Theorem le2msqi

Description: The square function on nonnegative reals is monotonic. (Contributed by NM, 2-Aug-1999)

	Ref	Expression
Hypotheses	ltplus1.1	$$⊢A\in ℝ$$
	prodgt0.2	$$⊢B\in ℝ$$
Assertion	le2msqi	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A\le B\leftrightarrow AA\le BB\right)$$

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	$$⊢A\in ℝ$$
2		prodgt0.2	$$⊢B\in ℝ$$
3		le2msq	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(A\le B\leftrightarrow AA\le BB\right)$$
4	2 3	mpanr1	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge 0\le B\right)\to \left(A\le B\leftrightarrow AA\le BB\right)$$
5	1 4	mpanl1	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A\le B\leftrightarrow AA\le BB\right)$$