Theorem hhmet

Description: The induced metric 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	hhmet	|- D e. ( Met ` ~H )

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	4 2	imsmet	|- ( U e. NrmCVec -> D e. ( Met ` ~H ) )
6	3 5	ax-mp	|- D e. ( Met ` ~H )