Metamath Proof Explorer


Theorem ltexp1d

Description: 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 ) ) )