Theorem immul2d

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

	Ref	Expression
Hypotheses	crred.1	$$⊢\phi \to A\in ℝ$$
	remul2d.2	$$⊢\phi \to B\in ℂ$$
Assertion	immul2d	$$⊢\phi \to \Im \left(AB\right)=A\Im \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		crred.1	$$⊢\phi \to A\in ℝ$$
2		remul2d.2	$$⊢\phi \to B\in ℂ$$
3		immul2	$$⊢\left(A\in ℝ\wedge B\in ℂ\right)\to \Im \left(AB\right)=A\Im \left(B\right)$$
4	1 2 3	syl2anc	$$⊢\phi \to \Im \left(AB\right)=A\Im \left(B\right)$$