Theorem sqrtmuli

Description: Square root distributes over multiplication. (Contributed by NM, 30-Jul-1999)

	Ref	Expression
Hypotheses	sqrtthi.1	$$⊢A\in ℝ$$
	sqr11.1	$$⊢B\in ℝ$$
Assertion	sqrtmuli	$$⊢\left(0\le A\wedge 0\le B\right)\to \sqrt{AB}=\sqrt{A}\sqrt{B}$$

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	$$⊢A\in ℝ$$
2		sqr11.1	$$⊢B\in ℝ$$
3		sqrtmul	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \sqrt{AB}=\sqrt{A}\sqrt{B}$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge 0\le B\right)\to \sqrt{AB}=\sqrt{A}\sqrt{B}$$
5	1 4	mpanl1	$$⊢\left(0\le A\wedge 0\le B\right)\to \sqrt{AB}=\sqrt{A}\sqrt{B}$$