Theorem mscl

Description: Closure of the distance function of a metric space. (Contributed by NM, 30-Aug-2006) (Revised by Mario Carneiro, 2-Oct-2015)

	Ref	Expression
Hypotheses	mscl.x	|- X = ( Base ` M )
	mscl.d	|- D = ( dist ` M )
Assertion	mscl	|- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )

Proof

Step	Hyp	Ref	Expression
1		mscl.x	|- X = ( Base ` M )
2		mscl.d	|- D = ( dist ` M )
3		ovres	|- ( ( A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
4	3	3adant1	|- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) = ( A D B ) )
5	1 2	msmet2	|- ( M e. MetSp -> ( D |` ( X X. X ) ) e. ( Met ` X ) )
6		metcl	|- ( ( ( D |` ( X X. X ) ) e. ( Met ` X ) /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) e. RR )
7	5 6	syl3an1	|- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A ( D |` ( X X. X ) ) B ) e. RR )
8	4 7	eqeltrrd	|- ( ( M e. MetSp /\ A e. X /\ B e. X ) -> ( A D B ) e. RR )