Theorem cjmulvald

Description: A complex number times its conjugate. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	recld.1	$$⊢\phi \to A\in ℂ$$
Assertion	cjmulvald	$$⊢\phi \to A\bar{A}={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$

Proof

Step	Hyp	Ref	Expression
1		recld.1	$$⊢\phi \to A\in ℂ$$
2		cjmulval	$$⊢A\in ℂ\to A\bar{A}={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$
3	1 2	syl	$$⊢\phi \to A\bar{A}={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$