Theorem reabsifnneg

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

	Ref	Expression
Assertion	reabsifnneg	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 0 ≤ 𝐴 , 𝐴 , - 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		absid	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( abs ‘ 𝐴 ) = 𝐴 )
2	1	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → 𝐴 = ( abs ‘ 𝐴 ) )
3		0re	⊢ 0 ∈ ℝ
4		ltnle	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 < 0 ↔ ¬ 0 ≤ 𝐴 ) )
5		ltle	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 < 0 → 𝐴 ≤ 0 ) )
6	4 5	sylbird	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( ¬ 0 ≤ 𝐴 → 𝐴 ≤ 0 ) )
7	3 6	mpan2	⊢ ( 𝐴 ∈ ℝ → ( ¬ 0 ≤ 𝐴 → 𝐴 ≤ 0 ) )
8	7	imdistani	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 0 ≤ 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) )
9		absnid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
10	8 9	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 0 ≤ 𝐴 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
11	10	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 0 ≤ 𝐴 ) → - 𝐴 = ( abs ‘ 𝐴 ) )
12	2 11	ifeqda	⊢ ( 𝐴 ∈ ℝ → if ( 0 ≤ 𝐴 , 𝐴 , - 𝐴 ) = ( abs ‘ 𝐴 ) )
13	12	eqcomd	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 0 ≤ 𝐴 , 𝐴 , - 𝐴 ) )