Step |
Hyp |
Ref |
Expression |
1 |
|
rpcn |
|- ( A e. RR+ -> A e. CC ) |
2 |
1
|
mulid2d |
|- ( A e. RR+ -> ( 1 x. A ) = A ) |
3 |
2
|
eqcomd |
|- ( A e. RR+ -> A = ( 1 x. A ) ) |
4 |
3
|
adantr |
|- ( ( A e. RR+ /\ B e. RR ) -> A = ( 1 x. A ) ) |
5 |
4
|
breq1d |
|- ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> ( 1 x. A ) <_ B ) ) |
6 |
|
1red |
|- ( ( A e. RR+ /\ B e. RR ) -> 1 e. RR ) |
7 |
|
simpr |
|- ( ( A e. RR+ /\ B e. RR ) -> B e. RR ) |
8 |
|
rpregt0 |
|- ( A e. RR+ -> ( A e. RR /\ 0 < A ) ) |
9 |
8
|
adantr |
|- ( ( A e. RR+ /\ B e. RR ) -> ( A e. RR /\ 0 < A ) ) |
10 |
|
lemuldiv |
|- ( ( 1 e. RR /\ B e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( ( 1 x. A ) <_ B <-> 1 <_ ( B / A ) ) ) |
11 |
6 7 9 10
|
syl3anc |
|- ( ( A e. RR+ /\ B e. RR ) -> ( ( 1 x. A ) <_ B <-> 1 <_ ( B / A ) ) ) |
12 |
5 11
|
bitrd |
|- ( ( A e. RR+ /\ B e. RR ) -> ( A <_ B <-> 1 <_ ( B / A ) ) ) |