Metamath Proof Explorer
Description: The distance function, suitably truncated, is an extended metric on
X . (Contributed by Mario Carneiro, 2-Sep-2015)
|
|
Ref |
Expression |
|
Hypotheses |
msf.x |
⊢ 𝑋 = ( Base ‘ 𝑀 ) |
|
|
msf.d |
⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) ) |
|
Assertion |
xmsxmet |
⊢ ( 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
|
isxms2 |
⊢ ( 𝑀 ∈ ∞MetSp ↔ ( 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ∧ ( TopOpen ‘ 𝑀 ) = ( MetOpen ‘ 𝐷 ) ) ) |
5 |
4
|
simplbi |
⊢ ( 𝑀 ∈ ∞MetSp → 𝐷 ∈ ( ∞Met ‘ 𝑋 ) ) |