Theorem absnidi

Description: A negative number is the negative of its own absolute value. (Contributed by NM, 2-Aug-1999)

	Ref	Expression
Hypothesis	sqrtthi.1	⊢ 𝐴 ∈ ℝ
Assertion	absnidi	⊢ ( 𝐴 ≤ 0 → ( abs ‘ 𝐴 ) = - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sqrtthi.1	⊢ 𝐴 ∈ ℝ
2		absnid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
3	1 2	mpan	⊢ ( 𝐴 ≤ 0 → ( abs ‘ 𝐴 ) = - 𝐴 )