Theorem immuli

Description: Imaginary part of a product. (Contributed by NM, 28-Jul-1999)

	Ref	Expression
Hypotheses	recl.1	$$⊢A\in ℂ$$
	readdi.2	$$⊢B\in ℂ$$
Assertion	immuli	$$⊢\Im \left(AB\right)=\Re \left(A\right)\Im \left(B\right)+\Im \left(A\right)\Re \left(B\right)$$

Proof

Step	Hyp	Ref	Expression
1		recl.1	$$⊢A\in ℂ$$
2		readdi.2	$$⊢B\in ℂ$$
3		immul	$$⊢\left(A\in ℂ\wedge B\in ℂ\right)\to \Im \left(AB\right)=\Re \left(A\right)\Im \left(B\right)+\Im \left(A\right)\Re \left(B\right)$$
4	1 2 3	mp2an	$$⊢\Im \left(AB\right)=\Re \left(A\right)\Im \left(B\right)+\Im \left(A\right)\Re \left(B\right)$$