Theorem le2sqi

Description: The square function on nonnegative reals is monotonic. (Contributed by NM, 12-Sep-1999)

	Ref	Expression
Hypotheses	resqcl.1	$$⊢A\in ℝ$$
	lt2sq.2	$$⊢B\in ℝ$$
Assertion	le2sqi	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A\le B\leftrightarrow A^2\le B^2\right)$$

Proof

Step	Hyp	Ref	Expression
1		resqcl.1	$$⊢A\in ℝ$$
2		lt2sq.2	$$⊢B\in ℝ$$
3	1 2	le2msqi	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A\le B\leftrightarrow AA\le BB\right)$$
4	1	recni	$$⊢A\in ℂ$$
5	4	sqvali	$$⊢A^2=AA$$
6	2	recni	$$⊢B\in ℂ$$
7	6	sqvali	$$⊢B^2=BB$$
8	5 7	breq12i	$$⊢A^2\le B^2\leftrightarrow AA\le BB$$
9	3 8	bitr4di	$$⊢\left(0\le A\wedge 0\le B\right)\to \left(A\le B\leftrightarrow A^2\le B^2\right)$$