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Metamath Proof Explorer

Theorem negrebd

Description: The negative of a real is real. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
negrebd.2 ⊢ ( 𝜑 → - 𝐴 ∈ ℝ )
Assertion negrebd ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step Hyp Ref Expression
1 negidd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 negrebd.2 ⊢ ( 𝜑 → - 𝐴 ∈ ℝ )
3 negreb ⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )
4 1 3 syl ⊢ ( 𝜑 → ( - 𝐴 ∈ ℝ ↔ 𝐴 ∈ ℝ ) )
5 2 4 mpbid ⊢ ( 𝜑 → 𝐴 ∈ ℝ )