Metamath Proof Explorer

Theorem reexplog

Description: Exponentiation of a positive real number to an integer power. (Contributed by Steve Rodriguez, 25-Nov-2007)

Ref Expression
Assertion reexplog 
|- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) )

Proof

Step Hyp Ref Expression
1 relogcl 
 |-  ( A e. RR+ -> ( log ` A ) e. RR )
2 1 recnd 
 |-  ( A e. RR+ -> ( log ` A ) e. CC )
3 efexp 
 |-  ( ( ( log ` A ) e. CC /\ N e. ZZ ) -> ( exp ` ( N x. ( log ` A ) ) ) = ( ( exp ` ( log ` A ) ) ^ N ) )
4 2 3 sylan 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> ( exp ` ( N x. ( log ` A ) ) ) = ( ( exp ` ( log ` A ) ) ^ N ) )
5 reeflog 
 |-  ( A e. RR+ -> ( exp ` ( log ` A ) ) = A )
6 5 oveq1d 
 |-  ( A e. RR+ -> ( ( exp ` ( log ` A ) ) ^ N ) = ( A ^ N ) )
7 6 adantr 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> ( ( exp ` ( log ` A ) ) ^ N ) = ( A ^ N ) )
8 4 7 eqtr2d 
 |-  ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) = ( exp ` ( N x. ( log ` A ) ) ) )