Description: A metric space is an extended metric space. (Contributed by Mario Carneiro, 26-Aug-2015)
Ref | Expression | ||
---|---|---|---|
Assertion | msxms | ⊢ ( 𝑀 ∈ MetSp → 𝑀 ∈ ∞MetSp ) |
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 ) |