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 ) |
Step | Hyp | Ref | Expression |
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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 ) |