Metamath Proof Explorer

Theorem expcl

Description: Closure law for nonnegative integer exponentiation. For integer exponents, see expclz . (Contributed by NM, 26-May-2005)

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

Proof

Step Hyp Ref Expression
1 ssid 
 |-  CC C_ CC
2 mulcl 
 |-  ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
3 ax-1cn 
 |-  1 e. CC
4 1 2 3 expcllem 
 |-  ( ( A e. CC /\ N e. NN0 ) -> ( A ^ N ) e. CC )