Theorem negreb

Description: The negative of a real is real. (Contributed by NM, 11-Aug-1999) (Revised by Mario Carneiro, 14-Jul-2014)

	Ref	Expression
Assertion	negreb	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )

Proof

Step	Hyp	Ref	Expression
1		renegcl	⊢ ( - 𝐴 ∈ ℝ → - - 𝐴 ∈ ℝ )
2		negneg	⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 )
3	2	eleq1d	⊢ ( 𝐴 ∈ ℂ → ( - - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )
4	1 3	syl5ib	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℝ → 𝐴 ∈ ℝ ) )
5		renegcl	⊢ ( 𝐴 ∈ ℝ → - 𝐴 ∈ ℝ )
6	4 5	impbid1	⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )