Theorem reabsifneg

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

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

Proof

Step	Hyp	Ref	Expression
1		0re	$$⊢0\in ℝ$$
2		ltle	$$⊢\left(A\in ℝ\wedge 0\in ℝ\right)\to \left(A<0\to A\le 0\right)$$
3	1 2	mpan2	$$⊢A\in ℝ\to \left(A<0\to A\le 0\right)$$
4	3	imdistani	$$⊢\left(A\in ℝ\wedge A<0\right)\to \left(A\in ℝ\wedge A\le 0\right)$$
5		absnid	$$⊢\left(A\in ℝ\wedge A\le 0\right)\to \left|A\right|=-A$$
6	4 5	syl	$$⊢\left(A\in ℝ\wedge A<0\right)\to \left|A\right|=-A$$
7	6	eqcomd	$$⊢\left(A\in ℝ\wedge A<0\right)\to -A=\left|A\right|$$
8		0red	$$⊢A\in ℝ\to 0\in ℝ$$
9		id	$$⊢A\in ℝ\to A\in ℝ$$
10	8 9	lenltd	$$⊢A\in ℝ\to \left(0\le A\leftrightarrow \neg A<0\right)$$
11	10	bicomd	$$⊢A\in ℝ\to \left(\neg A<0\leftrightarrow 0\le A\right)$$
12	11	pm5.32i	$$⊢\left(A\in ℝ\wedge \neg A<0\right)\leftrightarrow \left(A\in ℝ\wedge 0\le A\right)$$
13		absid	$$⊢\left(A\in ℝ\wedge 0\le A\right)\to \left|A\right|=A$$
14	12 13	sylbi	$$⊢\left(A\in ℝ\wedge \neg A<0\right)\to \left|A\right|=A$$
15	14	eqcomd	$$⊢\left(A\in ℝ\wedge \neg A<0\right)\to A=\left|A\right|$$
16	7 15	ifeqda	$$⊢A\in ℝ\to if\left(A<0,-A,A\right)=\left|A\right|$$
17	16	eqcomd	$$⊢A\in ℝ\to \left|A\right|=if\left(A<0,-A,A\right)$$