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Theorem xmullem2 11502
Description: Lemma for xmulneg1 11506. (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 11369 . . . . . . . . . . . 12  |- -oo  =/= +oo
2 eqeq1 2432 . . . . . . . . . . . . 13  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
32necon3bbid 2638 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  ( -.  A  = +oo  <-> -oo  =/= +oo ) )
41, 3mpbiri 236 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  A  = +oo )
54con2i 123 . . . . . . . . . 10  |-  ( A  = +oo  ->  -.  A  = -oo )
65adantl 467 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  = -oo )
7 0xr 9638 . . . . . . . . . . . . 13  |-  0  e.  RR*
8 nltmnf 11382 . . . . . . . . . . . . 13  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  -.  0  < -oo
10 breq2 4370 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  (
0  <  A  <->  0  < -oo ) )
119, 10mtbiri 304 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  0  <  A )
1211con2i 123 . . . . . . . . . 10  |-  ( 0  <  A  ->  -.  A  = -oo )
1312adantr 466 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  A  = -oo )
146, 13jaoi 380 . . . . . . . 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 462 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
17 xrltnsym 11387 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1816, 7, 17sylancl 666 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1918adantrd 469 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  B ) )
20 breq2 4370 . . . . . . . . . . 11  |-  ( B  = -oo  ->  (
0  <  B  <->  0  < -oo ) )
219, 20mtbiri 304 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  0  <  B )
2221adantl 467 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  0  <  B
)
2322a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  B ) )
2419, 23jaod 381 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  B ) )
2515, 24orim12d 846 . . . . . 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 490 . . . . . . 7  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  0  < 
B  \/  -.  A  = -oo ) )
27 orcom 388 . . . . . . 7  |-  ( ( -.  0  <  B  \/  -.  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2826, 27bitri 252 . . . . . 6  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2925, 28syl6ibr 230 . . . . 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 118 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  B  ->  -.  B  <  0 ) )
3130adantrd 469 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  <  0 ) )
32 pnfnlt 11381 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -. +oo  <  0 )
337, 32ax-mp 5 . . . . . . . . . 10  |-  -. +oo  <  0
34 simpr 462 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  = +oo )  ->  B  = +oo )
3534breq1d 4376 . . . . . . . . . 10  |-  ( ( 0  <  A  /\  B  = +oo )  ->  ( B  <  0  <-> +oo 
<  0 ) )
3633, 35mtbiri 304 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0
)
3736a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0 ) )
3831, 37jaod 381 . . . . . . 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 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  A  = +oo ) )
41 breq1 4369 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( A  <  0  <-> +oo  <  0
) )
4233, 41mtbiri 304 . . . . . . . . . . 11  |-  ( A  = +oo  ->  -.  A  <  0 )
4342con2i 123 . . . . . . . . . 10  |-  ( A  <  0  ->  -.  A  = +oo )
4443adantr 466 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo )
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo ) )
4640, 45jaod 381 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  A  = +oo ) )
4738, 46orim12d 846 . . . . . 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 490 . . . . . 6  |-  ( -.  ( B  <  0  /\  A  = +oo ) 
<->  ( -.  B  <  0  \/  -.  A  = +oo ) )
4947, 48syl6ibr 230 . . . . 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 535 . . . 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 492 . . . 4  |-  ( -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  <->  ( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) )
5250, 51syl6ibr 230 . . 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 123 . . . . . . . . . 10  |-  ( 0  <  B  ->  -.  B  = -oo )
5453adantr 466 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo )
5554a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo ) )
56 pnfnemnf 11368 . . . . . . . . . . 11  |- +oo  =/= -oo
57 eqeq1 2432 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  = -oo  <-> +oo  = -oo ) )
5857necon3bbid 2638 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( -.  B  = -oo  <-> +oo  =/= -oo ) )
5956, 58mpbiri 236 . . . . . . . . . 10  |-  ( B  = +oo  ->  -.  B  = -oo )
6059adantl 467 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo )
6160a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo ) )
6255, 61jaod 381 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  B  = -oo ) )
6311adantl 467 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  0  <  A
)
6463a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  A ) )
65 simpl 458 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
66 xrltnsym 11387 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6765, 7, 66sylancl 666 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6867adantrd 469 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  A ) )
6964, 68jaod 381 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  A ) )
7062, 69orim12d 846 . . . . . 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 490 . . . . . . 7  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  0  < 
A  \/  -.  B  = -oo ) )
72 orcom 388 . . . . . . 7  |-  ( ( -.  0  <  A  \/  -.  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7371, 72bitri 252 . . . . . 6  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7470, 73syl6ibr 230 . . . . 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 467 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0
)
7675a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0 ) )
7767con2d 118 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  A  ->  -.  A  <  0 ) )
7877adantrd 469 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  A  <  0 ) )
7976, 78jaod 381 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  A  <  0 ) )
80 breq1 4369 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  <  0  <-> +oo  <  0
) )
8133, 80mtbiri 304 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -.  B  <  0 )
8281con2i 123 . . . . . . . . . 10  |-  ( B  <  0  ->  -.  B  = +oo )
8382adantr 466 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  B  = +oo )
8459con2i 123 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  B  = +oo )
8584adantl 467 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  B  = +oo )
8683, 85jaoi 380 . . . . . . . 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 846 . . . . . 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 490 . . . . . 6  |-  ( -.  ( A  <  0  /\  B  = +oo ) 
<->  ( -.  A  <  0  \/  -.  B  = +oo ) )
9088, 89syl6ibr 230 . . . . 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 535 . . . 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 492 . . . 4  |-  ( -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  <->  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) )
9391, 92syl6ibr 230 . . 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 535 . 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 530 . 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 492 . 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 273 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 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599   class class class wbr 4366   0cc0 9490   +oocpnf 9623   -oocmnf 9624   RR*cxr 9625    < clt 9626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pow 4545  ax-pr 4603  ax-un 6541  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-i2m1 9558  ax-1ne0 9559  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-nel 2602  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-csb 3339  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-pw 3926  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-mpt 4427  df-id 4711  df-po 4717  df-so 4718  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-f 5548  df-f1 5549  df-fo 5550  df-f1o 5551  df-fv 5552  df-ov 6252  df-er 7318  df-en 7525  df-dom 7526  df-sdom 7527  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631
This theorem is referenced by:  xmulneg1  11506
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