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Metamath Proof Explorer
Description: A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006) (Revised by Mario Carneiro, 29-Jan-2014)
|
|
Ref |
Expression |
|
Assertion |
cmetmet |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
eqid |
⊢ ( MetOpen ‘ 𝐷 ) = ( MetOpen ‘ 𝐷 ) |
| 2 |
1
|
iscmet |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ∀ 𝑓 ∈ ( CauFil ‘ 𝐷 ) ( ( MetOpen ‘ 𝐷 ) fLim 𝑓 ) ≠ ∅ ) ) |
| 3 |
2
|
simplbi |
⊢ ( 𝐷 ∈ ( CMet ‘ 𝑋 ) → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |