Theorem immul2d

Description: Imaginary part of a product. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	crred.1	|- ( ph -> A e. RR )
	remul2d.2	|- ( ph -> B e. CC )
Assertion	immul2d	|- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )

Proof

Step	Hyp	Ref	Expression
1		crred.1	|- ( ph -> A e. RR )
2		remul2d.2	|- ( ph -> B e. CC )
3		immul2	|- ( ( A e. RR /\ B e. CC ) -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )
4	1 2 3	syl2anc	|- ( ph -> ( Im ` ( A x. B ) ) = ( A x. ( Im ` B ) ) )