Theorem expne0

Description: Positive integer exponentiation is nonzero iff its mantissa is nonzero. (Contributed by NM, 6-May-2005)

	Ref	Expression
Assertion	expne0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑁 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )

Proof

Step	Hyp	Ref	Expression
1		expeq0	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑁 ) = 0 ↔ 𝐴 = 0 ) )
2	1	necon3bid	⊢ ( ( 𝐴 ∈ ℂ ∧ 𝑁 ∈ ℕ ) → ( ( 𝐴 ↑ 𝑁 ) ≠ 0 ↔ 𝐴 ≠ 0 ) )