Theorem sqrtsqd

Description: Square root of square. (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	sqrtsqd	$$⊢\phi \to \sqrt{A^2}=A$$

Proof

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