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
|
3adant3 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( A / B ) e. RR ) |
8 |
|
simp3 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> C e. RR ) |
9 |
|
simp2 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( B e. RR /\ 0 < B ) ) |
10 |
|
ltmul1 |
|- ( ( ( A / B ) e. RR /\ C e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) ) |
11 |
7 8 9 10
|
syl3anc |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ C e. RR ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) ) |
12 |
11
|
3adant3r |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( ( A / B ) x. B ) < ( C x. B ) ) ) |
13 |
|
recn |
|- ( A e. RR -> A e. CC ) |
14 |
13
|
adantr |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> A e. CC ) |
15 |
|
recn |
|- ( B e. RR -> B e. CC ) |
16 |
15
|
ad2antrl |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B e. CC ) |
17 |
2
|
adantl |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> B =/= 0 ) |
18 |
14 16 17
|
divcan1d |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) ) -> ( ( A / B ) x. B ) = A ) |
19 |
18
|
3adant3 |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) x. B ) = A ) |
20 |
19
|
breq1d |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( ( A / B ) x. B ) < ( C x. B ) <-> A < ( C x. B ) ) ) |
21 |
|
remulcl |
|- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR ) |
22 |
21
|
ancoms |
|- ( ( B e. RR /\ C e. RR ) -> ( C x. B ) e. RR ) |
23 |
22
|
adantrr |
|- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR ) |
24 |
23
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( C x. B ) e. RR ) |
25 |
|
ltdiv1 |
|- ( ( A e. RR /\ ( C x. B ) e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) ) |
26 |
24 25
|
syld3an2 |
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < ( ( C x. B ) / C ) ) ) |
27 |
|
recn |
|- ( C e. RR -> C e. CC ) |
28 |
27
|
adantr |
|- ( ( C e. RR /\ 0 < C ) -> C e. CC ) |
29 |
|
gt0ne0 |
|- ( ( C e. RR /\ 0 < C ) -> C =/= 0 ) |
30 |
28 29
|
jca |
|- ( ( C e. RR /\ 0 < C ) -> ( C e. CC /\ C =/= 0 ) ) |
31 |
|
divcan3 |
|- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B ) |
32 |
31
|
3expb |
|- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. B ) / C ) = B ) |
33 |
15 30 32
|
syl2an |
|- ( ( B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B ) |
34 |
33
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( C x. B ) / C ) = B ) |
35 |
34
|
breq2d |
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / C ) < ( ( C x. B ) / C ) <-> ( A / C ) < B ) ) |
36 |
26 35
|
bitrd |
|- ( ( A e. RR /\ B e. RR /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) ) |
37 |
36
|
3adant2r |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( A < ( C x. B ) <-> ( A / C ) < B ) ) |
38 |
12 20 37
|
3bitrd |
|- ( ( A e. RR /\ ( B e. RR /\ 0 < B ) /\ ( C e. RR /\ 0 < C ) ) -> ( ( A / B ) < C <-> ( A / C ) < B ) ) |