Metamath Proof Explorer

Theorem qexpcl

Description: Closure of exponentiation of rationals. For integer exponents, see qexpclz . (Contributed by NM, 16-Dec-2005)

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

Proof

Step Hyp Ref Expression
1 qsscn 
 |-  QQ C_ CC
2 qmulcl 
 |-  ( ( x e. QQ /\ y e. QQ ) -> ( x x. y ) e. QQ )
3 1q 
 |-  1 e. QQ
4 1 2 3 expcllem 
 |-  ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )