Theorem expne0i

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

	Ref	Expression
Assertion	expne0i	|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) =/= 0 )

Proof

Step	Hyp	Ref	Expression
1		expclzlem	|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. ( CC \ { 0 } ) )
2		eldifsni	|- ( ( A ^ N ) e. ( CC \ { 0 } ) -> ( A ^ N ) =/= 0 )
3	1 2	syl	|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) =/= 0 )