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Theorem xmullem2 11576
Description: Lemma for xmulneg1 11580. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmullem2  |-  ( ( 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 ) ) ) ) )

Proof of Theorem xmullem2
StepHypRef Expression
1 mnfnepnf 11441 . . . . . . . . . . . 12  |- -oo  =/= +oo
2 eqeq1 2475 . . . . . . . . . . . . 13  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
32necon3bbid 2680 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  ( -.  A  = +oo  <-> -oo  =/= +oo ) )
41, 3mpbiri 241 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  A  = +oo )
54con2i 124 . . . . . . . . . 10  |-  ( A  = +oo  ->  -.  A  = -oo )
65adantl 473 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  = -oo )
7 0xr 9705 . . . . . . . . . . . . 13  |-  0  e.  RR*
8 nltmnf 11454 . . . . . . . . . . . . 13  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  -.  0  < -oo
10 breq2 4399 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  (
0  <  A  <->  0  < -oo ) )
119, 10mtbiri 310 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  0  <  A )
1211con2i 124 . . . . . . . . . 10  |-  ( 0  <  A  ->  -.  A  = -oo )
1312adantr 472 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  A  = -oo )
146, 13jaoi 386 . . . . . . . 8  |-  ( ( ( 0  <  B  /\  A  = +oo )  \/  ( 0  <  A  /\  B  = +oo ) )  ->  -.  A  = -oo )
1514a1i 11 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  A  = -oo ) )
16 simpr 468 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
17 xrltnsym 11459 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1816, 7, 17sylancl 675 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1918adantrd 475 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  B ) )
20 breq2 4399 . . . . . . . . . . 11  |-  ( B  = -oo  ->  (
0  <  B  <->  0  < -oo ) )
219, 20mtbiri 310 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  0  <  B )
2221adantl 473 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  0  <  B
)
2322a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  B ) )
2419, 23jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  B ) )
2515, 24orim12d 856 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  A  = -oo  \/  -.  0  <  B ) ) )
26 ianor 496 . . . . . . 7  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  0  < 
B  \/  -.  A  = -oo ) )
27 orcom 394 . . . . . . 7  |-  ( ( -.  0  <  B  \/  -.  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2826, 27bitri 257 . . . . . 6  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2925, 28syl6ibr 235 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( 0  <  B  /\  A  = -oo ) ) )
3018con2d 119 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  B  ->  -.  B  <  0 ) )
3130adantrd 475 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  <  0 ) )
32 pnfnlt 11453 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -. +oo  <  0 )
337, 32ax-mp 5 . . . . . . . . . 10  |-  -. +oo  <  0
34 simpr 468 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  = +oo )  ->  B  = +oo )
3534breq1d 4405 . . . . . . . . . 10  |-  ( ( 0  <  A  /\  B  = +oo )  ->  ( B  <  0  <-> +oo 
<  0 ) )
3633, 35mtbiri 310 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0
)
3736a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0 ) )
3831, 37jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  B  <  0 ) )
394a1i 11 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  = -oo  ->  -.  A  = +oo )
)
4039adantld 474 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  A  = +oo ) )
41 breq1 4398 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( A  <  0  <-> +oo  <  0
) )
4233, 41mtbiri 310 . . . . . . . . . . 11  |-  ( A  = +oo  ->  -.  A  <  0 )
4342con2i 124 . . . . . . . . . 10  |-  ( A  <  0  ->  -.  A  = +oo )
4443adantr 472 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo )
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo ) )
4640, 45jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  A  = +oo ) )
4738, 46orim12d 856 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  B  <  0  \/  -.  A  = +oo ) ) )
48 ianor 496 . . . . . 6  |-  ( -.  ( B  <  0  /\  A  = +oo ) 
<->  ( -.  B  <  0  \/  -.  A  = +oo ) )
4947, 48syl6ibr 235 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( B  <  0  /\  A  = +oo ) ) )
5029, 49jcad 542 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) ) )
51 ioran 498 . . . 4  |-  ( -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  <->  ( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) )
5250, 51syl6ibr 235 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) ) ) )
5321con2i 124 . . . . . . . . . 10  |-  ( 0  <  B  ->  -.  B  = -oo )
5453adantr 472 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo )
5554a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo ) )
56 pnfnemnf 11440 . . . . . . . . . . 11  |- +oo  =/= -oo
57 eqeq1 2475 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  = -oo  <-> +oo  = -oo ) )
5857necon3bbid 2680 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( -.  B  = -oo  <-> +oo  =/= -oo ) )
5956, 58mpbiri 241 . . . . . . . . . 10  |-  ( B  = +oo  ->  -.  B  = -oo )
6059adantl 473 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo )
6160a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo ) )
6255, 61jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  B  = -oo ) )
6311adantl 473 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  0  <  A
)
6463a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  A ) )
65 simpl 464 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
66 xrltnsym 11459 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6765, 7, 66sylancl 675 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6867adantrd 475 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  A ) )
6964, 68jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  A ) )
7062, 69orim12d 856 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  B  = -oo  \/  -.  0  <  A ) ) )
71 ianor 496 . . . . . . 7  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  0  < 
A  \/  -.  B  = -oo ) )
72 orcom 394 . . . . . . 7  |-  ( ( -.  0  <  A  \/  -.  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7371, 72bitri 257 . . . . . 6  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7470, 73syl6ibr 235 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( 0  <  A  /\  B  = -oo ) ) )
7542adantl 473 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0
)
7675a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0 ) )
7767con2d 119 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  A  ->  -.  A  <  0 ) )
7877adantrd 475 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  A  <  0 ) )
7976, 78jaod 387 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  A  <  0 ) )
80 breq1 4398 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  <  0  <-> +oo  <  0
) )
8133, 80mtbiri 310 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -.  B  <  0 )
8281con2i 124 . . . . . . . . . 10  |-  ( B  <  0  ->  -.  B  = +oo )
8382adantr 472 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  B  = +oo )
8459con2i 124 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  B  = +oo )
8584adantl 473 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  B  = +oo )
8683, 85jaoi 386 . . . . . . . 8  |-  ( ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  B  = +oo )
8786a1i 11 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  B  = +oo ) )
8879, 87orim12d 856 . . . . . 6  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  A  <  0  \/  -.  B  = +oo ) ) )
89 ianor 496 . . . . . 6  |-  ( -.  ( A  <  0  /\  B  = +oo ) 
<->  ( -.  A  <  0  \/  -.  B  = +oo ) )
9088, 89syl6ibr 235 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( A  <  0  /\  B  = +oo ) ) )
9174, 90jcad 542 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) ) )
92 ioran 498 . . . 4  |-  ( -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  <->  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) )
9391, 92syl6ibr 235 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  ->  -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) )
9452, 93jcad 542 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  -> 
( -.  ( ( 0  <  B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  /\  -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) ) ) ) )
95 or4 537 . 2  |-  ( ( ( ( 0  < 
B  /\  A  = +oo )  \/  ( B  <  0  /\  A  = -oo ) )  \/  ( ( 0  < 
A  /\  B  = +oo )  \/  ( A  <  0  /\  B  = -oo ) ) )  <-> 
( ( ( 0  <  B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  \/  ( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) ) ) )
96 ioran 498 . 2  |-  ( -.  ( ( ( 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 ) ) ) )
9794, 95, 963imtr4g 278 1  |-  ( ( 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 ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   class class class wbr 4395   0cc0 9557   +oocpnf 9690   -oocmnf 9691   RR*cxr 9692    < clt 9693
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-i2m1 9625  ax-1ne0 9626  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632
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-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-ov 6311  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698
This theorem is referenced by:  xmulneg1  11580
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