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 ) ) ) |
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 ) ) ) |