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Theorem xmullem 12094
Description: Lemma for rexmul 12101. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmullem |- ( ( ( ( ( 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 )
Proof of Theorem xmullem
StepHypRef Expression
1 ioran 511 . . . 4 |- ( -. ( A = 0 \/ B = 0 ) <-> ( -. A = 0 /\ -. B = 0 ) )
21anbi2i 730 . . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ -. ( A = 0 \/ B = 0 ) ) <-> ( ( A e. RR* /\ B e. RR* ) /\ ( -. A = 0 /\ -. B = 0 ) ) )
3 ioran 511 . . . . 5 |- ( -. ( ( ( 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 511 . . . . . 6 |- ( -. ( ( 0 < B /\ A = +oo ) \/ ( B < 0 /\ A = -oo ) ) <-> ( -. ( 0 < B /\ A = +oo ) /\ -. ( B < 0 /\ A = -oo ) ) )
5 ioran 511 . . . . . 6 |- ( -. ( ( 0 < A /\ B = +oo ) \/ ( A < 0 /\ B = -oo ) ) <-> ( -. ( 0 < A /\ B = +oo ) /\ -. ( A < 0 /\ B = -oo ) ) )
64, 5anbi12i 733 . . . . 5 |- ( ( -. ( ( 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 ) ) ) )
73, 6bitri 264 . . . 4 |- ( -. ( ( ( 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 511 . . . . 5 |- ( -. ( ( ( 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 511 . . . . . 6 |- ( -. ( ( 0 < B /\ A = -oo ) \/ ( B < 0 /\ A = +oo ) ) <-> ( -. ( 0 < B /\ A = -oo ) /\ -. ( B < 0 /\ A = +oo ) ) )
10 ioran 511 . . . . . 6 |- ( -. ( ( 0 < A /\ B = -oo ) \/ ( A < 0 /\ B = +oo ) ) <-> ( -. ( 0 < A /\ B = -oo ) /\ -. ( A < 0 /\ B = +oo ) ) )
119, 10anbi12i 733 . . . . 5 |- ( ( -. ( ( 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 ) ) ) )
128, 11bitri 264 . . . 4 |- ( -. ( ( ( 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 ) ) ) )
137, 12anbi12i 733 . . 3 |- ( ( -. ( ( ( 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 798 . . . . 5 |- ( ( ( ( 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 11950 . . . . 5 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
1614, 15sylib 208 . . . 4 |- ( ( ( ( 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 24 . . . . 5 |- ( ( ( ( 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 803 . . . . . . . . 9 |- ( ( ( ( -. ( 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 ) )
1918adantl 482 . . . . . . . 8 |- ( ( ( ( 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 ) )
2019pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) )
2120expdimp 453 . . . . . 6 |- ( ( ( ( ( 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 801 . . . . . . . 8 |- ( ( ( ( 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 )
2322pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) ) )
2423imp 445 . . . . . 6 |- ( ( ( ( ( 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 798 . . . . . . . . 9 |- ( ( ( ( -. ( 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 ) )
2625adantl 482 . . . . . . . 8 |- ( ( ( ( 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 ) )
2726pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) )
2827expdimp 453 . . . . . 6 |- ( ( ( ( ( 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 799 . . . . . . 7 |- ( ( ( ( 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 10086 . . . . . . 7 |- 0 e. RR*
31 xrltso 11974 . . . . . . . 8 |- < Or RR*
32 solin 5058 . . . . . . . 8 |- ( ( < Or RR* /\ ( B e. RR* /\ 0 e. RR* ) ) -> ( B < 0 \/ B = 0 \/ 0 < B ) )
3331, 32mpan 706 . . . . . . 7 |- ( ( B e. RR* /\ 0 e. RR* ) -> ( B < 0 \/ B = 0 \/ 0 < B ) )
3429, 30, 33sylancl 694 . . . . . 6 |- ( ( ( ( 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 ) )
3521, 24, 28, 34mpjao3dan 1395 . . . . 5 |- ( ( ( ( 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 799 . . . . . . . . 9 |- ( ( ( ( -. ( 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 ) )
3736adantl 482 . . . . . . . 8 |- ( ( ( ( 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 ) )
3837pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) )
3938expdimp 453 . . . . . 6 |- ( ( ( ( ( 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 ) )
4022pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) ) )
4140imp 445 . . . . . 6 |- ( ( ( ( ( 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 802 . . . . . . . . 9 |- ( ( ( ( -. ( 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 ) )
4342adantl 482 . . . . . . . 8 |- ( ( ( ( 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 ) )
4443pm2.21d 118 . . . . . . 7 |- ( ( ( ( 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 ) )
4544expdimp 453 . . . . . 6 |- ( ( ( ( ( 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 ) )
4639, 41, 45, 34mpjao3dan 1395 . . . . 5 |- ( ( ( ( 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 ) )
4717, 35, 463jaod 1392 . . . 4 |- ( ( ( ( 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 ) )
4816, 47mpd 15 . . 3 |- ( ( ( ( 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 )
492, 13, 48syl2anb 496 . 2 |- ( ( ( ( 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 )
5049anassrs 680 1 |- ( ( ( ( ( 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 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   \/ w3o 1036   = wceq 1483   e. wcel 1990   class class class wbr 4653   Or wor 5034   RRcr 9935   0cc0 9936   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-i2m1 10004  ax-1ne0 10005  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079
This theorem is referenced by:  xmulcom  12096  xmulneg1  12099  xmulf  12102
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