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Metamath Proof Explorer

Theorem cxpexpzd

Description: Relate the complex power function to the integer power function. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses cxp0d.1 ⊢ φ → A ∈ ℂ
cxpefd.2 ⊢ φ → A ≠ 0
cxpexpzd.3 ⊢ φ → B ∈ ℤ
Assertion cxpexpzd ⊢ φ → A B = A B

Proof

Step Hyp Ref Expression
1 cxp0d.1 ⊢ φ → A ∈ ℂ
2 cxpefd.2 ⊢ φ → A ≠ 0
3 cxpexpzd.3 ⊢ φ → B ∈ ℤ
4 cxpexpz ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ B ∈ ℤ → A B = A B
5 1 2 3 4 syl3anc ⊢ φ → A B = A B