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 )