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Metamath Proof Explorer
Description: Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006) (Revised by Mario Carneiro, 24-Dec-2013)
|
|
Ref |
Expression |
|
Assertion |
caufpm |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
iscau |
⊢ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) → ( 𝐹 ∈ ( Cau ‘ 𝐷 ) ↔ ( 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ ℤ ( 𝐹 ↾ ( ℤ≥ ‘ 𝑦 ) ) : ( ℤ≥ ‘ 𝑦 ) ⟶ ( ( 𝐹 ‘ 𝑦 ) ( ball ‘ 𝐷 ) 𝑥 ) ) ) ) |
| 2 |
1
|
simprbda |
⊢ ( ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ 𝐹 ∈ ( Cau ‘ 𝐷 ) ) → 𝐹 ∈ ( 𝑋 ↑pm ℂ ) ) |