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 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 𝐴 < 0 , - 𝐴 , 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		0re	⊢ 0 ∈ ℝ
2		ltle	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ∈ ℝ ) → ( 𝐴 < 0 → 𝐴 ≤ 0 ) )
3	1 2	mpan2	⊢ ( 𝐴 ∈ ℝ → ( 𝐴 < 0 → 𝐴 ≤ 0 ) )
4	3	imdistani	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) )
5		absnid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
6	4 5	syl	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
7	6	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 < 0 ) → - 𝐴 = ( abs ‘ 𝐴 ) )
8		0red	⊢ ( 𝐴 ∈ ℝ → 0 ∈ ℝ )
9		id	⊢ ( 𝐴 ∈ ℝ → 𝐴 ∈ ℝ )
10	8 9	lenltd	⊢ ( 𝐴 ∈ ℝ → ( 0 ≤ 𝐴 ↔ ¬ 𝐴 < 0 ) )
11	10	bicomd	⊢ ( 𝐴 ∈ ℝ → ( ¬ 𝐴 < 0 ↔ 0 ≤ 𝐴 ) )
12	11	pm5.32i	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 < 0 ) ↔ ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) )
13		absid	⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ) → ( abs ‘ 𝐴 ) = 𝐴 )
14	12 13	sylbi	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 < 0 ) → ( abs ‘ 𝐴 ) = 𝐴 )
15	14	eqcomd	⊢ ( ( 𝐴 ∈ ℝ ∧ ¬ 𝐴 < 0 ) → 𝐴 = ( abs ‘ 𝐴 ) )
16	7 15	ifeqda	⊢ ( 𝐴 ∈ ℝ → if ( 𝐴 < 0 , - 𝐴 , 𝐴 ) = ( abs ‘ 𝐴 ) )
17	16	eqcomd	⊢ ( 𝐴 ∈ ℝ → ( abs ‘ 𝐴 ) = if ( 𝐴 < 0 , - 𝐴 , 𝐴 ) )