Theorem expne0i

Description: An integer power is nonzero if its base is nonzero. (Contributed by NM, 2-Aug-2006) (Revised by Mario Carneiro, 4-Jun-2014)

	Ref	Expression
Assertion	expne0i	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 )

Proof

Step	Hyp	Ref	Expression
1		expclzlem	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ∈ ( ℂ ∖ { 0 } ) )
2		eldifsni	⊢ ( ( 𝐴 ↑ 𝑁 ) ∈ ( ℂ ∖ { 0 } ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 )
3	1 2	syl	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ ) → ( 𝐴 ↑ 𝑁 ) ≠ 0 )