Theorem reabsifnneg

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

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

Proof

Step	Hyp	Ref	Expression
1		absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$
2	1	eqcomd	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to A=\left|A\right|$$
3		0re	$$⊢0\in ℝ$$
4		ltnle	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(A<0\leftrightarrow \neg 0\le A\right)$$
5		ltle	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(A<0\to A\le 0\right)$$
6	4 5	sylbird	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(\neg 0\le A\to A\le 0\right)$$
7	3 6	mpan2	$$⊢A\in ℝ\to \left(\neg 0\le A\to A\le 0\right)$$
8	7	imdistani	$$⊢\left(A\in ℝ\wedge \neg 0\le A\right)\to \left(A\in ℝ\wedge A\le 0\right)$$
9		absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$
10	8 9	syl	$$⊢\left(A\in ℝ\wedge \neg 0\le A\right)\to \left|A\right|=-A$$
11	10	eqcomd	$$⊢\left(A\in ℝ\wedge \neg 0\le A\right)\to -A=\left|A\right|$$
12	2 11	ifeqda	$$⊢A\in ℝ\to if\left(0\le A,A,-A\right)=\left|A\right|$$
13	12	eqcomd	$$⊢A\in ℝ\to \left|A\right|=if\left(0\le A,A,-A\right)$$