Theorem remsqsqrt

Description: Square of square root. (Contributed by Mario Carneiro, 10-Jul-2013)

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

Proof

Step	Hyp	Ref	Expression
1		resqrtcl	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A}\in ℝ$$
2	1	recnd	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A}\in ℂ$$
3	2	sqvald	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to {\sqrt{A}}^2=\sqrt{A}\sqrt{A}$$
4		resqrtth	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to {\sqrt{A}}^2=A$$
5	3 4	eqtr3d	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \sqrt{A}\sqrt{A}=A$$