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 ) )