Metamath Proof Explorer


Theorem expne0i

Description: Nonnegative integer exponentiation 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 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 )