Metamath Proof Explorer

Theorem zexpcl

Description: Closure of exponentiation of integers. (Contributed by NM, 16-Dec-2005)

Ref Expression
Assertion zexpcl 
|- ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )

Proof

Step Hyp Ref Expression
1 zsscn 
 |-  ZZ C_ CC
2 zmulcl 
 |-  ( ( x e. ZZ /\ y e. ZZ ) -> ( x x. y ) e. ZZ )
3 1z 
 |-  1 e. ZZ
4 1 2 3 expcllem 
 |-  ( ( A e. ZZ /\ N e. NN0 ) -> ( A ^ N ) e. ZZ )