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Metamath Proof Explorer

Theorem sqrtmsq

Description: Square root of square. (Contributed by NM, 2-Aug-1999) (Revised by Mario Carneiro, 29-May-2016)

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

Step	Hyp	Ref	Expression
1		simpl	$⊢ (A \in ℝ \land 0 \leq A) \to A \in ℝ$
2	1	recnd	$⊢ (A \in ℝ \land 0 \leq A) \to A \in ℂ$
3	2	sqvald	$⊢ (A \in ℝ \land 0 \leq A) \to A^{2} = A A$
4	3	fveq2d	$⊢ (A \in ℝ \land 0 \leq A) \to \sqrt{A^{2}} = \sqrt{A A}$
5		sqrtsq	$⊢ (A \in ℝ \land 0 \leq A) \to \sqrt{A^{2}} = A$
6	4 5	eqtr3d	$⊢ (A \in ℝ \land 0 \leq A) \to \sqrt{A A} = A$