Theorem xnegrecl2d

Description: If the extended real negative is real, then the number itself is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)

	Ref	Expression
Hypotheses	xnegrecl2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
	xnegrecl2d.2	⊢ ( 𝜑 → -𝑒 𝐴 ∈ ℝ )
Assertion	xnegrecl2d	⊢ ( 𝜑 → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		xnegrecl2d.1	⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
2		xnegrecl2d.2	⊢ ( 𝜑 → -𝑒 𝐴 ∈ ℝ )
3		xnegrecl2	⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ )
4	1 2 3	syl2anc	⊢ ( 𝜑 → 𝐴 ∈ ℝ )