Theorem sqrtrege0d

Description: The real part of the square root function is nonnegative. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypothesis	abscld.1	$$⊢\phi \to A\in ℂ$$
Assertion	sqrtrege0d	$$⊢\phi \to 0\le \Re \left(\sqrt{A}\right)$$

Proof

Step	Hyp	Ref	Expression
1		abscld.1	$$⊢\phi \to A\in ℂ$$
2		sqrtrege0	$$⊢A\in ℂ\to 0\le \Re \left(\sqrt{A}\right)$$
3	1 2	syl	$$⊢\phi \to 0\le \Re \left(\sqrt{A}\right)$$