Metamath Proof Explorer


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 ( 𝜑𝐴 ∈ ℝ* )
xnegrecl2d.2 ( 𝜑 → -𝑒 𝐴 ∈ ℝ )
Assertion xnegrecl2d ( 𝜑𝐴 ∈ ℝ )

Proof

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