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Metamath Proof Explorer

Theorem absord

Description: The absolute value of a real number is either that number or its negative. (Contributed by Mario Carneiro, 29-May-2016)

Ref Expression
Hypothesis resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
Assertion absord ( 𝜑 → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) )

Proof

Step Hyp Ref Expression
1 resqrcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 absor ⊢ ( 𝐴 ∈ ℝ → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ( abs ‘ 𝐴 ) = 𝐴 ∨ ( abs ‘ 𝐴 ) = - 𝐴 ) )