Theorem abssuble0d

Description: Absolute value of a nonpositive difference. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	absltd.1	$$⊢\phi \to A\in ℝ$$
	absltd.2	$$⊢\phi \to B\in ℝ$$
	abssubge0d.2	$$⊢\phi \to A\le B$$
Assertion	abssuble0d	$$⊢\phi \to \left|A-B\right|=B-A$$

Proof

Step	Hyp	Ref	Expression
1		absltd.1	$$⊢\phi \to A\in ℝ$$
2		absltd.2	$$⊢\phi \to B\in ℝ$$
3		abssubge0d.2	$$⊢\phi \to A\le B$$
4		abssuble0	$$⊢\left(A\in ℝ\wedge B\in ℝ\wedge A\le B\right)\to \left|A-B\right|=B-A$$
5	1 2 3 4	syl3anc	$$⊢\phi \to \left|A-B\right|=B-A$$