Theorem hhmetdval

Description: Value of the distance function of the metric space of Hilbert space. (Contributed by NM, 10-Apr-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	hhnv.1	|- U = <. <. +h , .h >. , normh >.
	hhims2.2	|- D = ( IndMet ` U )
Assertion	hhmetdval	|- ( ( A e. ~H /\ B e. ~H ) -> ( A D B ) = ( normh ` ( A -h B ) ) )

Proof

Step	Hyp	Ref	Expression
1		hhnv.1	|- U = <. <. +h , .h >. , normh >.
2		hhims2.2	|- D = ( IndMet ` U )
3	1	hhnv	|- U e. NrmCVec
4	1	hhba	|- ~H = ( BaseSet ` U )
5	1 3 4 2	h2hmetdval	|- ( ( A e. ~H /\ B e. ~H ) -> ( A D B ) = ( normh ` ( A -h B ) ) )