Theorem absor

Description: The absolute value of a real number is either that number or its negative. (Contributed by NM, 27-Feb-2005)

	Ref	Expression
Assertion	absor	$$⊢A\in ℝ\to \left(\left|A\right|=A\vee \left|A\right|=-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		letric	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0\le A\vee A\le 0\right)$$
3	1 2	mpan	$$⊢A\in ℝ\to \left(0\le A\vee A\le 0\right)$$
4		absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$
5	4	ex	$$⊢A\in ℝ\to \left(0\le A\to \left|A\right|=A\right)$$
6		absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$
7	6	ex	$$⊢A\in ℝ\to \left(A\le 0\to \left|A\right|=-A\right)$$
8	5 7	orim12d	$$⊢A\in ℝ\to \left(\left(0\le A\vee A\le 0\right)\to \left(\left|A\right|=A\vee \left|A\right|=-A\right)\right)$$
9	3 8	mpd	$$⊢A\in ℝ\to \left(\left|A\right|=A\vee \left|A\right|=-A\right)$$