Metamath Proof Explorer


Theorem ltexp2rd

Description: The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpexpcld.1
|- ( ph -> A e. RR+ )
rpexpcld.2
|- ( ph -> N e. ZZ )
ltexp2rd.3
|- ( ph -> M e. ZZ )
ltexp2rd.4
|- ( ph -> A < 1 )
Assertion ltexp2rd
|- ( ph -> ( M < N <-> ( A ^ N ) < ( A ^ M ) ) )

Proof

Step Hyp Ref Expression
1 rpexpcld.1
 |-  ( ph -> A e. RR+ )
2 rpexpcld.2
 |-  ( ph -> N e. ZZ )
3 ltexp2rd.3
 |-  ( ph -> M e. ZZ )
4 ltexp2rd.4
 |-  ( ph -> A < 1 )
5 ltexp2r
 |-  ( ( ( A e. RR+ /\ M e. ZZ /\ N e. ZZ ) /\ A < 1 ) -> ( M < N <-> ( A ^ N ) < ( A ^ M ) ) )
6 1 3 2 4 5 syl31anc
 |-  ( ph -> ( M < N <-> ( A ^ N ) < ( A ^ M ) ) )