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Metamath Proof Explorer

Theorem cxpcld

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

Ref Expression
Hypotheses cxp0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
cxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
Assertion cxpcld ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )

Proof

Step Hyp Ref Expression
1 cxp0d.1 ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
2 cxpcld.2 ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
3 cxpcl ⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )
4 1 2 3 syl2anc ⊢ ( 𝜑 → ( 𝐴 ↑𝑐 𝐵 ) ∈ ℂ )