Metamath Proof Explorer


Theorem msxms

Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015)

Ref Expression
Assertion msxms ( 𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp )

Proof

Step Hyp Ref Expression
1 eqid ( TopOpen ‘ 𝑀 ) = ( TopOpen ‘ 𝑀 )
2 eqid ( Base ‘ 𝑀 ) = ( Base ‘ 𝑀 )
3 eqid ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) ) = ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) )
4 1 2 3 isms ( 𝑀 ∈ MetSp ↔ ( 𝑀 ∈ ∞MetSp ∧ ( ( dist ‘ 𝑀 ) ↾ ( ( Base ‘ 𝑀 ) × ( Base ‘ 𝑀 ) ) ) ∈ ( Met ‘ ( Base ‘ 𝑀 ) ) ) )
5 4 simplbi ( 𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp )