Theorem le2sqd

Description: The square function on nonnegative reals is monotonic. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	sqgt0d.1	$$⊢\phi \to A\in ℝ$$
	lt2sqd.2	$$⊢\phi \to B\in ℝ$$
	lt2sqd.3	$$⊢\phi \to 0\le A$$
	lt2sqd.4	$$⊢\phi \to 0\le B$$
Assertion	le2sqd	$$⊢\phi \to \left(A\le B\leftrightarrow A^2\le B^2\right)$$

Proof

Step	Hyp	Ref	Expression
1		sqgt0d.1	$$⊢\phi \to A\in ℝ$$
2		lt2sqd.2	$$⊢\phi \to B\in ℝ$$
3		lt2sqd.3	$$⊢\phi \to 0\le A$$
4		lt2sqd.4	$$⊢\phi \to 0\le B$$
5		le2sq	$$⊢\left(\left(A\in ℝ\wedge 0\le A\right)\wedge \left(B\in ℝ\wedge 0\le B\right)\right)\to \left(A\le B\leftrightarrow A^2\le B^2\right)$$
6	1 3 2 4 5	syl22anc	$$⊢\phi \to \left(A\le B\leftrightarrow A^2\le B^2\right)$$