Theorem sqrtsq

Description: Square root of square. (Contributed by NM, 14-Jan-2006) (Revised by Mario Carneiro, 29-May-2016)

	Ref	Expression
Assertion	sqrtsq	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A^2}=A$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢A^2=A^2$$
2		resqcl	$$⊢A\in ℝ\to A^2\in ℝ$$
3		sqge0	$$⊢A\in ℝ\to 0\le A^2$$
4	2 3	jca	$$⊢A\in ℝ\to \left(A^2\in ℝ\wedge 0\le A^2\right)$$
5	4	adantr	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left(A^2\in ℝ\wedge 0\le A^2\right)$$
6		sqrtsq2	$$⊢\left(\left(A^2\in ℝ\wedge 0\le A^2\right)\wedge \left(A\in ℝ\wedge 0\le A\right)\right)\to \left(\sqrt{A^2}=A\leftrightarrow A^2=A^2\right)$$
7	5 6	mpancom	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left(\sqrt{A^2}=A\leftrightarrow A^2=A^2\right)$$
8	1 7	mpbiri	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A^2}=A$$