ ( 𝑛 ∈ ℕ0𝑦 ∈ ( 𝐹𝑛 ) ∧ ( 𝑦 𝐵 𝑛 ) ∈ 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 ) ( 𝑦 𝐵 𝑛 ) )
heibor.9 ( 𝜑 → ∀ 𝑥𝐺 ( ( 𝑇𝑥 ) 𝐺 ( ( 2nd𝑥 ) + 1 ) ∧ ( ( 𝐵𝑥 ) ∩ ( ( 𝑇𝑥 ) 𝐵 ( ( 2nd𝑥 ) + 1 ) ) ) ∈ 𝐾 ) )
heibor.10 ( 𝜑𝐶 𝐺 0 )
heibor.11 𝑆 = seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) )
heibor.12 𝑀 = ( 𝑛 ∈ ℕ ↦ ⟨ ( 𝑆𝑛 ) , ( 3 / ( 2 ↑ 𝑛 ) ) ⟩ )
Assertion heiborlem7 𝑟 ∈ ℝ+𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀𝑘 ) ) < 𝑟

Proof

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 ∈ ℕ0𝑦 ∈ ( 𝐹𝑛 ) ∧ ( 𝑦 𝐵 𝑛 ) ∈ 𝐾 ) }
4 heibor.5 𝐵 = ( 𝑧𝑋 , 𝑚 ∈ ℕ0 ↦ ( 𝑧 ( ball ‘ 𝐷 ) ( 1 / ( 2 ↑ 𝑚 ) ) ) )
5 heibor.6 ( 𝜑𝐷 ∈ ( CMet ‘ 𝑋 ) )
6 heibor.7 ( 𝜑𝐹 : ℕ0 ⟶ ( 𝒫 𝑋 ∩ Fin ) )
7 heibor.8 ( 𝜑 → ∀ 𝑛 ∈ ℕ0 𝑋 = 𝑦 ∈ ( 𝐹𝑛 ) ( 𝑦 𝐵 𝑛 ) )
8 heibor.9 ( 𝜑 → ∀ 𝑥𝐺 ( ( 𝑇𝑥 ) 𝐺 ( ( 2nd𝑥 ) + 1 ) ∧ ( ( 𝐵𝑥 ) ∩ ( ( 𝑇𝑥 ) 𝐵 ( ( 2nd𝑥 ) + 1 ) ) ) ∈ 𝐾 ) )
9 heibor.10 ( 𝜑𝐶 𝐺 0 )
10 heibor.11 𝑆 = seq 0 ( 𝑇 , ( 𝑚 ∈ ℕ0 ↦ if ( 𝑚 = 0 , 𝐶 , ( 𝑚 − 1 ) ) ) )
11 heibor.12 𝑀 = ( 𝑛 ∈ ℕ ↦ ⟨ ( 𝑆𝑛 ) , ( 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 → ( 𝑆𝑛 ) = ( 𝑆𝑘 ) )
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 → ⟨ ( 𝑆𝑛 ) , ( 3 / ( 2 ↑ 𝑛 ) ) ⟩ = ⟨ ( 𝑆𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) ⟩ )
49 opex ⟨ ( 𝑆𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) ⟩ ∈ V
50 48 11 49 fvmpt ( 𝑘 ∈ ℕ → ( 𝑀𝑘 ) = ⟨ ( 𝑆𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) ⟩ )
51 50 fveq2d ( 𝑘 ∈ ℕ → ( 2nd ‘ ( 𝑀𝑘 ) ) = ( 2nd ‘ ⟨ ( 𝑆𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) ⟩ ) )
52 fvex ( 𝑆𝑘 ) ∈ V
53 ovex ( 3 / ( 2 ↑ 𝑘 ) ) ∈ V
54 52 53 op2nd ( 2nd ‘ ⟨ ( 𝑆𝑘 ) , ( 3 / ( 2 ↑ 𝑘 ) ) ⟩ ) = ( 3 / ( 2 ↑ 𝑘 ) )
55 51 54 eqtrdi ( 𝑘 ∈ ℕ → ( 2nd ‘ ( 𝑀𝑘 ) ) = ( 3 / ( 2 ↑ 𝑘 ) ) )
56 55 breq1d ( 𝑘 ∈ ℕ → ( ( 2nd ‘ ( 𝑀𝑘 ) ) < 𝑟 ↔ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 ) )
57 56 rexbiia ( ∃ 𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀𝑘 ) ) < 𝑟 ↔ ∃ 𝑘 ∈ ℕ ( 3 / ( 2 ↑ 𝑘 ) ) < 𝑟 )
58 44 57 sylibr ( 𝑟 ∈ ℝ+ → ∃ 𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀𝑘 ) ) < 𝑟 )
59 58 rgen 𝑟 ∈ ℝ+𝑘 ∈ ℕ ( 2nd ‘ ( 𝑀𝑘 ) ) < 𝑟