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 ) )