Theorem sqrtlei

Description: Square root is monotonic. (Contributed by NM, 3-Aug-1999)

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

Proof

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