Metamath Proof Explorer
Description: Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016)
|
|
Ref |
Expression |
|
Hypotheses |
recxpcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
|
|
recxpcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
|
|
recxpcld.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
|
Assertion |
recxpcld |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
recxpcld.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
recxpcld.2 |
⊢ ( 𝜑 → 0 ≤ 𝐴 ) |
3 |
|
recxpcld.3 |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
recxpcl |
⊢ ( ( 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴 ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ ) |
5 |
1 2 3 4
|
syl3anc |
⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ ) |