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Theorem xmullem 11333
Description: Lemma for rexmul 11340. (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 490 . . . 4  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  ( -.  A  =  0  /\  -.  B  =  0 ) )
21anbi2i 694 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( -.  A  =  0  /\  -.  B  =  0 ) ) )
3 ioran 490 . . . . 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 490 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  <->  ( -.  ( 0  <  B  /\  A  = +oo )  /\  -.  ( B  <  0  /\  A  = -oo ) ) )
5 ioran 490 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  <->  ( -.  ( 0  <  A  /\  B  = +oo )  /\  -.  ( A  <  0  /\  B  = -oo ) ) )
64, 5anbi12i 697 . . . . 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 249 . . . 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 490 . . . . 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 490 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  <->  ( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) )
10 ioran 490 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  <->  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) )
119, 10anbi12i 697 . . . . 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 249 . . . 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 697 . . 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 757 . . . . 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 11202 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1614, 15sylib 196 . . . 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 simpllr 758 . . . . . . 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* )
19 0xr 9536 . . . . . . 7  |-  0  e.  RR*
20 xrltso 11224 . . . . . . . 8  |-  <  Or  RR*
21 solin 4767 . . . . . . . 8  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  0  e.  RR* ) )  -> 
( B  <  0  \/  B  =  0  \/  0  <  B ) )
2220, 21mpan 670 . . . . . . 7  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  \/  B  =  0  \/  0  <  B ) )
2318, 19, 22sylancl 662 . . . . . 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 ) )
24 simprlr 762 . . . . . . . . . 10  |-  ( ( ( ( -.  (
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 ) )
2524adantl 466 . . . . . . . . 9  |-  ( ( ( ( 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 ) )
2625pm2.21d 106 . . . . . . . 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 )  ->  A  e.  RR ) )
2726expdimp 437 . . . . . . 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 ) )
28 simplrr 760 . . . . . . . . 9  |-  ( ( ( ( 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 )
2928pm2.21d 106 . . . . . . . 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  ->  A  e.  RR ) ) )
3029imp 429 . . . . . . 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 ) )
31 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( -.  (
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 ) )
3231adantl 466 . . . . . . . . 9  |-  ( ( ( ( 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 )
)
3332pm2.21d 106 . . . . . . . 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 )  ->  A  e.  RR ) )
3433expdimp 437 . . . . . . 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 ) )
3527, 30, 343jaodan 1285 . . . . . 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 ) )  ->  ( A  = +oo  ->  A  e.  RR ) )
3623, 35mpdan 668 . . . . 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 ) )
37 simpllr 758 . . . . . . . . . 10  |-  ( ( ( ( -.  (
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 ) )
3837adantl 466 . . . . . . . . 9  |-  ( ( ( ( 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 ) )
3938pm2.21d 106 . . . . . . . 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 )  ->  A  e.  RR ) )
4039expdimp 437 . . . . . . 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 ) )
4128pm2.21d 106 . . . . . . . 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  ->  A  e.  RR ) ) )
4241imp 429 . . . . . . 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 ) )
43 simprll 761 . . . . . . . . . 10  |-  ( ( ( ( -.  (
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 ) )
4443adantl 466 . . . . . . . . 9  |-  ( ( ( ( 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 )
)
4544pm2.21d 106 . . . . . . . 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 )  ->  A  e.  RR ) )
4645expdimp 437 . . . . . . 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 ) )
4740, 42, 463jaodan 1285 . . . . . 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 ) )  ->  ( A  = -oo  ->  A  e.  RR ) )
4823, 47mpdan 668 . . . . 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 ) )
4917, 36, 483jaod 1283 . . . 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 ) )
5016, 49mpd 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 )
512, 13, 50syl2anb 479 . 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 )
5251anassrs 648 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 368    /\ wa 369    \/ w3o 964    = wceq 1370    e. wcel 1758   class class class wbr 4395    Or wor 4743   RRcr 9387   0cc0 9388   +oocpnf 9521   -oocmnf 9522   RR*cxr 9523    < clt 9524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634  ax-un 6477  ax-cnex 9444  ax-resscn 9445  ax-1cn 9446  ax-icn 9447  ax-addcl 9448  ax-addrcl 9449  ax-mulcl 9450  ax-mulrcl 9451  ax-i2m1 9456  ax-1ne0 9457  ax-rnegex 9459  ax-rrecex 9460  ax-cnre 9461  ax-pre-lttri 9462  ax-pre-lttrn 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-nel 2648  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-csb 3391  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4739  df-po 4744  df-so 4745  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-rn 4954  df-res 4955  df-ima 4956  df-iota 5484  df-fun 5523  df-fn 5524  df-f 5525  df-f1 5526  df-fo 5527  df-f1o 5528  df-fv 5529  df-ov 6198  df-er 7206  df-en 7416  df-dom 7417  df-sdom 7418  df-pnf 9526  df-mnf 9527  df-xr 9528  df-ltxr 9529
This theorem is referenced by:  xmulcom  11335  xmulneg1  11338  xmulf  11341
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