Step |
Hyp |
Ref |
Expression |
1 |
|
gt0ne0 |
|- ( ( B e. RR /\ 0 < B ) -> B =/= 0 ) |
2 |
|
rereccl |
|- ( ( B e. RR /\ B =/= 0 ) -> ( 1 / B ) e. RR ) |
3 |
1 2
|
syldan |
|- ( ( B e. RR /\ 0 < B ) -> ( 1 / B ) e. RR ) |
4 |
|
recgt0 |
|- ( ( B e. RR /\ 0 < B ) -> 0 < ( 1 / B ) ) |
5 |
3 4
|
jca |
|- ( ( B e. RR /\ 0 < B ) -> ( ( 1 / B ) e. RR /\ 0 < ( 1 / B ) ) ) |
6 |
|
lerec |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( ( 1 / B ) e. RR /\ 0 < ( 1 / B ) ) ) -> ( A <_ ( 1 / B ) <-> ( 1 / ( 1 / B ) ) <_ ( 1 / A ) ) ) |
7 |
5 6
|
sylan2 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( 1 / B ) <-> ( 1 / ( 1 / B ) ) <_ ( 1 / A ) ) ) |
8 |
|
recn |
|- ( B e. RR -> B e. CC ) |
9 |
|
recrec |
|- ( ( B e. CC /\ B =/= 0 ) -> ( 1 / ( 1 / B ) ) = B ) |
10 |
8 1 9
|
syl2an2r |
|- ( ( B e. RR /\ 0 < B ) -> ( 1 / ( 1 / B ) ) = B ) |
11 |
10
|
adantl |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / ( 1 / B ) ) = B ) |
12 |
11
|
breq1d |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( 1 / ( 1 / B ) ) <_ ( 1 / A ) <-> B <_ ( 1 / A ) ) ) |
13 |
7 12
|
bitrd |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ ( 1 / B ) <-> B <_ ( 1 / A ) ) ) |