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Metamath Proof Explorer

Theorem recxpcld

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 )