Metamath Proof Explorer

Theorem nn0expcl

Description: Closure of exponentiation of nonnegative integers. (Contributed by NM, 14-Dec-2005)

Ref Expression
Assertion nn0expcl 
|- ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )

Proof

Step Hyp Ref Expression
1 nn0sscn 
 |-  NN0 C_ CC
2 nn0mulcl 
 |-  ( ( x e. NN0 /\ y e. NN0 ) -> ( x x. y ) e. NN0 )
3 1nn0 
 |-  1 e. NN0
4 1 2 3 expcllem 
 |-  ( ( A e. NN0 /\ N e. NN0 ) -> ( A ^ N ) e. NN0 )