Theorem absval2d

Description: Value of absolute value function. Definition 10.36 of Gleason p. 133. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	$$⊢\phi \to A\in ℂ$$
Assertion	absval2d	$$⊢\phi \to \left|A\right|=\sqrt{{\Re \left(A\right)}^2+{\Im \left(A\right)}^2}$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		absval2	$$⊢A\in ℂ\to \left|A\right|=\sqrt{{\Re \left(A\right)}^2+{\Im \left(A\right)}^2}$$
3	1 2	syl	$$⊢\phi \to \left|A\right|=\sqrt{{\Re \left(A\right)}^2+{\Im \left(A\right)}^2}$$