sqrtsq

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

		Ref	Expression
	Assertion	sqrtsq	$⊢ (A \in ℝ \land 0 \leq A) \to \sqrt{A^{2}} = A$

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

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