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 ) |
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 ) |