MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xmullem2 Structured version   Unicode version

Theorem xmullem2 11460
Description: Lemma for xmulneg1 11464. (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 11330 . . . . . . . . . . . 12  |- -oo  =/= +oo
2 eqeq1 2458 . . . . . . . . . . . . 13  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
32necon3bbid 2701 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  ( -.  A  = +oo  <-> -oo  =/= +oo ) )
41, 3mpbiri 233 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  A  = +oo )
54con2i 120 . . . . . . . . . 10  |-  ( A  = +oo  ->  -.  A  = -oo )
65adantl 464 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  A  = -oo )
7 0xr 9629 . . . . . . . . . . . . 13  |-  0  e.  RR*
8 nltmnf 11341 . . . . . . . . . . . . 13  |-  ( 0  e.  RR*  ->  -.  0  < -oo )
97, 8ax-mp 5 . . . . . . . . . . . 12  |-  -.  0  < -oo
10 breq2 4443 . . . . . . . . . . . 12  |-  ( A  = -oo  ->  (
0  <  A  <->  0  < -oo ) )
119, 10mtbiri 301 . . . . . . . . . . 11  |-  ( A  = -oo  ->  -.  0  <  A )
1211con2i 120 . . . . . . . . . 10  |-  ( 0  <  A  ->  -.  A  = -oo )
1312adantr 463 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  A  = -oo )
146, 13jaoi 377 . . . . . . . 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 459 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  B  e.  RR* )
17 xrltnsym 11346 . . . . . . . . . 10  |-  ( ( B  e.  RR*  /\  0  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1816, 7, 17sylancl 660 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( B  <  0  ->  -.  0  <  B ) )
1918adantrd 466 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  B ) )
20 breq2 4443 . . . . . . . . . . 11  |-  ( B  = -oo  ->  (
0  <  B  <->  0  < -oo ) )
219, 20mtbiri 301 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  0  <  B )
2221adantl 464 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  0  <  B
)
2322a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  B ) )
2419, 23jaod 378 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  B ) )
2515, 24orim12d 836 . . . . . 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 486 . . . . . . 7  |-  ( -.  ( 0  <  B  /\  A  = -oo ) 
<->  ( -.  0  < 
B  \/  -.  A  = -oo ) )
27 orcom 385 . . . . . . 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 466 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  <  0 ) )
32 pnfnlt 11340 . . . . . . . . . . 11  |-  ( 0  e.  RR*  ->  -. +oo  <  0 )
337, 32ax-mp 5 . . . . . . . . . 10  |-  -. +oo  <  0
34 simpr 459 . . . . . . . . . . 11  |-  ( ( 0  <  A  /\  B  = +oo )  ->  B  = +oo )
3534breq1d 4449 . . . . . . . . . 10  |-  ( ( 0  <  A  /\  B  = +oo )  ->  ( B  <  0  <-> +oo 
<  0 ) )
3633, 35mtbiri 301 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0
)
3736a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  <  0 ) )
3831, 37jaod 378 . . . . . . 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 465 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  A  = +oo ) )
41 breq1 4442 . . . . . . . . . . . 12  |-  ( A  = +oo  ->  ( A  <  0  <-> +oo  <  0
) )
4233, 41mtbiri 301 . . . . . . . . . . 11  |-  ( A  = +oo  ->  -.  A  <  0 )
4342con2i 120 . . . . . . . . . 10  |-  ( A  <  0  ->  -.  A  = +oo )
4443adantr 463 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo )
4544a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  A  = +oo ) )
4640, 45jaod 378 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  A  = +oo ) )
4738, 46orim12d 836 . . . . . 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 486 . . . . . 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 531 . . . 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 488 . . . 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 463 . . . . . . . . 9  |-  ( ( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo )
5554a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  B  /\  A  = +oo )  ->  -.  B  = -oo ) )
56 pnfnemnf 11329 . . . . . . . . . . 11  |- +oo  =/= -oo
57 eqeq1 2458 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  = -oo  <-> +oo  = -oo ) )
5857necon3bbid 2701 . . . . . . . . . . 11  |-  ( B  = +oo  ->  ( -.  B  = -oo  <-> +oo  =/= -oo ) )
5956, 58mpbiri 233 . . . . . . . . . 10  |-  ( B  = +oo  ->  -.  B  = -oo )
6059adantl 464 . . . . . . . . 9  |-  ( ( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo )
6160a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  B  = -oo ) )
6255, 61jaod 378 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  B  = -oo ) )
6311adantl 464 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  0  <  A
)
6463a1i 11 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( B  <  0  /\  A  = -oo )  ->  -.  0  <  A ) )
65 simpl 455 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  A  e.  RR* )
66 xrltnsym 11346 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  0  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6765, 7, 66sylancl 660 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <  0  ->  -.  0  <  A ) )
6867adantrd 466 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( A  <  0  /\  B  = -oo )  ->  -.  0  <  A ) )
6964, 68jaod 378 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( B  <  0  /\  A  = -oo )  \/  ( A  <  0  /\  B  = -oo ) )  ->  -.  0  <  A ) )
7062, 69orim12d 836 . . . . . 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 486 . . . . . . 7  |-  ( -.  ( 0  <  A  /\  B  = -oo ) 
<->  ( -.  0  < 
A  \/  -.  B  = -oo ) )
72 orcom 385 . . . . . . 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 464 . . . . . . . . 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 466 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( 0  <  A  /\  B  = +oo )  ->  -.  A  <  0 ) )
7976, 78jaod 378 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
( ( 0  < 
B  /\  A  = +oo )  \/  (
0  <  A  /\  B  = +oo )
)  ->  -.  A  <  0 ) )
80 breq1 4442 . . . . . . . . . . . 12  |-  ( B  = +oo  ->  ( B  <  0  <-> +oo  <  0
) )
8133, 80mtbiri 301 . . . . . . . . . . 11  |-  ( B  = +oo  ->  -.  B  <  0 )
8281con2i 120 . . . . . . . . . 10  |-  ( B  <  0  ->  -.  B  = +oo )
8382adantr 463 . . . . . . . . 9  |-  ( ( B  <  0  /\  A  = -oo )  ->  -.  B  = +oo )
8459con2i 120 . . . . . . . . . 10  |-  ( B  = -oo  ->  -.  B  = +oo )
8584adantl 464 . . . . . . . . 9  |-  ( ( A  <  0  /\  B  = -oo )  ->  -.  B  = +oo )
8683, 85jaoi 377 . . . . . . . 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 836 . . . . . 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 486 . . . . . 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 531 . . . 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 488 . . . 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 531 . 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 526 . 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 488 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   class class class wbr 4439   0cc0 9481   +oocpnf 9614   -oocmnf 9615   RR*cxr 9616    < clt 9617
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-i2m1 9549  ax-1ne0 9550  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-po 4789  df-so 4790  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622
This theorem is referenced by:  xmulneg1  11464
  Copyright terms: Public domain W3C validator