Theorem abslei

Description: Absolute value and 'less than or equal to' relation. (Contributed by NM, 6-Apr-2005)

	Ref	Expression
Hypotheses	sqrtthi.1	$$⊢A\in ℝ$$
	sqr11.1	$$⊢B\in ℝ$$
Assertion	abslei	$$⊢\left|A\right|\le B\leftrightarrow \left(-B\le A\wedge A\le B\right)$$

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	$$⊢A\in ℝ$$
2		sqr11.1	$$⊢B\in ℝ$$
3		absle	$$⊢\left(A\in ℝ\wedge B\in ℝ\right)\to \left(\left|A\right|\le B\leftrightarrow \left(-B\le A\wedge A\le B\right)\right)$$
4	1 2 3	mp2an	$$⊢\left|A\right|\le B\leftrightarrow \left(-B\le A\wedge A\le B\right)$$