Theorem reabsifpos

Description: Alternate expression for the absolute value of a real number. (Contributed by RP, 11-May-2024)

	Ref	Expression
Assertion	reabsifpos	$$⊢A\in ℝ\to \left|A\right|=if\left(0<A,A,-A\right)$$

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltle	$$⊢\left(0\in ℝ\wedge A\in ℝ\right)\to \left(0<A\to 0\le A\right)$$
3	1 2	mpan	$$⊢A\in ℝ\to \left(0<A\to 0\le A\right)$$
4	3	imdistani	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left(A\in ℝ\wedge 0\le A\right)$$
5		absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$
6	4 5	syl	$$⊢\left(A\in ℝ\wedge 0<A\right)\to \left|A\right|=A$$
7	6	eqcomd	$$⊢\left(A\in ℝ\wedge 0<A\right)\to A=\left|A\right|$$
8		id	$$⊢A\in ℝ\to A\in ℝ$$
9		0red	$$⊢A\in ℝ\to 0\in ℝ$$
10	8 9	lenltd	$$⊢A\in ℝ\to \left(A\le 0\leftrightarrow \neg 0<A\right)$$
11	10	pm5.32i	$$⊢\left(A\in ℝ\wedge A\le 0\right)\leftrightarrow \left(A\in ℝ\wedge \neg 0<A\right)$$
12		absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$
13	11 12	sylbir	$$⊢\left(A\in ℝ\wedge \neg 0<A\right)\to \left|A\right|=-A$$
14	13	eqcomd	$$⊢\left(A\in ℝ\wedge \neg 0<A\right)\to -A=\left|A\right|$$
15	7 14	ifeqda	$$⊢A\in ℝ\to if\left(0<A,A,-A\right)=\left|A\right|$$
16	15	eqcomd	$$⊢A\in ℝ\to \left|A\right|=if\left(0<A,A,-A\right)$$