Theorem absnidd

Description: A negative number is the negative of its own absolute value. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resqrcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
	absnidd.2	⊢ ( 𝜑 → 𝐴 ≤ 0 )
Assertion	absnidd	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = - 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		resqrcld.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2		absnidd.2	⊢ ( 𝜑 → 𝐴 ≤ 0 )
3		absnid	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐴 ≤ 0 ) → ( abs ‘ 𝐴 ) = - 𝐴 )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( abs ‘ 𝐴 ) = - 𝐴 )