Metamath Proof Explorer

Theorem reexpcl

Description: Closure of exponentiation of reals. For integer exponents, see reexpclz . (Contributed by NM, 14-Dec-2005)

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

Proof

Step Hyp Ref Expression
1 ax-resscn 
 |-  RR C_ CC
2 remulcl 
 |-  ( ( x e. RR /\ y e. RR ) -> ( x x. y ) e. RR )
3 1re 
 |-  1 e. RR
4 1 2 3 expcllem 
 |-  ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )