Theorem cphip0l

Description: Inner product with a zero first argument. Part of proof of Theorem 6.44 of Ponnusamy p. 361. Complex version of ip0l . (Contributed by Mario Carneiro, 16-Oct-2015)

	Ref	Expression
Hypotheses	cphipcj.h	|- ., = ( .i ` W )
	cphipcj.v	|- V = ( Base ` W )
	cphip0l.z	|- .0. = ( 0g ` W )
Assertion	cphip0l	|- ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = 0 )

Proof

Step	Hyp	Ref	Expression
1		cphipcj.h	|- ., = ( .i ` W )
2		cphipcj.v	|- V = ( Base ` W )
3		cphip0l.z	|- .0. = ( 0g ` W )
4		cphphl	|- ( W e. CPreHil -> W e. PreHil )
5		eqid	|- ( Scalar ` W ) = ( Scalar ` W )
6		eqid	|- ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
7	5 1 2 6 3	ip0l	|- ( ( W e. PreHil /\ A e. V ) -> ( .0. ., A ) = ( 0g ` ( Scalar ` W ) ) )
8	4 7	sylan	|- ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = ( 0g ` ( Scalar ` W ) ) )
9		cphclm	|- ( W e. CPreHil -> W e. CMod )
10	5	clm0	|- ( W e. CMod -> 0 = ( 0g ` ( Scalar ` W ) ) )
11	9 10	syl	|- ( W e. CPreHil -> 0 = ( 0g ` ( Scalar ` W ) ) )
12	11	adantr	|- ( ( W e. CPreHil /\ A e. V ) -> 0 = ( 0g ` ( Scalar ` W ) ) )
13	8 12	eqtr4d	|- ( ( W e. CPreHil /\ A e. V ) -> ( .0. ., A ) = 0 )