Step |
Hyp |
Ref |
Expression |
1 |
|
simpl |
|- ( ( B e. RR /\ 0 < B ) -> B e. RR ) |
2 |
|
gt0ne0 |
|- ( ( B e. RR /\ 0 < B ) -> B =/= 0 ) |
3 |
1 2
|
jca |
|- ( ( B e. RR /\ 0 < B ) -> ( B e. RR /\ B =/= 0 ) ) |
4 |
|
redivcl |
|- ( ( A e. RR /\ B e. RR /\ B =/= 0 ) -> ( A / B ) e. RR ) |
5 |
4
|
3expb |
|- ( ( A e. RR /\ ( B e. RR /\ B =/= 0 ) ) -> ( A / B ) e. RR ) |
6 |
3 5
|
sylan2 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR ) |
7 |
6
|
adantlr |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A / B ) e. RR ) |
8 |
|
divgt0 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> 0 < ( A / B ) ) |
9 |
7 8
|
jca |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) e. RR /\ 0 < ( A / B ) ) ) |
10 |
|
simpl |
|- ( ( D e. RR /\ 0 < D ) -> D e. RR ) |
11 |
|
gt0ne0 |
|- ( ( D e. RR /\ 0 < D ) -> D =/= 0 ) |
12 |
10 11
|
jca |
|- ( ( D e. RR /\ 0 < D ) -> ( D e. RR /\ D =/= 0 ) ) |
13 |
|
redivcl |
|- ( ( C e. RR /\ D e. RR /\ D =/= 0 ) -> ( C / D ) e. RR ) |
14 |
13
|
3expb |
|- ( ( C e. RR /\ ( D e. RR /\ D =/= 0 ) ) -> ( C / D ) e. RR ) |
15 |
12 14
|
sylan2 |
|- ( ( C e. RR /\ ( D e. RR /\ 0 < D ) ) -> ( C / D ) e. RR ) |
16 |
15
|
adantlr |
|- ( ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) -> ( C / D ) e. RR ) |
17 |
|
divgt0 |
|- ( ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) -> 0 < ( C / D ) ) |
18 |
16 17
|
jca |
|- ( ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) -> ( ( C / D ) e. RR /\ 0 < ( C / D ) ) ) |
19 |
|
lerec |
|- ( ( ( ( A / B ) e. RR /\ 0 < ( A / B ) ) /\ ( ( C / D ) e. RR /\ 0 < ( C / D ) ) ) -> ( ( A / B ) <_ ( C / D ) <-> ( 1 / ( C / D ) ) <_ ( 1 / ( A / B ) ) ) ) |
20 |
9 18 19
|
syl2an |
|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A / B ) <_ ( C / D ) <-> ( 1 / ( C / D ) ) <_ ( 1 / ( A / B ) ) ) ) |
21 |
|
recn |
|- ( C e. RR -> C e. CC ) |
22 |
21
|
adantr |
|- ( ( C e. RR /\ 0 < C ) -> C e. CC ) |
23 |
|
gt0ne0 |
|- ( ( C e. RR /\ 0 < C ) -> C =/= 0 ) |
24 |
22 23
|
jca |
|- ( ( C e. RR /\ 0 < C ) -> ( C e. CC /\ C =/= 0 ) ) |
25 |
|
recn |
|- ( D e. RR -> D e. CC ) |
26 |
25
|
adantr |
|- ( ( D e. RR /\ 0 < D ) -> D e. CC ) |
27 |
26 11
|
jca |
|- ( ( D e. RR /\ 0 < D ) -> ( D e. CC /\ D =/= 0 ) ) |
28 |
|
recdiv |
|- ( ( ( C e. CC /\ C =/= 0 ) /\ ( D e. CC /\ D =/= 0 ) ) -> ( 1 / ( C / D ) ) = ( D / C ) ) |
29 |
24 27 28
|
syl2an |
|- ( ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) -> ( 1 / ( C / D ) ) = ( D / C ) ) |
30 |
|
recn |
|- ( A e. RR -> A e. CC ) |
31 |
30
|
adantr |
|- ( ( A e. RR /\ 0 < A ) -> A e. CC ) |
32 |
|
gt0ne0 |
|- ( ( A e. RR /\ 0 < A ) -> A =/= 0 ) |
33 |
31 32
|
jca |
|- ( ( A e. RR /\ 0 < A ) -> ( A e. CC /\ A =/= 0 ) ) |
34 |
|
recn |
|- ( B e. RR -> B e. CC ) |
35 |
34
|
adantr |
|- ( ( B e. RR /\ 0 < B ) -> B e. CC ) |
36 |
35 2
|
jca |
|- ( ( B e. RR /\ 0 < B ) -> ( B e. CC /\ B =/= 0 ) ) |
37 |
|
recdiv |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ B =/= 0 ) ) -> ( 1 / ( A / B ) ) = ( B / A ) ) |
38 |
33 36 37
|
syl2an |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( 1 / ( A / B ) ) = ( B / A ) ) |
39 |
29 38
|
breqan12rd |
|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( 1 / ( C / D ) ) <_ ( 1 / ( A / B ) ) <-> ( D / C ) <_ ( B / A ) ) ) |
40 |
20 39
|
bitrd |
|- ( ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) /\ ( ( C e. RR /\ 0 < C ) /\ ( D e. RR /\ 0 < D ) ) ) -> ( ( A / B ) <_ ( C / D ) <-> ( D / C ) <_ ( B / A ) ) ) |