Theorem msmet

Description: The distance function, suitably truncated, is a metric on X . (Contributed by Mario Carneiro, 12-Nov-2013)

	Ref	Expression
Hypotheses	msf.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
	msf.d	⊢ 𝐷 = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
Assertion	msmet	⊢ ( 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	isms2	⊢ ( 𝑀 ∈ MetSp ↔ ( 𝐷 ∈ ( Met ‘ 𝑋 ) ∧ ( TopOpen ‘ 𝑀 ) = ( MetOpen ‘ 𝐷 ) ) )
5	4	simplbi	⊢ ( 𝑀 ∈ MetSp → 𝐷 ∈ ( Met ‘ 𝑋 ) )