Theorem sqrtnegd

Description: The square root of a negative number. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	$$⊢\phi \to A\in ℝ$$
	resqrcld.2	$$⊢\phi \to 0\le A$$
Assertion	sqrtnegd	$$⊢\phi \to \sqrt{-A}=\mathrm{i}\sqrt{A}$$

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	$$⊢\phi \to A\in ℝ$$
2		resqrcld.2	$$⊢\phi \to 0\le A$$
3		sqrtneg	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{-A}=\mathrm{i}\sqrt{A}$$
4	1 2 3	syl2anc	$$⊢\phi \to \sqrt{-A}=\mathrm{i}\sqrt{A}$$