Metamath Proof Explorer


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 )