Theorem ipcni

Description: Standard inner product on complex numbers. (Contributed by NM, 2-Oct-1999)

	Ref	Expression
Hypotheses	recl.1	|- A e. CC
	readdi.2	|- B e. CC
Assertion	ipcni	|- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		recl.1	|- A e. CC
2		readdi.2	|- B e. CC
3		ipcnval	|- ( ( A e. CC /\ B e. CC ) -> ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) ) )
4	1 2 3	mp2an	|- ( Re ` ( A x. ( * ` B ) ) ) = ( ( ( Re ` A ) x. ( Re ` B ) ) + ( ( Im ` A ) x. ( Im ` B ) ) )