Metamath Proof Explorer
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 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
|
|
xnegrecl2d.2 |
⊢ ( 𝜑 → -𝑒 𝐴 ∈ ℝ ) |
|
Assertion |
xnegrecl2d |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
xnegrecl2d.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
2 |
|
xnegrecl2d.2 |
⊢ ( 𝜑 → -𝑒 𝐴 ∈ ℝ ) |
3 |
|
xnegrecl2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ -𝑒 𝐴 ∈ ℝ ) → 𝐴 ∈ ℝ ) |
4 |
1 2 3
|
syl2anc |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |