Metamath Proof Explorer

Theorem rpexpcl

Description: Closure law for integer exponentiation of positive reals. (Contributed by NM, 24-Feb-2008) (Revised by Mario Carneiro, 9-Sep-2014)

Ref Expression
Assertion rpexpcl 
|- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )

Proof

Step Hyp Ref Expression
1 simpl 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> A e. RR+ )
2 rpne0 
 |-  ( A e. RR+ -> A =/= 0 )
3 2 adantr 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> A =/= 0 )
4 simpr 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> N e. ZZ )
5 rpssre 
 |-  RR+ C_ RR
6 ax-resscn 
 |-  RR C_ CC
7 5 6 sstri 
 |-  RR+ C_ CC
8 rpmulcl 
 |-  ( ( x e. RR+ /\ y e. RR+ ) -> ( x x. y ) e. RR+ )
9 1rp 
 |-  1 e. RR+
10 rpreccl 
 |-  ( x e. RR+ -> ( 1 / x ) e. RR+ )
11 10 adantr 
 |-  ( ( x e. RR+ /\ x =/= 0 ) -> ( 1 / x ) e. RR+ )
12 7 8 9 11 expcl2lem 
 |-  ( ( A e. RR+ /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )
13 1 3 4 12 syl3anc 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )