Theorem absled

Description: Absolute value and 'less than or equal to' relation. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	$$⊢\phi \to A\in ℝ$$
	absltd.2	$$⊢\phi \to B\in ℝ$$
Assertion	absled	$$⊢\phi \to \left(\left|A\right|\le B\leftrightarrow \left(-B\le A\wedge A\le B\right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		absltd.1	$$⊢\phi \to A\in ℝ$$
2		absltd.2	$$⊢\phi \to B\in ℝ$$
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	syl2anc	$$⊢\phi \to \left(\left|A\right|\le B\leftrightarrow \left(-B\le A\wedge A\le B\right)\right)$$