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 β†’ 𝑀 ∈ ∞MetSp )

Proof

Step Hyp Ref Expression
1 eqid ⊒ ( TopOpen β€˜ 𝑀 ) = ( TopOpen β€˜ 𝑀 )
2 eqid ⊒ ( Base β€˜ 𝑀 ) = ( Base β€˜ 𝑀 )
3 eqid ⊒ ( ( dist β€˜ 𝑀 ) β†Ύ ( ( Base β€˜ 𝑀 ) Γ— ( Base β€˜ 𝑀 ) ) ) = ( ( dist β€˜ 𝑀 ) β†Ύ ( ( Base β€˜ 𝑀 ) Γ— ( Base β€˜ 𝑀 ) ) )
4 1 2 3 isms ⊒ ( 𝑀 ∈ MetSp ↔ ( 𝑀 ∈ ∞MetSp ∧ ( ( dist β€˜ 𝑀 ) β†Ύ ( ( Base β€˜ 𝑀 ) Γ— ( Base β€˜ 𝑀 ) ) ) ∈ ( Met β€˜ ( Base β€˜ 𝑀 ) ) ) )
5 4 simplbi ⊒ ( 𝑀 ∈ MetSp β†’ 𝑀 ∈ ∞MetSp )