Theorem relogexp

Description: The natural logarithm of positive A raised to an integer power. Property 4 of Cohen p. 301-302, restricted to natural logarithms and integer powers N . (Contributed by Steve Rodriguez, 25-Nov-2007)

	Ref	Expression
Assertion	relogexp	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( log ` ( A ^ N ) ) = ( 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	eqtrd	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( exp ` ( N x. ( log ` A ) ) ) = ( A ^ N ) )
9	8	fveq2d	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( log ` ( exp ` ( N x. ( log ` A ) ) ) ) = ( log ` ( A ^ N ) ) )
10		zre	|- ( N e. ZZ -> N e. RR )
11		remulcl	|- ( ( N e. RR /\ ( log ` A ) e. RR ) -> ( N x. ( log ` A ) ) e. RR )
12	10 1 11	syl2anr	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( N x. ( log ` A ) ) e. RR )
13		relogef	|- ( ( N x. ( log ` A ) ) e. RR -> ( log ` ( exp ` ( N x. ( log ` A ) ) ) ) = ( N x. ( log ` A ) ) )
14	12 13	syl	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( log ` ( exp ` ( N x. ( log ` A ) ) ) ) = ( N x. ( log ` A ) ) )
15	9 14	eqtr3d	|- ( ( A e. RR+ /\ N e. ZZ ) -> ( log ` ( A ^ N ) ) = ( N x. ( log ` A ) ) )