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Theorem xmullem2 11332
Description: Lemma for xmulneg1 11336. (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 11202 . . . . . . . . . . . 12  |- -oo  =/= +oo
2 eqeq1 2455 . . . . . . . . . . . . 13  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
32necon3bbid 2695 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  ( -.  A  = +oo  <-> -oo  =/= +oo ) )
41, 3mpbiri 233 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  A  = +oo )
54con2i 120 . . . . . . . . . 10  |-  ( A  = +oo  ->  -.  A  = -oo )
65adantl 466 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  = -oo )
7 0xr 9534 . . . . . . . . . . . . 13  |-  0  e.  RR*
8 nltmnf 11213 . . . . . . . . . . . . 13  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  -.  0  < -oo
10 breq2 4397 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  (
0  <  A  <->  0  < -oo ) )
119, 10mtbiri 303 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  0  <  A )
1211con2i 120 . . . . . . . . . 10  |-  ( 0  <  A  ->  -.  A  = -oo )
1312adantr 465 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  A  = -oo )
146, 13jaoi 379 . . . . . . . 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 461 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
17 xrltnsym 11218 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1816, 7, 17sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1918adantrd 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  B ) )
20 breq2 4397 . . . . . . . . . . 11  |-  ( B  = -oo  ->  (
0  <  B  <->  0  < -oo ) )
219, 20mtbiri 303 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  0  <  B )
2221adantl 466 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  0  <  B
)
2322a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  B ) )
2419, 23jaod 380 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  B ) )
2515, 24orim12d 834 . . . . . 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 488 . . . . . . 7  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  0  < 
B  \/  -.  A  = -oo ) )
27 orcom 387 . . . . . . 7  |-  ( ( -.  0  <  B  \/  -.  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2826, 27bitri 249 . . . . . 6  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  A  = -oo  \/  -.  0  <  B ) )
2925, 28syl6ibr 227 . . . . 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 115 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  B  ->  -.  B  <  0 ) )
3130adantrd 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  <  0 ) )
32 pnfnlt 11212 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -. +oo  <  0 )
337, 32ax-mp 5 . . . . . . . . . 10  |-  -. +oo  <  0
34 simpr 461 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  = +oo )  ->  B  = +oo )
3534breq1d 4403 . . . . . . . . . 10  |-  ( ( 0  <  A  /\  B  = +oo )  ->  ( B  <  0  <-> +oo 
<  0 ) )
3633, 35mtbiri 303 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0
)
3736a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0 ) )
3831, 37jaod 380 . . . . . . 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 467 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  A  = +oo ) )
41 breq1 4396 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( A  <  0  <-> +oo  <  0
) )
4233, 41mtbiri 303 . . . . . . . . . . 11  |-  ( A  = +oo  ->  -.  A  <  0 )
4342con2i 120 . . . . . . . . . 10  |-  ( A  <  0  ->  -.  A  = +oo )
4443adantr 465 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo )
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo ) )
4640, 45jaod 380 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  A  = +oo ) )
4738, 46orim12d 834 . . . . . 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 488 . . . . . 6  |-  ( -.  ( B  <  0  /\  A  = +oo ) 
<->  ( -.  B  <  0  \/  -.  A  = +oo ) )
4947, 48syl6ibr 227 . . . . 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 533 . . . 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 490 . . . 4  |-  ( -.  ( ( 0  < 
B  /\  A  = -oo )  \/  ( B  <  0  /\  A  = +oo ) )  <->  ( -.  ( 0  <  B  /\  A  = -oo )  /\  -.  ( B  <  0  /\  A  = +oo ) ) )
5250, 51syl6ibr 227 . . 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 120 . . . . . . . . . 10  |-  ( 0  <  B  ->  -.  B  = -oo )
5453adantr 465 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo )
5554a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo ) )
56 pnfnemnf 11201 . . . . . . . . . . 11  |- +oo  =/= -oo
57 eqeq1 2455 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  = -oo  <-> +oo  = -oo ) )
5857necon3bbid 2695 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( -.  B  = -oo  <-> +oo  =/= -oo ) )
5956, 58mpbiri 233 . . . . . . . . . 10  |-  ( B  = +oo  ->  -.  B  = -oo )
6059adantl 466 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo )
6160a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo ) )
6255, 61jaod 380 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  B  = -oo ) )
6311adantl 466 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  0  <  A
)
6463a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  A ) )
65 simpl 457 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
66 xrltnsym 11218 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6765, 7, 66sylancl 662 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6867adantrd 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  A ) )
6964, 68jaod 380 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  A ) )
7062, 69orim12d 834 . . . . . 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 488 . . . . . . 7  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  0  < 
A  \/  -.  B  = -oo ) )
72 orcom 387 . . . . . . 7  |-  ( ( -.  0  <  A  \/  -.  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7371, 72bitri 249 . . . . . 6  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  B  = -oo  \/  -.  0  <  A ) )
7470, 73syl6ibr 227 . . . . 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 466 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0
)
7675a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  A  <  0 ) )
7767con2d 115 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
0  <  A  ->  -.  A  <  0 ) )
7877adantrd 468 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  A  <  0 ) )
7976, 78jaod 380 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  A  <  0 ) )
80 breq1 4396 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  <  0  <-> +oo  <  0
) )
8133, 80mtbiri 303 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -.  B  <  0 )
8281con2i 120 . . . . . . . . . 10  |-  ( B  <  0  ->  -.  B  = +oo )
8382adantr 465 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  B  = +oo )
8459con2i 120 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  B  = +oo )
8584adantl 466 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  B  = +oo )
8683, 85jaoi 379 . . . . . . . 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 834 . . . . . 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 488 . . . . . 6  |-  ( -.  ( A  <  0  /\  B  = +oo ) 
<->  ( -.  A  <  0  \/  -.  B  = +oo ) )
9088, 89syl6ibr 227 . . . . 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 533 . . . 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 490 . . . 4  |-  ( -.  ( ( 0  < 
A  /\  B  = -oo )  \/  ( A  <  0  /\  B  = +oo ) )  <->  ( -.  ( 0  <  A  /\  B  = -oo )  /\  -.  ( A  <  0  /\  B  = +oo ) ) )
9391, 92syl6ibr 227 . . 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 533 . 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 528 . 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 490 . 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 270 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 368    /\ wa 369    = wceq 1370    e. wcel 1758    =/= wne 2644   class class class wbr 4393   0cc0 9386   +oocpnf 9519   -oocmnf 9520   RR*cxr 9521    < clt 9522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-i2m1 9454  ax-1ne0 9455  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527
This theorem is referenced by:  xmulneg1  11336
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