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Theorem xmullem 10799
Description: Lemma for rexmul 10806. (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 477 . . . 4  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  ( -.  A  =  0  /\  -.  B  =  0 ) )
21anbi2i 676 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( -.  A  =  0  /\  -.  B  =  0 ) ) )
3 ioran 477 . . . . 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 477 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  =  +oo )  \/  ( B  <  0  /\  A  =  -oo ) )  <->  ( -.  ( 0  <  B  /\  A  =  +oo )  /\  -.  ( B  <  0  /\  A  =  -oo ) ) )
5 ioran 477 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  =  +oo )  \/  ( A  <  0  /\  B  =  -oo ) )  <->  ( -.  ( 0  <  A  /\  B  =  +oo )  /\  -.  ( A  <  0  /\  B  =  -oo ) ) )
64, 5anbi12i 679 . . . . 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 241 . . . 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 477 . . . . 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 477 . . . . . 6  |-  ( -.  ( ( 0  < 
B  /\  A  =  -oo )  \/  ( B  <  0  /\  A  =  +oo ) )  <->  ( -.  ( 0  <  B  /\  A  =  -oo )  /\  -.  ( B  <  0  /\  A  =  +oo ) ) )
10 ioran 477 . . . . . 6  |-  ( -.  ( ( 0  < 
A  /\  B  =  -oo )  \/  ( A  <  0  /\  B  =  +oo ) )  <->  ( -.  ( 0  <  A  /\  B  =  -oo )  /\  -.  ( A  <  0  /\  B  =  +oo ) ) )
119, 10anbi12i 679 . . . . 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 241 . . . 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 679 . . 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 735 . . . . 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 10672 . . . . 5  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
1614, 15sylib 189 . . . 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 22 . . . . 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 736 . . . . . . 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 9087 . . . . . . 7  |-  0  e.  RR*
20 xrltso 10690 . . . . . . . 8  |-  <  Or  RR*
21 solin 4486 . . . . . . . 8  |-  ( (  <  Or  RR*  /\  ( B  e.  RR*  /\  0  e.  RR* ) )  -> 
( B  <  0  \/  B  =  0  \/  0  <  B ) )
2220, 21mpan 652 . . . . . . 7  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  \/  B  =  0  \/  0  <  B ) )
2318, 19, 22sylancl 644 . . . . . 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 740 . . . . . . . . . 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 453 . . . . . . . . 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 100 . . . . . . . 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 427 . . . . . . 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 738 . . . . . . . . 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 100 . . . . . . . 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 419 . . . . . . 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 735 . . . . . . . . . 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 453 . . . . . . . . 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 100 . . . . . . . 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 427 . . . . . . 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 1250 . . . . . 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 650 . . . . 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 736 . . . . . . . . . 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 453 . . . . . . . . 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 100 . . . . . . . 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 427 . . . . . . 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 100 . . . . . . . 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 419 . . . . . . 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 739 . . . . . . . . . 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 453 . . . . . . . . 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 100 . . . . . . . 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 427 . . . . . . 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 1250 . . . . . 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 650 . . . . 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 1248 . . . 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 466 . 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 630 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 set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1721   class class class wbr 4172    Or wor 4462   RRcr 8945   0cc0 8946    +oocpnf 9073    -oocmnf 9074   RR*cxr 9075    < clt 9076
This theorem is referenced by:  xmulcom  10801  xmulneg1  10804  xmulf  10807
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-i2m1 9014  ax-1ne0 9015  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081
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