This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem negelrpd

Description: The negation of a negative number is in the positive real numbers. (Contributed by Glauco Siliprandi, 26-Jun-2021)

Ref Expression
Hypotheses negelrpd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
negelrpd.2 ⊢ ( 𝜑 → 𝐴 < 0 )
Assertion negelrpd ( 𝜑 → - 𝐴 ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 negelrpd.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
2 negelrpd.2 ⊢ ( 𝜑 → 𝐴 < 0 )
3 negelrp ⊢ ( 𝐴 ∈ ℝ → ( - 𝐴 ∈ ℝ+ ↔ 𝐴 < 0 ) )
4 1 3 syl ⊢ ( 𝜑 → ( - 𝐴 ∈ ℝ+ ↔ 𝐴 < 0 ) )
5 2 4 mpbird ⊢ ( 𝜑 → - 𝐴 ∈ ℝ+ )