Description: The reciprocal of both sides of 'less than'. (Contributed by Mario Carneiro, 28-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rpred.1 | |- ( ph -> A e. RR+ ) |
|
rpaddcld.1 | |- ( ph -> B e. RR+ ) |
||
Assertion | ltrecd | |- ( ph -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | |- ( ph -> A e. RR+ ) |
|
2 | rpaddcld.1 | |- ( ph -> B e. RR+ ) |
|
3 | 1 | rpregt0d | |- ( ph -> ( A e. RR /\ 0 < A ) ) |
4 | 2 | rpregt0d | |- ( ph -> ( B e. RR /\ 0 < B ) ) |
5 | ltrec | |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) |
|
6 | 3 4 5 | syl2anc | |- ( ph -> ( A < B <-> ( 1 / B ) < ( 1 / A ) ) ) |