Theorem ipz

Description: The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of Kreyszig p. 129. (Contributed by NM, 24-Jan-2008) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dip0r.1	|- X = ( BaseSet ` U )
	dip0r.5	|- Z = ( 0vec ` U )
	dip0r.7	|- P = ( .iOLD ` U )
Assertion	ipz	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A P A ) = 0 <-> A = Z ) )

Proof

Step	Hyp	Ref	Expression
1		dip0r.1	|- X = ( BaseSet ` U )
2		dip0r.5	|- Z = ( 0vec ` U )
3		dip0r.7	|- P = ( .iOLD ` U )
4		eqid	|- ( normCV ` U ) = ( normCV ` U )
5	1 4 3	ipidsq	|- ( ( U e. NrmCVec /\ A e. X ) -> ( A P A ) = ( ( ( normCV ` U ) ` A ) ^ 2 ) )
6	5	eqeq1d	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A P A ) = 0 <-> ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 ) )
7	1 4	nvcl	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( normCV ` U ) ` A ) e. RR )
8	7	recnd	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( normCV ` U ) ` A ) e. CC )
9		sqeq0	|- ( ( ( normCV ` U ) ` A ) e. CC -> ( ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 <-> ( ( normCV ` U ) ` A ) = 0 ) )
10	8 9	syl	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( ( ( normCV ` U ) ` A ) ^ 2 ) = 0 <-> ( ( normCV ` U ) ` A ) = 0 ) )
11	1 2 4	nvz	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( ( normCV ` U ) ` A ) = 0 <-> A = Z ) )
12	6 10 11	3bitrd	|- ( ( U e. NrmCVec /\ A e. X ) -> ( ( A P A ) = 0 <-> A = Z ) )