Theorem absvalsq2d

Description: Square of value of absolute value function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	$$⊢\phi \to A\in ℂ$$
Assertion	absvalsq2d	$$⊢\phi \to {\left|A\right|}^2={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		absvalsq2	$$⊢A\in ℂ\to {\left|A\right|}^2={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$
3	1 2	syl	$$⊢\phi \to {\left|A\right|}^2={\Re \left(A\right)}^2+{\Im \left(A\right)}^2$$