Theorem msmet2

Description: The distance function, suitably truncated, is a metric on X . (Contributed by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mscl.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
	mscl.d	⊢ 𝐷 = ( dist ‘ 𝑀 )
Assertion	msmet2	⊢ ( 𝑀 ∈ MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( Met ‘ 𝑋 ) )

Proof

Step	Hyp	Ref	Expression
1		mscl.x	⊢ 𝑋 = ( Base ‘ 𝑀 )
2		mscl.d	⊢ 𝐷 = ( dist ‘ 𝑀 )
3	2	reseq1i	⊢ ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) = ( ( dist ‘ 𝑀 ) ↾ ( 𝑋 × 𝑋 ) )
4	1 3	msmet	⊢ ( 𝑀 ∈ MetSp → ( 𝐷 ↾ ( 𝑋 × 𝑋 ) ) ∈ ( Met ‘ 𝑋 ) )