Description: Reciprocal swap in a 'less than' relation. (Contributed by Mario Carneiro, 28-May-2016)
Ref | Expression | ||
---|---|---|---|
Hypotheses | rpred.1 | |- ( ph -> A e. RR+ ) |
|
rpaddcld.1 | |- ( ph -> B e. RR+ ) |
||
ltrec1d.2 | |- ( ph -> ( 1 / A ) < B ) |
||
Assertion | ltrec1d | |- ( ph -> ( 1 / B ) < A ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpred.1 | |- ( ph -> A e. RR+ ) |
|
2 | rpaddcld.1 | |- ( ph -> B e. RR+ ) |
|
3 | ltrec1d.2 | |- ( ph -> ( 1 / A ) < B ) |
|
4 | 1 | rpregt0d | |- ( ph -> ( A e. RR /\ 0 < A ) ) |
5 | 2 | rpregt0d | |- ( ph -> ( B e. RR /\ 0 < B ) ) |
6 | ltrec1 | |- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) ) |
|
7 | 4 5 6 | syl2anc | |- ( ph -> ( ( 1 / A ) < B <-> ( 1 / B ) < A ) ) |
8 | 3 7 | mpbid | |- ( ph -> ( 1 / B ) < A ) |