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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 → 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 → A = − A
4 1 2 3 syl2anc ⊢ φ → A = − A