Theorem cmsms

Description: A complete metric space is a metric space. (Contributed by Mario Carneiro, 15-Oct-2015)

	Ref	Expression
Assertion	cmsms	⊢ ( 𝐺 ∈ CMetSp → 𝐺 ∈ MetSp )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ( Base ‘ 𝐺 ) = ( Base ‘ 𝐺 )
2		eqid	⊢ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) = ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) )
3	1 2	iscms	⊢ ( 𝐺 ∈ CMetSp ↔ ( 𝐺 ∈ MetSp ∧ ( ( dist ‘ 𝐺 ) ↾ ( ( Base ‘ 𝐺 ) × ( Base ‘ 𝐺 ) ) ) ∈ ( CMet ‘ ( Base ‘ 𝐺 ) ) ) )
4	3	simplbi	⊢ ( 𝐺 ∈ CMetSp → 𝐺 ∈ MetSp )