Theorem absnid

Description: A negative number is the negative of its own absolute value. (Contributed by NM, 27-Feb-2005)

	Ref	Expression
Assertion	absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$

Proof

Step	Hyp	Ref	Expression
1		le0neg1	$$⊢A\in ℝ\to \left(A\le 0\leftrightarrow 0\le -A\right)$$
2		recn	$$⊢A\in ℝ\to A\in ℂ$$
3		absneg	$$⊢A\in ℂ\to \left|-A\right|=\left|A\right|$$
4	2 3	syl	$$⊢A\in ℝ\to \left|-A\right|=\left|A\right|$$
5	4	adantr	$$⊢\left(A\in ℝ\wedge 0\le -A\right)\to \left|-A\right|=\left|A\right|$$
6		renegcl	$$⊢A\in ℝ\to -A\in ℝ$$
7		absid	$$⊢\left(-A\in ℝ\wedge 0\le -A\right)\to \left|-A\right|=-A$$
8	6 7	sylan	$$⊢\left(A\in ℝ\wedge 0\le -A\right)\to \left|-A\right|=-A$$
9	5 8	eqtr3d	$$⊢\left(A\in ℝ\wedge 0\le -A\right)\to \left|A\right|=-A$$
10	9	ex	$$⊢A\in ℝ\to \left(0\le -A\to \left|A\right|=-A\right)$$
11	1 10	sylbid	$$⊢A\in ℝ\to \left(A\le 0\to \left|A\right|=-A\right)$$
12	11	imp	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$