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Metamath Proof Explorer

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 )