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Theorem xmulneg1 11473
Description: Extended real version of mulneg1 10005. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmulneg1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )

Proof of Theorem xmulneg1
StepHypRef Expression
1 xneg0 11423 . . . . . . . . 9  |-  -e 0  =  0
21eqeq2i 2485 . . . . . . . 8  |-  (  -e A  =  -e 0  <->  -e A  =  0 )
3 0xr 9652 . . . . . . . . 9  |-  0  e.  RR*
4 xneg11 11426 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  (  -e A  =  -e 0  <->  A  = 
0 ) )
53, 4mpan2 671 . . . . . . . 8  |-  ( A  e.  RR*  ->  (  -e A  =  -e 0  <->  A  =  0
) )
62, 5syl5bbr 259 . . . . . . 7  |-  ( A  e.  RR*  ->  (  -e A  =  0  <->  A  =  0 ) )
76adantr 465 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  0  <-> 
A  =  0 ) )
87orbi1d 702 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
(  -e A  =  0  \/  B  =  0 )  <->  ( A  =  0  \/  B  =  0 ) ) )
98ifbid 3967 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( (  -e
A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) ) ) )
10 xnegpnf 11420 . . . . . . . . . . . . . 14  |-  -e +oo  = -oo
1110eqeq2i 2485 . . . . . . . . . . . . 13  |-  (  -e A  =  -e +oo 
<-> 
-e A  = -oo )
12 simpll 753 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  A  e.  RR* )
13 pnfxr 11333 . . . . . . . . . . . . . 14  |- +oo  e.  RR*
14 xneg11 11426 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A  =  -e +oo  <->  A  = +oo ) )
1512, 13, 14sylancl 662 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = 
-e +oo  <->  A  = +oo ) )
1611, 15syl5bbr 259 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = -oo  <->  A  = +oo ) )
1716anbi2d 703 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  < 
B  /\  -e A  = -oo )  <->  ( 0  <  B  /\  A  = +oo ) ) )
18 xnegmnf 11421 . . . . . . . . . . . . . 14  |-  -e -oo  = +oo
1918eqeq2i 2485 . . . . . . . . . . . . 13  |-  (  -e A  =  -e -oo 
<-> 
-e A  = +oo )
20 mnfxr 11335 . . . . . . . . . . . . . 14  |- -oo  e.  RR*
21 xneg11 11426 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\ -oo  e.  RR* )  ->  (  -e A  =  -e -oo  <->  A  = -oo ) )
2212, 20, 21sylancl 662 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = 
-e -oo  <->  A  = -oo ) )
2319, 22syl5bbr 259 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = +oo  <->  A  = -oo ) )
2423anbi2d 703 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( B  <  0  /\  -e
A  = +oo )  <->  ( B  <  0  /\  A  = -oo )
) )
2517, 24orbi12d 709 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  <->  ( (
0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
26 xlt0neg1 11430 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
2726ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( A  <  0  <->  0  <  -e A ) )
2827bicomd 201 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( 0  <  -e
A  <->  A  <  0
) )
2928anbi1d 704 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  <  -e A  /\  B  = -oo )  <->  ( A  <  0  /\  B  = -oo )
) )
30 xlt0neg2 11431 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
3130ad2antrr 725 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( 0  <  A  <->  -e A  <  0
) )
3231bicomd 201 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  <  0  <->  0  <  A
) )
3332anbi1d 704 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( (  -e
A  <  0  /\  B  = +oo )  <->  ( 0  <  A  /\  B  = +oo )
) )
3429, 33orbi12d 709 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) )  <->  ( ( A  <  0  /\  B  = -oo )  \/  (
0  <  A  /\  B  = +oo )
) ) )
35 orcom 387 . . . . . . . . . . 11  |-  ( ( ( A  <  0  /\  B  = -oo )  \/  ( 0  <  A  /\  B  = +oo ) )  <->  ( (
0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )
3634, 35syl6bb 261 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) )  <->  ( (
0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )
3725, 36orbi12d 709 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) )  <->  ( (
( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) ) )
3837biimpar 485 . . . . . . . 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  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) )
39 iftrue 3951 . . . . . . . 8  |-  ( ( ( ( 0  < 
B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) )  = -oo )
4038, 39syl 16 . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) )  = -oo )
41 xmullem2 11469 . . . . . . . . . . 11  |-  ( ( 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 ) ) ) ) )
4241adantr 465 . . . . . . . . . 10  |-  ( ( ( 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 ) ) ) ) )
4323anbi2d 703 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  < 
B  /\  -e A  = +oo )  <->  ( 0  <  B  /\  A  = -oo ) ) )
4416anbi2d 703 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( B  <  0  /\  -e
A  = -oo )  <->  ( B  <  0  /\  A  = +oo )
) )
4543, 44orbi12d 709 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  <->  ( (
0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
4628anbi1d 704 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  <  -e A  /\  B  = +oo )  <->  ( A  <  0  /\  B  = +oo )
) )
4732anbi1d 704 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( (  -e
A  <  0  /\  B  = -oo )  <->  ( 0  <  A  /\  B  = -oo )
) )
4846, 47orbi12d 709 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) )  <->  ( ( A  <  0  /\  B  = +oo )  \/  (
0  <  A  /\  B  = -oo )
) ) )
49 orcom 387 . . . . . . . . . . . . 13  |-  ( ( ( A  <  0  /\  B  = +oo )  \/  ( 0  <  A  /\  B  = -oo ) )  <->  ( (
0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )
5048, 49syl6bb 261 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) )  <->  ( (
0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )
5145, 50orbi12d 709 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) )  <->  ( (
( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) ) )
5251notbid 294 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( -.  ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) )  <->  -.  (
( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) ) )
5342, 52sylibrd 234 . . . . . . . . 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  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) ) )
5453imp 429 . . . . . . . 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  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) )
55 iffalse 3954 . . . . . . . 8  |-  ( -.  ( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  =  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )
5654, 55syl 16 . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  =  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )
57 iftrue 3951 . . . . . . . . . 10  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = +oo )
5857adantl 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = +oo )
59 xnegeq 11418 . . . . . . . . 9  |-  ( if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = +oo  -> 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e +oo )
6058, 59syl 16 . . . . . . . 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e +oo )
6160, 10syl6eq 2524 . . . . . . 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = -oo )
6240, 56, 613eqtr4d 2518 . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
6351biimpar 485 . . . . . . . . . 10  |-  ( ( ( ( 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  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) )
64 iftrue 3951 . . . . . . . . . 10  |-  ( ( ( ( 0  < 
B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = +oo )
6563, 64syl 16 . . . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = +oo )
6642con2d 115 . . . . . . . . . . . . . 14  |-  ( ( ( 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 ) ) ) ) )
6766imp 429 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) ) ) )
68 iffalse 3954 . . . . . . . . . . . . 13  |-  ( -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )
6967, 68syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )
70 iftrue 3951 . . . . . . . . . . . . 13  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  = -oo )
7170adantl 466 . . . . . . . . . . . 12  |-  ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  = -oo )
7269, 71eqtrd 2508 . . . . . . . . . . 11  |-  ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = -oo )
73 xnegeq 11418 . . . . . . . . . . 11  |-  ( if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = -oo  -> 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e -oo )
7472, 73syl 16 . . . . . . . . . 10  |-  ( ( ( ( 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e -oo )
7574, 18syl6eq 2524 . . . . . . . . 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  = +oo )
7665, 75eqtr4d 2511 . . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
7776adantlr 714 . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
7837notbid 294 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( -.  ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) )  <->  -.  (
( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) ) )
7978biimpar 485 . . . . . . . . . 10  |-  ( ( ( ( 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  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) )
8079adantr 465 . . . . . . . . 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  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) )
81 iffalse 3954 . . . . . . . . 9  |-  ( -.  ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) )  =  (  -e A  x.  B ) )
8280, 81syl 16 . . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) )  =  (  -e A  x.  B ) )
8352biimpar 485 . . . . . . . . . 10  |-  ( ( ( ( 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  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) )
8483adantlr 714 . . . . . . . . 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  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) )
8584, 55syl 16 . . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  =  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )
8668ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )
87 iffalse 3954 . . . . . . . . . . . 12  |-  ( -.  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  ->  if ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  =  ( A  x.  B ) )
8887adantl 466 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) )  =  ( A  x.  B ) )
8986, 88eqtrd 2508 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  ( A  x.  B ) )
90 xnegeq 11418 . . . . . . . . . 10  |-  ( if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  ( A  x.  B )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e ( A  x.  B ) )
9189, 90syl 16 . . . . . . . . 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  -e ( A  x.  B ) )
92 xmullem 11468 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 )
9392recnd 9634 . . . . . . . . . . 11  |-  ( ( ( ( ( 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.  CC )
94 ancom 450 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  <->  ( B  e.  RR*  /\  A  e. 
RR* ) )
95 orcom 387 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  0  \/  B  =  0 )  <-> 
( B  =  0  \/  A  =  0 ) )
9695notbii 296 . . . . . . . . . . . . . . 15  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  -.  ( B  =  0  \/  A  =  0 ) )
9794, 96anbi12i 697 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  ( B  =  0  \/  A  =  0
) ) )
98 orcom 387 . . . . . . . . . . . . . . 15  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
9998notbii 296 . . . . . . . . . . . . . 14  |-  ( -.  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <->  -.  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) )
10097, 99anbi12i 697 . . . . . . . . . . . . 13  |-  ( ( ( ( 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 ) ) ) )  <->  ( ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  ( B  =  0  \/  A  =  0
) )  /\  -.  ( ( ( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) )  \/  ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) ) ) ) )
101 orcom 387 . . . . . . . . . . . . . 14  |-  ( ( ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  <-> 
( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
102101notbii 296 . . . . . . . . . . . . 13  |-  ( -.  ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )  <->  -.  ( ( ( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  \/  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
103 xmullem 11468 . . . . . . . . . . . . 13  |-  ( ( ( ( ( B  e.  RR*  /\  A  e. 
RR* )  /\  -.  ( B  =  0  \/  A  =  0
) )  /\  -.  ( ( ( 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 ) ) ) )  ->  B  e.  RR )
104100, 102, 103syl2anb 479 . . . . . . . . . . . 12  |-  ( ( ( ( ( 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 )
105104recnd 9634 . . . . . . . . . . 11  |-  ( ( ( ( ( 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.  CC )
10693, 105mulneg1d 10021 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) ) ) )  ->  ( -u A  x.  B )  =  -u ( A  x.  B
) )
107 rexneg 11422 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
10892, 107syl 16 . . . . . . . . . . 11  |-  ( ( ( ( ( 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 ) ) ) )  ->  -e A  =  -u A )
109108oveq1d 6310 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) ) ) )  ->  (  -e
A  x.  B )  =  ( -u A  x.  B ) )
11092, 104remulcld 9636 . . . . . . . . . . 11  |-  ( ( ( ( ( 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  x.  B )  e.  RR )
111 rexneg 11422 . . . . . . . . . . 11  |-  ( ( A  x.  B )  e.  RR  ->  -e
( A  x.  B
)  =  -u ( A  x.  B )
)
112110, 111syl 16 . . . . . . . . . 10  |-  ( ( ( ( ( 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 ) ) ) )  ->  -e ( A  x.  B )  =  -u ( A  x.  B ) )
113106, 109, 1123eqtr4d 2518 . . . . . . . . 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 ) ) ) )  ->  (  -e
A  x.  B )  =  -e ( A  x.  B ) )
11491, 113eqtr4d 2511 . . . . . . . 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 ) ) ) )  ->  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  =  ( 
-e A  x.  B ) )
11582, 85, 1143eqtr4d 2518 . . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
11677, 115pm2.61dan 789 . . . . . 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 ) ) ) )  ->  if (
( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
11762, 116pm2.61dan 789 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) )  =  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
118117ifeq2da 3976 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  -e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
1199, 118eqtrd 2508 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( (  -e
A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 , 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
120 xnegeq 11418 . . . . 5  |-  ( if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  0  ->  -e if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  = 
-e 0 )
121120, 1syl6eq 2524 . . . 4  |-  ( if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  0  ->  -e if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  0 )
122 xnegeq 11418 . . . 4  |-  ( if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) )  ->  -e
if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  = 
-e if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
123121, 122ifsb 3958 . . 3  |-  -e
if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  -e
if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )
124119, 123syl6eqr 2526 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  if ( (  -e
A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) ) )  =  -e
if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
125 xnegcl 11424 . . 3  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
126 xmulval 11436 . . 3  |-  ( ( 
-e A  e. 
RR*  /\  B  e.  RR* )  ->  (  -e
A xe B )  =  if ( (  -e A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e
A  = +oo )  \/  ( B  <  0  /\  -e A  = -oo ) )  \/  ( ( 0  <  -e A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e A  = -oo )  \/  ( B  <  0  /\  -e
A  = +oo )
)  \/  ( ( 0  <  -e
A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  (  -e A  x.  B
) ) ) ) )
127125, 126sylan 471 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  if ( (  -e
A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  -e A  = +oo )  \/  ( B  <  0  /\  -e
A  = -oo )
)  \/  ( ( 0  <  -e
A  /\  B  = +oo )  \/  (  -e A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  -e
A  = -oo )  \/  ( B  <  0  /\  -e A  = +oo ) )  \/  ( ( 0  <  -e A  /\  B  = -oo )  \/  (  -e A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( 
-e A  x.  B ) ) ) ) )
128 xmulval 11436 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A xe B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  (
( 0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
129 xnegeq 11418 . . 3  |-  ( ( A xe B )  =  if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) )  ->  -e ( A xe B )  = 
-e if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
130128, 129syl 16 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  -e
( A xe B )  =  -e if ( ( A  =  0  \/  B  =  0 ) ,  0 ,  if ( ( ( ( 0  <  B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) , +oo ,  if ( ( ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  \/  (
( 0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) , -oo ,  ( A  x.  B ) ) ) ) )
131124, 127, 1303eqtr4d 2518 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A xe B )  =  -e ( A xe B ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   ifcif 3945   class class class wbr 4453  (class class class)co 6295   RRcr 9503   0cc0 9504    x. cmul 9509   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640   -ucneg 9818    -ecxne 11327   xecxmu 11329
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-xneg 11330  df-xmul 11332
This theorem is referenced by:  xmulneg2  11474  xmulpnf1n  11482  xmulm1  11485  xmulass  11491  xadddi  11499  xadddi2  11501  xrsmulgzz  27490
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