Metamath Proof Explorer


Theorem xmsxmet

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 ) )