Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( A e. RR /\ B e. RR+ ) -> A e. RR ) |
2 |
|
rpregt0 |
|- ( B e. RR+ -> ( B e. RR /\ 0 < B ) ) |
3 |
2
|
adantl |
|- ( ( A e. RR /\ B e. RR+ ) -> ( B e. RR /\ 0 < B ) ) |
4 |
|
1re |
|- 1 e. RR |
5 |
|
0lt1 |
|- 0 < 1 |
6 |
4 5
|
pm3.2i |
|- ( 1 e. RR /\ 0 < 1 ) |
7 |
6
|
a1i |
|- ( ( A e. RR /\ B e. RR+ ) -> ( 1 e. RR /\ 0 < 1 ) ) |
8 |
|
ltdiv23 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( 1 e. RR /\ 0 < 1 ) ) -> ( ( A / B ) < 1 <-> ( A / 1 ) < B ) ) |
9 |
1 3 7 8
|
syl3anc |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> ( A / 1 ) < B ) ) |
10 |
|
recn |
|- ( A e. RR -> A e. CC ) |
11 |
10
|
div1d |
|- ( A e. RR -> ( A / 1 ) = A ) |
12 |
11
|
adantr |
|- ( ( A e. RR /\ B e. RR+ ) -> ( A / 1 ) = A ) |
13 |
12
|
breq1d |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / 1 ) < B <-> A < B ) ) |
14 |
9 13
|
bitrd |
|- ( ( A e. RR /\ B e. RR+ ) -> ( ( A / B ) < 1 <-> A < B ) ) |