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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 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 B ∈ ℂ
4 1 2 3 syl2anc ⊢ φ → A B ∈ ℂ