Step |
Hyp |
Ref |
Expression |
1 |
|
ltdiv23neg.1 |
|- ( ph -> A e. RR ) |
2 |
|
ltdiv23neg.2 |
|- ( ph -> B e. RR ) |
3 |
|
ltdiv23neg.3 |
|- ( ph -> B < 0 ) |
4 |
|
ltdiv23neg.4 |
|- ( ph -> C e. RR ) |
5 |
|
ltdiv23neg.5 |
|- ( ph -> C < 0 ) |
6 |
2 3
|
ltned |
|- ( ph -> B =/= 0 ) |
7 |
1 2 6
|
redivcld |
|- ( ph -> ( A / B ) e. RR ) |
8 |
7 4 2 3
|
ltmulneg |
|- ( ph -> ( ( A / B ) < C <-> ( C x. B ) < ( ( A / B ) x. B ) ) ) |
9 |
|
recn |
|- ( A e. RR -> A e. CC ) |
10 |
1 9
|
syl |
|- ( ph -> A e. CC ) |
11 |
|
recn |
|- ( B e. RR -> B e. CC ) |
12 |
2 11
|
syl |
|- ( ph -> B e. CC ) |
13 |
10 12 6
|
divcan1d |
|- ( ph -> ( ( A / B ) x. B ) = A ) |
14 |
13
|
breq2d |
|- ( ph -> ( ( C x. B ) < ( ( A / B ) x. B ) <-> ( C x. B ) < A ) ) |
15 |
|
remulcl |
|- ( ( C e. RR /\ B e. RR ) -> ( C x. B ) e. RR ) |
16 |
4 2 15
|
syl2anc |
|- ( ph -> ( C x. B ) e. RR ) |
17 |
4 5
|
ltned |
|- ( ph -> C =/= 0 ) |
18 |
4 17
|
rereccld |
|- ( ph -> ( 1 / C ) e. RR ) |
19 |
4 5
|
reclt0d |
|- ( ph -> ( 1 / C ) < 0 ) |
20 |
16 1 18 19
|
ltmulneg |
|- ( ph -> ( ( C x. B ) < A <-> ( A x. ( 1 / C ) ) < ( ( C x. B ) x. ( 1 / C ) ) ) ) |
21 |
|
recn |
|- ( C e. RR -> C e. CC ) |
22 |
4 21
|
syl |
|- ( ph -> C e. CC ) |
23 |
10 22 17
|
divrecd |
|- ( ph -> ( A / C ) = ( A x. ( 1 / C ) ) ) |
24 |
23
|
eqcomd |
|- ( ph -> ( A x. ( 1 / C ) ) = ( A / C ) ) |
25 |
22 12
|
mulcld |
|- ( ph -> ( C x. B ) e. CC ) |
26 |
25 22 17
|
divrecd |
|- ( ph -> ( ( C x. B ) / C ) = ( ( C x. B ) x. ( 1 / C ) ) ) |
27 |
|
divcan3 |
|- ( ( B e. CC /\ C e. CC /\ C =/= 0 ) -> ( ( C x. B ) / C ) = B ) |
28 |
27
|
3expb |
|- ( ( B e. CC /\ ( C e. CC /\ C =/= 0 ) ) -> ( ( C x. B ) / C ) = B ) |
29 |
12 22 17 28
|
syl12anc |
|- ( ph -> ( ( C x. B ) / C ) = B ) |
30 |
26 29
|
eqtr3d |
|- ( ph -> ( ( C x. B ) x. ( 1 / C ) ) = B ) |
31 |
24 30
|
breq12d |
|- ( ph -> ( ( A x. ( 1 / C ) ) < ( ( C x. B ) x. ( 1 / C ) ) <-> ( A / C ) < B ) ) |
32 |
20 31
|
bitrd |
|- ( ph -> ( ( C x. B ) < A <-> ( A / C ) < B ) ) |
33 |
8 14 32
|
3bitrd |
|- ( ph -> ( ( A / B ) < C <-> ( A / C ) < B ) ) |