Step |
Hyp |
Ref |
Expression |
1 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) |
2 |
|
fveq1 |
⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) |
3 |
1 2
|
oveq12d |
⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) = ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) |
4 |
3
|
fveq2d |
⊢ ( 𝑓 = 𝐹 → ( normℎ ‘ ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) ) = ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) ) |
5 |
4
|
breq1d |
⊢ ( 𝑓 = 𝐹 → ( ( normℎ ‘ ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) ) < 𝑥 ↔ ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
6 |
5
|
rexralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) ) < 𝑥 ↔ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
7 |
6
|
ralbidv |
⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) ) < 𝑥 ↔ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
8 |
|
df-hcau |
⊢ Cauchy = { 𝑓 ∈ ( ℋ ↑m ℕ ) ∣ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝑓 ‘ 𝑦 ) −ℎ ( 𝑓 ‘ 𝑧 ) ) ) < 𝑥 } |
9 |
7 8
|
elrab2 |
⊢ ( 𝐹 ∈ Cauchy ↔ ( 𝐹 ∈ ( ℋ ↑m ℕ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
10 |
|
ax-hilex |
⊢ ℋ ∈ V |
11 |
|
nnex |
⊢ ℕ ∈ V |
12 |
10 11
|
elmap |
⊢ ( 𝐹 ∈ ( ℋ ↑m ℕ ) ↔ 𝐹 : ℕ ⟶ ℋ ) |
13 |
12
|
anbi1i |
⊢ ( ( 𝐹 ∈ ( ℋ ↑m ℕ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ↔ ( 𝐹 : ℕ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |
14 |
9 13
|
bitri |
⊢ ( 𝐹 ∈ Cauchy ↔ ( 𝐹 : ℕ ⟶ ℋ ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ( ℤ≥ ‘ 𝑦 ) ( normℎ ‘ ( ( 𝐹 ‘ 𝑦 ) −ℎ ( 𝐹 ‘ 𝑧 ) ) ) < 𝑥 ) ) |