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Theorem xmulneg1 11580
Description: Extended real version of mulneg1 10076. (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 11528 . . . . . . . . 9  |-  -e 0  =  0
21eqeq2i 2483 . . . . . . . 8  |-  (  -e A  =  -e 0  <->  -e A  =  0 )
3 0xr 9705 . . . . . . . . 9  |-  0  e.  RR*
4 xneg11 11531 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  (  -e A  =  -e 0  <->  A  = 
0 ) )
53, 4mpan2 685 . . . . . . . 8  |-  ( A  e.  RR*  ->  (  -e A  =  -e 0  <->  A  =  0
) )
62, 5syl5bbr 267 . . . . . . 7  |-  ( A  e.  RR*  ->  (  -e A  =  0  <->  A  =  0 ) )
76adantr 472 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (  -e A  =  0  <-> 
A  =  0 ) )
87orbi1d 717 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
(  -e A  =  0  \/  B  =  0 )  <->  ( A  =  0  \/  B  =  0 ) ) )
98ifbid 3894 . . . 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 11525 . . . . . . . . . . . . . 14  |-  -e +oo  = -oo
1110eqeq2i 2483 . . . . . . . . . . . . 13  |-  (  -e A  =  -e +oo 
<-> 
-e A  = -oo )
12 simpll 768 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  ->  A  e.  RR* )
13 pnfxr 11435 . . . . . . . . . . . . . 14  |- +oo  e.  RR*
14 xneg11 11531 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\ +oo  e.  RR* )  ->  (  -e A  =  -e +oo  <->  A  = +oo ) )
1512, 13, 14sylancl 675 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = 
-e +oo  <->  A  = +oo ) )
1611, 15syl5bbr 267 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = -oo  <->  A  = +oo ) )
1716anbi2d 718 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  < 
B  /\  -e A  = -oo )  <->  ( 0  <  B  /\  A  = +oo ) ) )
18 xnegmnf 11526 . . . . . . . . . . . . . 14  |-  -e -oo  = +oo
1918eqeq2i 2483 . . . . . . . . . . . . 13  |-  (  -e A  =  -e -oo 
<-> 
-e A  = +oo )
20 mnfxr 11437 . . . . . . . . . . . . . 14  |- -oo  e.  RR*
21 xneg11 11531 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\ -oo  e.  RR* )  ->  (  -e A  =  -e -oo  <->  A  = -oo ) )
2212, 20, 21sylancl 675 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = 
-e -oo  <->  A  = -oo ) )
2319, 22syl5bbr 267 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  = +oo  <->  A  = -oo ) )
2423anbi2d 718 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( B  <  0  /\  -e
A  = +oo )  <->  ( B  <  0  /\  A  = -oo )
) )
2517, 24orbi12d 724 . . . . . . . . . 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 11535 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR*  ->  ( A  <  0  <->  0  <  -e A ) )
2726ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( A  <  0  <->  0  <  -e A ) )
2827bicomd 206 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( 0  <  -e
A  <->  A  <  0
) )
2928anbi1d 719 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  <  -e A  /\  B  = -oo )  <->  ( A  <  0  /\  B  = -oo )
) )
30 xlt0neg2 11536 . . . . . . . . . . . . . . 15  |-  ( A  e.  RR*  ->  ( 0  <  A  <->  -e A  <  0 ) )
3130ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( 0  <  A  <->  -e A  <  0
) )
3231bicomd 206 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
(  -e A  <  0  <->  0  <  A
) )
3332anbi1d 719 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( (  -e
A  <  0  /\  B  = +oo )  <->  ( 0  <  A  /\  B  = +oo )
) )
3429, 33orbi12d 724 . . . . . . . . . . 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 394 . . . . . . . . . . 11  |-  ( ( ( A  <  0  /\  B  = -oo )  \/  ( 0  <  A  /\  B  = +oo ) )  <->  ( (
0  <  A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )
3634, 35syl6bb 269 . . . . . . . . . 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 724 . . . . . . . . 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 493 . . . . . . . 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 ) ) ) )
3938iftrued 3880 . . . . . . 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 )
40 xmullem2 11576 . . . . . . . . . . 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 ) ) ) ) )
4140adantr 472 . . . . . . . . . 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 ) ) ) ) )
4223anbi2d 718 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  < 
B  /\  -e A  = +oo )  <->  ( 0  <  B  /\  A  = -oo ) ) )
4316anbi2d 718 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( B  <  0  /\  -e
A  = -oo )  <->  ( B  <  0  /\  A  = +oo )
) )
4442, 43orbi12d 724 . . . . . . . . . . . 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 ) ) ) )
4528anbi1d 719 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( ( 0  <  -e A  /\  B  = +oo )  <->  ( A  <  0  /\  B  = +oo )
) )
4632anbi1d 719 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  -> 
( (  -e
A  <  0  /\  B  = -oo )  <->  ( 0  <  A  /\  B  = -oo )
) )
4745, 46orbi12d 724 . . . . . . . . . . . . 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 )
) ) )
48 orcom 394 . . . . . . . . . . . . 13  |-  ( ( ( A  <  0  /\  B  = +oo )  \/  ( 0  <  A  /\  B  = -oo ) )  <->  ( (
0  <  A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) )
4947, 48syl6bb 269 . . . . . . . . . . . 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 ) ) ) )
5044, 49orbi12d 724 . . . . . . . . . . 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 ) ) ) ) )
5150notbid 301 . . . . . . . . . 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 ) ) ) ) )
5241, 51sylibrd 242 . . . . . . . . 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 ) ) ) ) )
5352imp 436 . . . . . . . 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 ) ) ) )
5453iffalsed 3883 . . . . . . 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
) ) )
55 iftrue 3878 . . . . . . . . . 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 )
5655adantl 473 . . . . . . . . 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 )
57 xnegeq 11523 . . . . . . . . 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 )
5856, 57syl 17 . . . . . . . 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 )
5958, 10syl6eq 2521 . . . . . . 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 )
6039, 54, 593eqtr4d 2515 . . . . . 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 ) ) ) )
6150biimpar 493 . . . . . . . . . 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 ) ) ) )
6261iftrued 3880 . . . . . . . . 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 )
6341con2d 119 . . . . . . . . . . . . . 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 ) ) ) ) )
6463imp 436 . . . . . . . . . . . . 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 ) ) ) )
6564iffalsed 3883 . . . . . . . . . . . 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 ) ) )
66 iftrue 3878 . . . . . . . . . . . . 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 )
6766adantl 473 . . . . . . . . . . . 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 )
6865, 67eqtrd 2505 . . . . . . . . . . 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 )
69 xnegeq 11523 . . . . . . . . . . 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 )
7068, 69syl 17 . . . . . . . . . 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 )
7170, 18syl6eq 2521 . . . . . . . . 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 )
7262, 71eqtr4d 2508 . . . . . . . 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 ) ) ) )
7372adantlr 729 . . . . . . 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 ) ) ) )
7437notbid 301 . . . . . . . . . . 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 ) ) ) ) )
7574biimpar 493 . . . . . . . . . 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 ) ) ) )
7675adantr 472 . . . . . . . . 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 ) ) ) )
7776iffalsed 3883 . . . . . . . 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 ) )
7851biimpar 493 . . . . . . . . . 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 ) ) ) )
7978adantlr 729 . . . . . . . . 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 ) ) ) )
8079iffalsed 3883 . . . . . . . 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
) ) )
81 iffalse 3881 . . . . . . . . . . . 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 ,  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 ) ) )
8281ad2antlr 741 . . . . . . . . . . 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 ) ) )
83 iffalse 3881 . . . . . . . . . . . 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 ) )
8483adantl 473 . . . . . . . . . . 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 ) )
8582, 84eqtrd 2505 . . . . . . . . . 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 ) )
86 xnegeq 11523 . . . . . . . . . 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 ) )
8785, 86syl 17 . . . . . . . . 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 ) )
88 xmullem 11575 . . . . . . . . . . . 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 )
8988recnd 9687 . . . . . . . . . . 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 )
90 ancom 457 . . . . . . . . . . . . . . 15  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  <->  ( B  e.  RR*  /\  A  e. 
RR* ) )
91 orcom 394 . . . . . . . . . . . . . . . 16  |-  ( ( A  =  0  \/  B  =  0 )  <-> 
( B  =  0  \/  A  =  0 ) )
9291notbii 303 . . . . . . . . . . . . . . 15  |-  ( -.  ( A  =  0  \/  B  =  0 )  <->  -.  ( B  =  0  \/  A  =  0 ) )
9390, 92anbi12i 711 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  -.  ( A  =  0  \/  B  =  0 ) )  <->  ( ( B  e.  RR*  /\  A  e.  RR* )  /\  -.  ( B  =  0  \/  A  =  0
) ) )
94 orcom 394 . . . . . . . . . . . . . . 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 ) ) ) )
9594notbii 303 . . . . . . . . . . . . . 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 ) ) ) )
9693, 95anbi12i 711 . . . . . . . . . . . . 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 ) ) ) ) )
97 orcom 394 . . . . . . . . . . . . . 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 ) ) ) )
9897notbii 303 . . . . . . . . . . . . 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 ) ) ) )
99 xmullem 11575 . . . . . . . . . . . . 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 )
10096, 98, 99syl2anb 487 . . . . . . . . . . . 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 )
101100recnd 9687 . . . . . . . . . . 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 )
10289, 101mulneg1d 10092 . . . . . . . . . 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
) )
103 rexneg 11527 . . . . . . . . . . . 12  |-  ( A  e.  RR  ->  -e
A  =  -u A
)
10488, 103syl 17 . . . . . . . . . . 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 )
105104oveq1d 6323 . . . . . . . . . 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 ) )
10688, 100remulcld 9689 . . . . . . . . . . 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 )
107 rexneg 11527 . . . . . . . . . . 11  |-  ( ( A  x.  B )  e.  RR  ->  -e
( A  x.  B
)  =  -u ( A  x.  B )
)
108106, 107syl 17 . . . . . . . . . 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 ) )
109102, 105, 1083eqtr4d 2515 . . . . . . . . 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 ) )
11087, 109eqtr4d 2508 . . . . . . . 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 ) )
11177, 80, 1103eqtr4d 2515 . . . . . . 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 ) ) ) )
11273, 111pm2.61dan 808 . . . . . 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 ) ) ) )
11360, 112pm2.61dan 808 . . . . 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 ) ) ) )
114113ifeq2da 3903 . . . 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 ) ) ) ) )
1159, 114eqtrd 2505 . . 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 ) ) ) ) )
116 xnegeq 11523 . . . . 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 )
117116, 1syl6eq 2521 . . . 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 )
118 xnegeq 11523 . . . 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 ) ) ) )
119117, 118ifsb 3885 . . 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 ) ) ) )
120115, 119syl6eqr 2523 . 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 ) ) ) ) )
121 xnegcl 11529 . . 3  |-  ( A  e.  RR*  ->  -e
A  e.  RR* )
122 xmulval 11541 . . 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
) ) ) ) )
123121, 122sylan 479 . 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 ) ) ) ) )
124 xmulval 11541 . . 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 ) ) ) ) )
125 xnegeq 11523 . . 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 ) ) ) ) )
126124, 125syl 17 . 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 ) ) ) ) )
127120, 123, 1263eqtr4d 2515 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 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   ifcif 3872   class class class wbr 4395  (class class class)co 6308   RRcr 9556   0cc0 9557    x. cmul 9562   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693   -ucneg 9881    -ecxne 11429   xecxmu 11431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-xneg 11432  df-xmul 11434
This theorem is referenced by:  xmulneg2  11581  xmulpnf1n  11589  xmulm1  11592  xmulass  11598  xadddi  11606  xadddi2  11608  xrsmulgzz  28515
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