Step |
Hyp |
Ref |
Expression |
1 |
|
0red |
|- ( ( A e. RR /\ B e. RR ) -> 0 e. RR ) |
2 |
|
simpl |
|- ( ( A e. RR /\ B e. RR ) -> A e. RR ) |
3 |
1 2
|
leloed |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ A <-> ( 0 < A \/ 0 = A ) ) ) |
4 |
|
simpll |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A e. RR ) |
5 |
|
simplr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> B e. RR ) |
6 |
4 5
|
remulcld |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. RR ) |
7 |
|
simprl |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < A ) |
8 |
7
|
gt0ne0d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A =/= 0 ) |
9 |
4 8
|
rereccld |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( 1 / A ) e. RR ) |
10 |
|
simprr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( A x. B ) ) |
11 |
|
recgt0 |
|- ( ( A e. RR /\ 0 < A ) -> 0 < ( 1 / A ) ) |
12 |
11
|
ad2ant2r |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( 1 / A ) ) |
13 |
6 9 10 12
|
mulgt0d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < ( ( A x. B ) x. ( 1 / A ) ) ) |
14 |
6
|
recnd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( A x. B ) e. CC ) |
15 |
4
|
recnd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> A e. CC ) |
16 |
14 15 8
|
divrecd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) / A ) = ( ( A x. B ) x. ( 1 / A ) ) ) |
17 |
|
simpr |
|- ( ( A e. RR /\ B e. RR ) -> B e. RR ) |
18 |
17
|
recnd |
|- ( ( A e. RR /\ B e. RR ) -> B e. CC ) |
19 |
18
|
adantr |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> B e. CC ) |
20 |
19 15 8
|
divcan3d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) / A ) = B ) |
21 |
16 20
|
eqtr3d |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> ( ( A x. B ) x. ( 1 / A ) ) = B ) |
22 |
13 21
|
breqtrd |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 < A /\ 0 < ( A x. B ) ) ) -> 0 < B ) |
23 |
22
|
exp32 |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < A -> ( 0 < ( A x. B ) -> 0 < B ) ) ) |
24 |
|
0re |
|- 0 e. RR |
25 |
24
|
ltnri |
|- -. 0 < 0 |
26 |
18
|
mul02d |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 x. B ) = 0 ) |
27 |
26
|
breq2d |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( 0 x. B ) <-> 0 < 0 ) ) |
28 |
25 27
|
mtbiri |
|- ( ( A e. RR /\ B e. RR ) -> -. 0 < ( 0 x. B ) ) |
29 |
28
|
pm2.21d |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 < ( 0 x. B ) -> 0 < B ) ) |
30 |
|
oveq1 |
|- ( 0 = A -> ( 0 x. B ) = ( A x. B ) ) |
31 |
30
|
breq2d |
|- ( 0 = A -> ( 0 < ( 0 x. B ) <-> 0 < ( A x. B ) ) ) |
32 |
31
|
imbi1d |
|- ( 0 = A -> ( ( 0 < ( 0 x. B ) -> 0 < B ) <-> ( 0 < ( A x. B ) -> 0 < B ) ) ) |
33 |
29 32
|
syl5ibcom |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 = A -> ( 0 < ( A x. B ) -> 0 < B ) ) ) |
34 |
23 33
|
jaod |
|- ( ( A e. RR /\ B e. RR ) -> ( ( 0 < A \/ 0 = A ) -> ( 0 < ( A x. B ) -> 0 < B ) ) ) |
35 |
3 34
|
sylbid |
|- ( ( A e. RR /\ B e. RR ) -> ( 0 <_ A -> ( 0 < ( A x. B ) -> 0 < B ) ) ) |
36 |
35
|
imp32 |
|- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ 0 < ( A x. B ) ) ) -> 0 < B ) |