Theorem cxpgt0d

Description: A positive real raised to a real power is positive. (Contributed by SN, 6-Apr-2023)

	Ref	Expression
Hypotheses	cxpgt0d.1	|- ( ph -> A e. RR+ )
	cxpgt0d.2	|- ( ph -> N e. RR )
Assertion	cxpgt0d	|- ( ph -> 0 < ( A ^c N ) )

Proof

Step	Hyp	Ref	Expression
1		cxpgt0d.1	|- ( ph -> A e. RR+ )
2		cxpgt0d.2	|- ( ph -> N e. RR )
3	1	relogcld	|- ( ph -> ( log ` A ) e. RR )
4	2 3	remulcld	|- ( ph -> ( N x. ( log ` A ) ) e. RR )
5		efgt0	|- ( ( N x. ( log ` A ) ) e. RR -> 0 < ( exp ` ( N x. ( log ` A ) ) ) )
6	4 5	syl	|- ( ph -> 0 < ( exp ` ( N x. ( log ` A ) ) ) )
7	1	rpcnd	|- ( ph -> A e. CC )
8	1	rpne0d	|- ( ph -> A =/= 0 )
9	2	recnd	|- ( ph -> N e. CC )
10	7 8 9	cxpefd	|- ( ph -> ( A ^c N ) = ( exp ` ( N x. ( log ` A ) ) ) )
11	6 10	breqtrrd	|- ( ph -> 0 < ( A ^c N ) )