Theorem ltexp1d

Description: Elevating to a positive power does not affect inequalities. Similar to ltmul1d for exponentiation of positive reals. (Contributed by Steven Nguyen, 22-Aug-2023)

	Ref	Expression
Hypotheses	ltexp1d.1	|- ( ph -> A e. RR+ )
	ltexp1d.2	|- ( ph -> B e. RR+ )
	ltexp1d.3	|- ( ph -> N e. NN )
Assertion	ltexp1d	|- ( ph -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltexp1d.1	|- ( ph -> A e. RR+ )
2		ltexp1d.2	|- ( ph -> B e. RR+ )
3		ltexp1d.3	|- ( ph -> N e. NN )
4		rpexpmord	|- ( ( N e. NN /\ A e. RR+ /\ B e. RR+ ) -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )
5	3 1 2 4	syl3anc	|- ( ph -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )