Theorem ltexp1dd

Description: Raising both sides of 'less than' to the same positive integer preserves ordering. (Contributed by Steven Nguyen, 24-Aug-2023)

	Ref	Expression
Hypotheses	ltexp1d.1	|- ( ph -> A e. RR+ )
	ltexp1d.2	|- ( ph -> B e. RR+ )
	ltexp1d.3	|- ( ph -> N e. NN )
	ltexp1dd.4	|- ( ph -> A < B )
Assertion	ltexp1dd	|- ( ph -> ( 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		ltexp1dd.4	|- ( ph -> A < B )
5	1 2 3	ltexp1d	|- ( ph -> ( A < B <-> ( A ^ N ) < ( B ^ N ) ) )
6	4 5	mpbid	|- ( ph -> ( A ^ N ) < ( B ^ N ) )