Step |
Hyp |
Ref |
Expression |
1 |
|
mnfnepnf |
|- -oo =/= +oo |
2 |
|
eqeq1 |
|- ( A = -oo -> ( A = +oo <-> -oo = +oo ) ) |
3 |
2
|
necon3bbid |
|- ( A = -oo -> ( -. A = +oo <-> -oo =/= +oo ) ) |
4 |
1 3
|
mpbiri |
|- ( A = -oo -> -. A = +oo ) |
5 |
4
|
con2i |
|- ( A = +oo -> -. A = -oo ) |
6 |
5
|
adantl |
|- ( ( 0 < B /\ A = +oo ) -> -. A = -oo ) |
7 |
|
0xr |
|- 0 e. RR* |
8 |
|
nltmnf |
|- ( 0 e. RR* -> -. 0 < -oo ) |
9 |
7 8
|
ax-mp |
|- -. 0 < -oo |
10 |
|
breq2 |
|- ( A = -oo -> ( 0 < A <-> 0 < -oo ) ) |
11 |
9 10
|
mtbiri |
|- ( A = -oo -> -. 0 < A ) |
12 |
11
|
con2i |
|- ( 0 < A -> -. A = -oo ) |
13 |
12
|
adantr |
|- ( ( 0 < A /\ B = +oo ) -> -. A = -oo ) |
14 |
6 13
|
jaoi |
|- ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) -> -. A = -oo ) |
15 |
14
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) -> -. A = -oo ) ) |
16 |
|
simpr |
|- ( ( A e. RR* /\ B e. RR* ) -> B e. RR* ) |
17 |
|
xrltnsym |
|- ( ( B e. RR* /\ 0 e. RR* ) -> ( B < 0 -> -. 0 < B ) ) |
18 |
16 7 17
|
sylancl |
|- ( ( A e. RR* /\ B e. RR* ) -> ( B < 0 -> -. 0 < B ) ) |
19 |
18
|
adantrd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( B < 0 /\ A = -oo ) -> -. 0 < B ) ) |
20 |
|
breq2 |
|- ( B = -oo -> ( 0 < B <-> 0 < -oo ) ) |
21 |
9 20
|
mtbiri |
|- ( B = -oo -> -. 0 < B ) |
22 |
21
|
adantl |
|- ( ( A < 0 /\ B = -oo ) -> -. 0 < B ) |
23 |
22
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A < 0 /\ B = -oo ) -> -. 0 < B ) ) |
24 |
19 23
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) -> -. 0 < B ) ) |
25 |
15 24
|
orim12d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. A = -oo \/ -. 0 < B ) ) ) |
26 |
|
ianor |
|- ( -. ( 0 < B /\ A = -oo ) <-> ( -. 0 < B \/ -. A = -oo ) ) |
27 |
|
orcom |
|- ( ( -. 0 < B \/ -. A = -oo ) <-> ( -. A = -oo \/ -. 0 < B ) ) |
28 |
26 27
|
bitri |
|- ( -. ( 0 < B /\ A = -oo ) <-> ( -. A = -oo \/ -. 0 < B ) ) |
29 |
25 28
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( 0 < B /\ A = -oo ) ) ) |
30 |
18
|
con2d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 < B -> -. B < 0 ) ) |
31 |
30
|
adantrd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < B /\ A = +oo ) -> -. B < 0 ) ) |
32 |
|
pnfnlt |
|- ( 0 e. RR* -> -. +oo < 0 ) |
33 |
7 32
|
ax-mp |
|- -. +oo < 0 |
34 |
|
simpr |
|- ( ( 0 < A /\ B = +oo ) -> B = +oo ) |
35 |
34
|
breq1d |
|- ( ( 0 < A /\ B = +oo ) -> ( B < 0 <-> +oo < 0 ) ) |
36 |
33 35
|
mtbiri |
|- ( ( 0 < A /\ B = +oo ) -> -. B < 0 ) |
37 |
36
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < A /\ B = +oo ) -> -. B < 0 ) ) |
38 |
31 37
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) -> -. B < 0 ) ) |
39 |
4
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A = -oo -> -. A = +oo ) ) |
40 |
39
|
adantld |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( B < 0 /\ A = -oo ) -> -. A = +oo ) ) |
41 |
|
breq1 |
|- ( A = +oo -> ( A < 0 <-> +oo < 0 ) ) |
42 |
33 41
|
mtbiri |
|- ( A = +oo -> -. A < 0 ) |
43 |
42
|
con2i |
|- ( A < 0 -> -. A = +oo ) |
44 |
43
|
adantr |
|- ( ( A < 0 /\ B = -oo ) -> -. A = +oo ) |
45 |
44
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A < 0 /\ B = -oo ) -> -. A = +oo ) ) |
46 |
40 45
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) -> -. A = +oo ) ) |
47 |
38 46
|
orim12d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. B < 0 \/ -. A = +oo ) ) ) |
48 |
|
ianor |
|- ( -. ( B < 0 /\ A = +oo ) <-> ( -. B < 0 \/ -. A = +oo ) ) |
49 |
47 48
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( B < 0 /\ A = +oo ) ) ) |
50 |
29 49
|
jcad |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) ) ) |
51 |
|
ioran |
|- ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) <-> ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) ) |
52 |
50 51
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) ) ) |
53 |
21
|
con2i |
|- ( 0 < B -> -. B = -oo ) |
54 |
53
|
adantr |
|- ( ( 0 < B /\ A = +oo ) -> -. B = -oo ) |
55 |
54
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < B /\ A = +oo ) -> -. B = -oo ) ) |
56 |
|
pnfnemnf |
|- +oo =/= -oo |
57 |
|
eqeq1 |
|- ( B = +oo -> ( B = -oo <-> +oo = -oo ) ) |
58 |
57
|
necon3bbid |
|- ( B = +oo -> ( -. B = -oo <-> +oo =/= -oo ) ) |
59 |
56 58
|
mpbiri |
|- ( B = +oo -> -. B = -oo ) |
60 |
59
|
adantl |
|- ( ( 0 < A /\ B = +oo ) -> -. B = -oo ) |
61 |
60
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < A /\ B = +oo ) -> -. B = -oo ) ) |
62 |
55 61
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) -> -. B = -oo ) ) |
63 |
11
|
adantl |
|- ( ( B < 0 /\ A = -oo ) -> -. 0 < A ) |
64 |
63
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( B < 0 /\ A = -oo ) -> -. 0 < A ) ) |
65 |
|
simpl |
|- ( ( A e. RR* /\ B e. RR* ) -> A e. RR* ) |
66 |
|
xrltnsym |
|- ( ( A e. RR* /\ 0 e. RR* ) -> ( A < 0 -> -. 0 < A ) ) |
67 |
65 7 66
|
sylancl |
|- ( ( A e. RR* /\ B e. RR* ) -> ( A < 0 -> -. 0 < A ) ) |
68 |
67
|
adantrd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( A < 0 /\ B = -oo ) -> -. 0 < A ) ) |
69 |
64 68
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) -> -. 0 < A ) ) |
70 |
62 69
|
orim12d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. B = -oo \/ -. 0 < A ) ) ) |
71 |
|
ianor |
|- ( -. ( 0 < A /\ B = -oo ) <-> ( -. 0 < A \/ -. B = -oo ) ) |
72 |
|
orcom |
|- ( ( -. 0 < A \/ -. B = -oo ) <-> ( -. B = -oo \/ -. 0 < A ) ) |
73 |
71 72
|
bitri |
|- ( -. ( 0 < A /\ B = -oo ) <-> ( -. B = -oo \/ -. 0 < A ) ) |
74 |
70 73
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( 0 < A /\ B = -oo ) ) ) |
75 |
42
|
adantl |
|- ( ( 0 < B /\ A = +oo ) -> -. A < 0 ) |
76 |
75
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < B /\ A = +oo ) -> -. A < 0 ) ) |
77 |
67
|
con2d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( 0 < A -> -. A < 0 ) ) |
78 |
77
|
adantrd |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( 0 < A /\ B = +oo ) -> -. A < 0 ) ) |
79 |
76 78
|
jaod |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) -> -. A < 0 ) ) |
80 |
|
breq1 |
|- ( B = +oo -> ( B < 0 <-> +oo < 0 ) ) |
81 |
33 80
|
mtbiri |
|- ( B = +oo -> -. B < 0 ) |
82 |
81
|
con2i |
|- ( B < 0 -> -. B = +oo ) |
83 |
82
|
adantr |
|- ( ( B < 0 /\ A = -oo ) -> -. B = +oo ) |
84 |
59
|
con2i |
|- ( B = -oo -> -. B = +oo ) |
85 |
84
|
adantl |
|- ( ( A < 0 /\ B = -oo ) -> -. B = +oo ) |
86 |
83 85
|
jaoi |
|- ( ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) -> -. B = +oo ) |
87 |
86
|
a1i |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) -> -. B = +oo ) ) |
88 |
79 87
|
orim12d |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. A < 0 \/ -. B = +oo ) ) ) |
89 |
|
ianor |
|- ( -. ( A < 0 /\ B = +oo ) <-> ( -. A < 0 \/ -. B = +oo ) ) |
90 |
88 89
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( A < 0 /\ B = +oo ) ) ) |
91 |
74 90
|
jcad |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) ) |
92 |
|
ioran |
|- ( -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) <-> ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) ) |
93 |
91 92
|
syl6ibr |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) |
94 |
52 93
|
jcad |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) -> ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) /\ -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) ) ) ) |
95 |
|
or4 |
|- ( ( ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) \/ ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) ) <-> ( ( ( 0 < B /\ A = +oo ) \/ ( 0 < A /\ B = +oo ) ) \/ ( ( B < 0 /\ A = -oo ) \/ ( A < 0 /\ B = -oo ) ) ) ) |
96 |
|
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 ) ) ) ) |
97 |
94 95 96
|
3imtr4g |
|- ( ( A e. RR* /\ B e. RR* ) -> ( ( ( ( 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 ) ) ) ) ) |