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Theorem apneg 8366
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 7755 . . 3  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
21adantl 275 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) ) )
3 cnre 7755 . . . . . 6  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
43ad3antrrr 483 . . . . 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 109 . . . . . . . . 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 523 . . . . . . . . 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 3937 . . . . . . . 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 524 . . . . . . . . 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 525 . . . . . . . . 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 520 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
z  e.  RR )
1110ad3antrrr 483 . . . . . . . . 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 521 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  ->  w  e.  RR )
1312ad3antrrr 483 . . . . . . . . 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 8358 . . . . . . . . 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 1217 . . . . . . . 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 8135 . . . . . . . . . 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 8135 . . . . . . . . . 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 8135 . . . . . . . . . 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 8135 . . . . . . . . . 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 8358 . . . . . . . . . 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 1217 . . . . . . . . 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 7787 . . . . . . . . . . . 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 7708 . . . . . . . . . . . . . 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 7787 . . . . . . . . . . . . 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 7779 . . . . . . . . . . . 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 8079 . . . . . . . . . . 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 7950 . . . . . . . . . . 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 8167 . . . . . . . . . . . 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 5783 . . . . . . . . . . 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 2180 . . . . . . . . . 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 7787 . . . . . . . . . . . 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 7787 . . . . . . . . . . . . 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 7779 . . . . . . . . . . . 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 8079 . . . . . . . . . . 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 7950 . . . . . . . . . . 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 8167 . . . . . . . . . . . 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 5783 . . . . . . . . . . 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 2180 . . . . . . . . . 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 3937 . . . . . . . . 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 8352 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x #  z  <->  -u x #  -u z ) )
428, 11, 41syl2anc 408 . . . . . . . . . 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 8352 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y #  w  <->  -u y #  -u w ) )
449, 13, 43syl2anc 408 . . . . . . . . . 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 782 . . . . . . . . 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 220 . . . . . . . 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 213 . . . . . . 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 114 . . . . . 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 2555 . . . . 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 114 . . 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 2555 . 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 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   E.wrex 2415   class class class wbr 3924  (class class class)co 5767   CCcc 7611   RRcr 7612   _ici 7615    + caddc 7616    x. cmul 7618   -ucneg 7927   # cap 8336
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-ltxr 7798  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337
This theorem is referenced by:  mulext1  8367  negap0  8385  div2subap  8589  cjap  10671  geosergap  11268
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