Metamath Proof Explorer

Theorem qexpcl

Description: Closure of exponentiation of rationals. (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 1z 
 |-  1 e. ZZ
4 zq 
 |-  ( 1 e. ZZ -> 1 e. QQ )
5 3 4 ax-mp 
 |-  1 e. QQ
6 1 2 5 expcllem 
 |-  ( ( A e. QQ /\ N e. NN0 ) -> ( A ^ N ) e. QQ )