Theorem reabsifpos

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

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

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		ltle	⊢ ( ( 0 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 0 < 𝐴 → 0 ≤ 𝐴 ) )
3	1 2	mpan	⊢ ( 𝐴 ∈ ℝ → ( 0 < 𝐴 → 0 ≤ 𝐴 ) )
4	3	imdistani	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
5		absid	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( abs ‘ 𝐴 ) = 𝐴 )
6	4 5	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → ( abs ‘ 𝐴 ) = 𝐴 )
7	6	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 < 𝐴 ) → 𝐴 = ( abs ‘ 𝐴 ) )
8		id	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ )
9		0red	⊢ ( 𝐴 ∈ ℝ → 0 ∈ ℝ )
10	8 9	lenltd	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 ≤ 0 ↔ ¬ 0 < 𝐴 ) )
11	10	pm5.32i	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) ↔ ( 𝐴 ∈ ℝ ∧ ¬ 0 < 𝐴 ) )
12		absnid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
13	11 12	sylbir	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 0 < 𝐴 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
14	13	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 0 < 𝐴 ) → - 𝐴 = ( abs ‘ 𝐴 ) )
15	7 14	ifeqda	⊢ ( 𝐴 ∈ ℝ → if ( 0 < 𝐴 , 𝐴 , - 𝐴 ) = ( abs ‘ 𝐴 ) )
16	15	eqcomd	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 0 < 𝐴 , 𝐴 , - 𝐴 ) )