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Theorem apneg 7676
Description: Negation respects apartness. (Contributed by Jim Kingdon, 14-Feb-2020.)
Assertion
Ref Expression
apneg  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B
) )

Proof of Theorem apneg
Dummy variables  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7081 . . 3  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
21adantl 266 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) ) )
3 cnre 7081 . . . . . 6  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
43ad3antrrr 469 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
5 simpr 107 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
6 simpllr 494 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
75, 6breq12d 3805 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  <->  ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) ) ) )
8 simplrl 495 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  x  e.  RR )
9 simplrr 496 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  RR )
10 simprl 491 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
z  e.  RR )
1110ad3antrrr 469 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  z  e.  RR )
12 simprr 492 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  ->  w  e.  RR )
1312ad3antrrr 469 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  w  e.  RR )
14 apreim 7668 . . . . . . . . 9  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( z  +  ( _i  x.  w ) )  <->  ( x #  z  \/  y #  w
) ) )
158, 9, 11, 13, 14syl22anc 1147 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x  +  ( _i  x.  y ) ) #  ( z  +  ( _i  x.  w
) )  <->  ( x #  z  \/  y #  w
) ) )
168renegcld 7450 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u x  e.  RR )
179renegcld 7450 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u y  e.  RR )
1811renegcld 7450 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u z  e.  RR )
1913renegcld 7450 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u w  e.  RR )
20 apreim 7668 . . . . . . . . . 10  |-  ( ( ( -u x  e.  RR  /\  -u y  e.  RR )  /\  ( -u z  e.  RR  /\  -u w  e.  RR ) )  ->  ( ( -u x  +  ( _i  x.  -u y ) ) #  ( -u z  +  ( _i  x.  -u w
) )  <->  ( -u x #  -u z  \/  -u y #  -u w ) ) )
2116, 17, 18, 19, 20syl22anc 1147 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( -u x  +  ( _i  x.  -u y
) ) #  ( -u z  +  ( _i  x.  -u w ) )  <-> 
( -u x #  -u z  \/  -u y #  -u w
) ) )
228recnd 7113 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  x  e.  CC )
23 ax-icn 7037 . . . . . . . . . . . . . 14  |-  _i  e.  CC
2423a1i 9 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  _i  e.  CC )
259recnd 7113 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  y  e.  CC )
2624, 25mulcld 7105 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
_i  x.  y )  e.  CC )
2722, 26negdid 7398 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u (
x  +  ( _i  x.  y ) )  =  ( -u x  +  -u ( _i  x.  y ) ) )
285negeqd 7269 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u A  =  -u ( x  +  ( _i  x.  y
) ) )
2924, 25mulneg2d 7481 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
_i  x.  -u y )  =  -u ( _i  x.  y ) )
3029oveq2d 5556 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( -u x  +  ( _i  x.  -u y ) )  =  ( -u x  +  -u ( _i  x.  y ) ) )
3127, 28, 303eqtr4d 2098 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u A  =  ( -u x  +  ( _i  x.  -u y ) ) )
3211recnd 7113 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  z  e.  CC )
3313recnd 7113 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  w  e.  CC )
3424, 33mulcld 7105 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
_i  x.  w )  e.  CC )
3532, 34negdid 7398 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u (
z  +  ( _i  x.  w ) )  =  ( -u z  +  -u ( _i  x.  w ) ) )
366negeqd 7269 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u B  =  -u ( z  +  ( _i  x.  w
) ) )
3724, 33mulneg2d 7481 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
_i  x.  -u w )  =  -u ( _i  x.  w ) )
3837oveq2d 5556 . . . . . . . . . . 11  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( -u z  +  ( _i  x.  -u w ) )  =  ( -u z  +  -u ( _i  x.  w ) ) )
3935, 36, 383eqtr4d 2098 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  -u B  =  ( -u z  +  ( _i  x.  -u w ) ) )
4031, 39breq12d 3805 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( -u A #  -u B  <->  ( -u x  +  ( _i  x.  -u y ) ) #  (
-u z  +  ( _i  x.  -u w
) ) ) )
41 reapneg 7662 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x #  z  <->  -u x #  -u z ) )
428, 11, 41syl2anc 397 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
x #  z  <->  -u x #  -u z ) )
43 reapneg 7662 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y #  w  <->  -u y #  -u w ) )
449, 13, 43syl2anc 397 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
y #  w  <->  -u y #  -u w ) )
4542, 44orbi12d 717 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x #  z  \/  y #  w )  <->  ( -u x #  -u z  \/  -u y #  -u w ) ) )
4621, 40, 453bitr4rd 214 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  (
( x #  z  \/  y #  w )  <->  -u A #  -u B
) )
477, 15, 463bitrd 207 . . . . . . 7  |-  ( ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  /\  A  =  ( x  +  (
_i  x.  y )
) )  ->  ( A #  B  <->  -u A #  -u B
) )
4847ex 112 . . . . . 6  |-  ( ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  /\  (
x  e.  RR  /\  y  e.  RR )
)  ->  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A #  B  <->  -u A #  -u B
) ) )
4948rexlimdvva 2457 . . . . 5  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  ( A #  B  <->  -u A #  -u B
) ) )
504, 49mpd 13 . . . 4  |-  ( ( ( ( A  e.  CC  /\  B  e.  CC )  /\  (
z  e.  RR  /\  w  e.  RR )
)  /\  B  =  ( z  +  ( _i  x.  w ) ) )  ->  ( A #  B  <->  -u A #  -u B
) )
5150ex 112 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
( B  =  ( z  +  ( _i  x.  w ) )  ->  ( A #  B  <->  -u A #  -u B ) ) )
5251rexlimdvva 2457 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) )  ->  ( A #  B  <->  -u A #  -u B ) ) )
532, 52mpd 13 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A #  B  <->  -u A #  -u B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    \/ wo 639    = wceq 1259    e. wcel 1409   E.wrex 2324   class class class wbr 3792  (class class class)co 5540   CCcc 6945   RRcr 6946   _ici 6949    + caddc 6950    x. cmul 6952   -ucneg 7246   # cap 7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-ltxr 7124  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647
This theorem is referenced by:  mulext1  7677  negap0  7694  cjap  9734
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