Step |
Hyp |
Ref |
Expression |
1 |
|
ioran |
|- ( -. ( A = 0 \/ B = 0 ) <-> ( -. A = 0 /\ -. B = 0 ) ) |
2 |
1
|
anbi2i |
|- ( ( ( A e. RR* /\ B e. RR* ) /\ -. ( A = 0 \/ B = 0 ) ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) ) |
3 |
|
ioran |
|- ( -. ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) <-> ( -. ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) /\ -. ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) ) |
4 |
|
ioran |
|- ( -. ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) <-> ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) ) |
5 |
|
ioran |
|- ( -. ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) <-> ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) |
6 |
4 5
|
anbi12i |
|- ( ( -. ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) /\ -. ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) <-> ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) ) |
7 |
3 6
|
bitri |
|- ( -. ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) <-> ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) ) |
8 |
|
ioran |
|- ( -. ( ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) <-> ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) /\ -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) |
9 |
|
ioran |
|- ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) <-> ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) ) |
10 |
|
ioran |
|- ( -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) <-> ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) |
11 |
9 10
|
anbi12i |
|- ( ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) /\ -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) <-> ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) |
12 |
8 11
|
bitri |
|- ( -. ( ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) <-> ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) |
13 |
7 12
|
anbi12i |
|- ( ( -. ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) /\ -. ( ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) <-> ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) |
14 |
|
simplll |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> A e. RR* ) |
15 |
|
elxr |
|- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) ) |
16 |
14 15
|
sylib |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) ) |
17 |
|
idd |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( A e. RR -> A e. RR ) ) |
18 |
|
simprlr |
|- ( ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) -> -. ( B < 0 /\ A = +oo ) ) |
19 |
18
|
adantl |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> -. ( B < 0 /\ A = +oo ) ) |
20 |
19
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( ( B < 0 /\ A = +oo ) -> A e. RR ) ) |
21 |
20
|
expdimp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ B < 0 ) -> ( A = +oo -> A e. RR ) ) |
22 |
|
simplrr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> -. B = 0 ) |
23 |
22
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( B = 0 -> ( A = +oo -> A e. RR ) ) ) |
24 |
23
|
imp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ B = 0 ) -> ( A = +oo -> A e. RR ) ) |
25 |
|
simplll |
|- ( ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) -> -. ( 0 < B /\ A = +oo ) ) |
26 |
25
|
adantl |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> -. ( 0 < B /\ A = +oo ) ) |
27 |
26
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( ( 0 < B /\ A = +oo ) -> A e. RR ) ) |
28 |
27
|
expdimp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ 0 < B ) -> ( A = +oo -> A e. RR ) ) |
29 |
|
simpllr |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> B e. RR* ) |
30 |
|
0xr |
|- 0 e. RR* |
31 |
|
xrltso |
|- < Or RR* |
32 |
|
solin |
|- ( ( < Or RR* /\ ( B e. RR* /\ 0 e. RR* ) ) -> ( B < 0 \/ B = 0 \/ 0 < B ) ) |
33 |
31 32
|
mpan |
|- ( ( B e. RR* /\ 0 e. RR* ) -> ( B < 0 \/ B = 0 \/ 0 < B ) ) |
34 |
29 30 33
|
sylancl |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( B < 0 \/ B = 0 \/ 0 < B ) ) |
35 |
21 24 28 34
|
mpjao3dan |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( A = +oo -> A e. RR ) ) |
36 |
|
simpllr |
|- ( ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) -> -. ( B < 0 /\ A = -oo ) ) |
37 |
36
|
adantl |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> -. ( B < 0 /\ A = -oo ) ) |
38 |
37
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( ( B < 0 /\ A = -oo ) -> A e. RR ) ) |
39 |
38
|
expdimp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ B < 0 ) -> ( A = -oo -> A e. RR ) ) |
40 |
22
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( B = 0 -> ( A = -oo -> A e. RR ) ) ) |
41 |
40
|
imp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ B = 0 ) -> ( A = -oo -> A e. RR ) ) |
42 |
|
simprll |
|- ( ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) -> -. ( 0 < B /\ A = -oo ) ) |
43 |
42
|
adantl |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> -. ( 0 < B /\ A = -oo ) ) |
44 |
43
|
pm2.21d |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( ( 0 < B /\ A = -oo ) -> A e. RR ) ) |
45 |
44
|
expdimp |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) /\ 0 < B ) -> ( A = -oo -> A e. RR ) ) |
46 |
39 41 45 34
|
mpjao3dan |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( A = -oo -> A e. RR ) ) |
47 |
17 35 46
|
3jaod |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> ( ( A e. RR \/ A = +oo \/ A = -oo ) -> A e. RR ) ) |
48 |
16 47
|
mpd |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) /\ ( ( ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) /\ ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) ) /\ ( ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) /\ ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) ) -> A e. RR ) |
49 |
2 13 48
|
syl2anb |
|- ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. ( A = 0 \/ B = 0 ) ) /\ ( -. ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) /\ -. ( ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) ) -> A e. RR ) |
50 |
49
|
anassrs |
|- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ -. ( A = 0 \/ B = 0 ) ) /\ -. ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) ) /\ -. ( ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) \/ ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) -> A e. RR ) |