Metamath Proof Explorer

Theorem reexpclz

Description: Closure of exponentiation of reals. (Contributed by Mario Carneiro, 4-Jun-2014) (Revised by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion reexpclz 
|- ( ( A e. RR /\ A =/= 0 /\ N e. ZZ ) -> ( 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 rereccl 
 |-  ( ( x e. RR /\ x =/= 0 ) -> ( 1 / x ) e. RR )
5 1 2 3 4 expcl2lem 
 |-  ( ( A e. RR /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. RR )