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Metamath Proof Explorer

Theorem rpcxpcld

Description: Positive real closure of the complex power function. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses rpcxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
rpcxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
Assertion rpcxpcld ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )

Proof

Step Hyp Ref Expression
1 rpcxpcld.1 ⊢ ( 𝜑 → 𝐴 ∈ ℝ+ )
2 rpcxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
3 rpcxpcl ⊢ ( ( 𝐴 ∈ ℝ+ ∧ 𝐵 ∈ ℝ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℝ+ )