Step |
Hyp |
Ref |
Expression |
1 |
|
hcau |
⊢ ( 𝐹 ∈ Cauchy ↔ ( 𝐹 : ℕ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
2 |
1
|
simprbi |
⊢ ( 𝐹 ∈ Cauchy → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) |
3 |
|
breq2 |
⊢ ( 𝑥 = 𝐴 → ( ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ↔ ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝐴 ) ) |
4 |
3
|
rexralbidv |
⊢ ( 𝑥 = 𝐴 → ( ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ↔ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝐴 ) ) |
5 |
4
|
rspccva |
⊢ ( ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ∧ 𝐴 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝐴 ) |
6 |
2 5
|
sylan |
⊢ ( ( 𝐹 ∈ Cauchy ∧ 𝐴 ∈ ℝ+ ) → ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝐴 ) |