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 |
3
|
3ad2ant2 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / B ) e. RR ) |
5 |
|
gt0ne0 |
|- ( ( A e. RR /\ 0 < A ) -> A =/= 0 ) |
6 |
|
rereccl |
|- ( ( A e. RR /\ A =/= 0 ) -> ( 1 / A ) e. RR ) |
7 |
5 6
|
syldan |
|- ( ( A e. RR /\ 0 < A ) -> ( 1 / A ) e. RR ) |
8 |
7
|
3ad2ant1 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( 1 / A ) e. RR ) |
9 |
|
simp3l |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> C e. RR ) |
10 |
|
simp3r |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> 0 < C ) |
11 |
|
lemul2 |
|- ( ( ( 1 / B ) e. RR /\ ( 1 / A ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) <_ ( 1 / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) ) |
12 |
4 8 9 10 11
|
syl112anc |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( 1 / B ) <_ ( 1 / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) ) |
13 |
|
lerec |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) |
14 |
13
|
3adant3 |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( 1 / B ) <_ ( 1 / A ) ) ) |
15 |
|
recn |
|- ( C e. RR -> C e. CC ) |
16 |
|
recn |
|- ( B e. RR -> B e. CC ) |
17 |
16
|
adantr |
|- ( ( B e. RR /\ 0 < B ) -> B e. CC ) |
18 |
17 1
|
jca |
|- ( ( B e. RR /\ 0 < B ) -> ( B e. CC /\ B =/= 0 ) ) |
19 |
|
divrec |
|- ( ( C e. CC /\ B e. CC /\ B =/= 0 ) -> ( C / B ) = ( C x. ( 1 / B ) ) ) |
20 |
19
|
3expb |
|- ( ( C e. CC /\ ( B e. CC /\ B =/= 0 ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) ) |
21 |
15 18 20
|
syl2an |
|- ( ( C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) ) |
22 |
21
|
3adant2 |
|- ( ( C e. RR /\ ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( C / B ) = ( C x. ( 1 / B ) ) ) |
23 |
|
recn |
|- ( A e. RR -> A e. CC ) |
24 |
23
|
adantr |
|- ( ( A e. RR /\ 0 < A ) -> A e. CC ) |
25 |
24 5
|
jca |
|- ( ( A e. RR /\ 0 < A ) -> ( A e. CC /\ A =/= 0 ) ) |
26 |
|
divrec |
|- ( ( C e. CC /\ A e. CC /\ A =/= 0 ) -> ( C / A ) = ( C x. ( 1 / A ) ) ) |
27 |
26
|
3expb |
|- ( ( C e. CC /\ ( A e. CC /\ A =/= 0 ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) ) |
28 |
15 25 27
|
syl2an |
|- ( ( C e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) ) |
29 |
28
|
3adant3 |
|- ( ( C e. RR /\ ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( C / A ) = ( C x. ( 1 / A ) ) ) |
30 |
22 29
|
breq12d |
|- ( ( C e. RR /\ ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) ) -> ( ( C / B ) <_ ( C / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) ) |
31 |
30
|
3coml |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( ( C / B ) <_ ( C / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) ) |
32 |
31
|
3adant3r |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( C / B ) <_ ( C / A ) <-> ( C x. ( 1 / B ) ) <_ ( C x. ( 1 / A ) ) ) ) |
33 |
12 14 32
|
3bitr4d |
|- ( ( ( A e. RR /\ 0 < A ) /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A <_ B <-> ( C / B ) <_ ( C / A ) ) ) |