Description: An extended real is real if and only if its extended negative is real. (Contributed by Glauco Siliprandi, 2-Jan-2022)
Ref | Expression | ||
---|---|---|---|
Assertion | xnegre | ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ℝ ↔ -𝑒 𝐴 ∈ ℝ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xnegrecl | ⊢ ( 𝐴 ∈ ℝ → -𝑒 𝐴 ∈ ℝ ) | |
2 | 1 | adantl | ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐴 ∈ ℝ ) → -𝑒 𝐴 ∈ ℝ ) |
3 | xnegrecl2 | ⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ ) | |
4 | 2 3 | impbida | ⊢ ( 𝐴 ∈ ℝ* → ( 𝐴 ∈ ℝ ↔ -𝑒 𝐴 ∈ ℝ ) ) |