Theorem absnidd

Description: A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	$$⊢\phi \to A\in ℝ$$
	absnidd.2	$$⊢\phi \to A\le 0$$
Assertion	absnidd	$$⊢\phi \to \left|A\right|=-A$$

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	$$⊢\phi \to A\in ℝ$$
2		absnidd.2	$$⊢\phi \to A\le 0$$
3		absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$
4	1 2 3	syl2anc	$$⊢\phi \to \left|A\right|=-A$$