Theorem sqrtlti

Description: Square root is strictly monotonic. (Contributed by Roy F. Longton, 8-Aug-2005)

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

Proof

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