Theorem absvalsq2

Description: Square of value of absolute value function. (Contributed by NM, 1-Feb-2007)

	Ref	Expression
Assertion	absvalsq2	$$⊢A\in ℂ\to {\left|A\right|}^2={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$

Proof

Step	Hyp	Ref	Expression
1		absvalsq	$$⊢A\in ℂ\to {\left|A\right|}^2=A\bar{A}$$
2		cjmulval	$$⊢A\in ℂ\to A\bar{A}={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$
3	1 2	eqtrd	$$⊢A\in ℂ\to {\left|A\right|}^2={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$