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Metamath Proof Explorer
Description: The distance function, suitably truncated, is a metric on X .
(Contributed by Mario Carneiro, 12-Nov-2013)
|
|
Ref |
Expression |
|
Hypotheses |
msf.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
|
|
msf.d |
⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) |
|
Assertion |
msmet |
⊢ ( 𝑀 ∈ MetSp → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
msf.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
| 2 |
|
msf.d |
⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) |
| 3 |
|
eqid |
⊢ ( TopOpen ‘ 𝑀 ) = ( TopOpen ‘ 𝑀 ) |
| 4 |
3 1 2
|
isms2 |
⊢ ( 𝑀 ∈ MetSp ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( TopOpen ‘ 𝑀 ) = ( MetOpen ‘ 𝐷 ) ) ) |
| 5 |
4
|
simplbi |
⊢ ( 𝑀 ∈ MetSp → 𝐷 ∈ ( Met ‘ 𝑋 ) ) |