Theorem ipnm

Description: Norm expressed in terms of inner product. (Contributed by NM, 11-Sep-2007) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ipid.1	|- X = ( BaseSet ` U )
	ipid.6	|- N = ( normCV ` U )
	ipid.7	|- P = ( .iOLD ` U )
Assertion	ipnm	|- ( ( U e. NrmCVec /\ A e. X ) -> ( N ` A ) = ( sqrt ` ( A P A ) ) )

Proof

Step	Hyp	Ref	Expression
1		ipid.1	|- X = ( BaseSet ` U )
2		ipid.6	|- N = ( normCV ` U )
3		ipid.7	|- P = ( .iOLD ` U )
4	1 2 3	ipidsq	|- ( ( U e. NrmCVec /\ A e. X ) -> ( A P A ) = ( ( N ` A ) ^ 2 ) )
5	4	fveq2d	|- ( ( U e. NrmCVec /\ A e. X ) -> ( sqrt ` ( A P A ) ) = ( sqrt ` ( ( N ` A ) ^ 2 ) ) )
6	1 2	nvcl	|- ( ( U e. NrmCVec /\ A e. X ) -> ( N ` A ) e. RR )
7	1 2	nvge0	|- ( ( U e. NrmCVec /\ A e. X ) -> 0 <_ ( N ` A ) )
8	6 7	sqrtsqd	|- ( ( U e. NrmCVec /\ A e. X ) -> ( sqrt ` ( ( N ` A ) ^ 2 ) ) = ( N ` A ) )
9	5 8	eqtr2d	|- ( ( U e. NrmCVec /\ A e. X ) -> ( N ` A ) = ( sqrt ` ( A P A ) ) )