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Metamath Proof Explorer

Theorem expne0d

Description: A nonnegative integer power is nonzero if its base is nonzero. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses expcld.1 ⊢ φ → A ∈ ℂ
sqrecd.1 ⊢ φ → A ≠ 0
expclzd.3 ⊢ φ → N ∈ ℤ
Assertion expne0d ⊢ φ → A N ≠ 0

Proof

Step Hyp Ref Expression
1 expcld.1 ⊢ φ → A ∈ ℂ
2 sqrecd.1 ⊢ φ → A ≠ 0
3 expclzd.3 ⊢ φ → N ∈ ℤ
4 expne0i ⊢ A ∈ ℂ ∧ A ≠ 0 ∧ N ∈ ℤ → A N ≠ 0
5 1 2 3 4 syl3anc ⊢ φ → A N ≠ 0