Metamath Proof Explorer

Theorem nnexpcl

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

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

Proof

Step Hyp Ref Expression
1 nnsscn 
 |-  NN C_ CC
2 nnmulcl 
 |-  ( ( x e. NN /\ y e. NN ) -> ( x x. y ) e. NN )
3 1nn 
 |-  1 e. NN
4 1 2 3 expcllem 
 |-  ( ( A e. NN /\ N e. NN0 ) -> ( A ^ N ) e. NN )