Theorem xnegrecl2

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

	Ref	Expression
Assertion	xnegrecl2	⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ )

Proof

Step	Hyp	Ref	Expression
1		xnegneg	⊢ ( 𝐴 ∈ ℝ* → -𝑒 -𝑒 𝐴 = 𝐴 )
2	1	adantr	⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → -𝑒 -𝑒 𝐴 = 𝐴 )
3		xnegrecl	⊢ ( -𝑒 𝐴 ∈ ℝ → -𝑒 -𝑒 𝐴 ∈ ℝ )
4	3	adantl	⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → -𝑒 -𝑒 𝐴 ∈ ℝ )
5	2 4	eqeltrrd	⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ )