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Metamath Proof Explorer

Theorem xnegred

Description: An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

Ref Expression
Hypothesis xnegred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
Assertion xnegred ( 𝜑 → ( 𝐴 ∈ ℝ ↔ -𝑒 𝐴 ∈ ℝ ) )

Proof

Step Hyp Ref Expression
1 xnegred.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2 xnegre ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ℝ ↔ -𝑒 𝐴 ∈ ℝ ) )
3 1 2 syl ⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ↔ -𝑒 𝐴 ∈ ℝ ) )