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Theorem apneg 8503
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 7889 . . 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 7889 . . . . . 6  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
43ad3antrrr 484 . . . . 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 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 )
) )  ->  B  =  ( z  +  ( _i  x.  w
) ) )
75, 6breq12d 3992 . . . . . . . 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 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 )
) )  ->  x  e.  RR )
9 simplrr 526 . . . . . . . . 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 521 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
z  e.  RR )
1110ad3antrrr 484 . . . . . . . . 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 522 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  ->  w  e.  RR )
1312ad3antrrr 484 . . . . . . . . 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 8495 . . . . . . . . 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 1228 . . . . . . . 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 8272 . . . . . . . . . 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 8272 . . . . . . . . . 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 8272 . . . . . . . . . 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 8272 . . . . . . . . . 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 8495 . . . . . . . . . 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 1228 . . . . . . . . 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 7921 . . . . . . . . . . . 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 7842 . . . . . . . . . . . . . 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 7921 . . . . . . . . . . . . 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 7913 . . . . . . . . . . . 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 8216 . . . . . . . . . . 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 8087 . . . . . . . . . . 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 8304 . . . . . . . . . . . 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 5855 . . . . . . . . . . 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 2207 . . . . . . . . . 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 7921 . . . . . . . . . . . 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 7921 . . . . . . . . . . . . 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 7913 . . . . . . . . . . . 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 8216 . . . . . . . . . . 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 8087 . . . . . . . . . . 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 8304 . . . . . . . . . . . 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 5855 . . . . . . . . . . 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 2207 . . . . . . . . . 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 3992 . . . . . . . . 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 8489 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x #  z  <->  -u x #  -u z ) )
428, 11, 41syl2anc 409 . . . . . . . . . 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 8489 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y #  w  <->  -u y #  -u w ) )
449, 13, 43syl2anc 409 . . . . . . . . . 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 783 . . . . . . . . 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 2589 . . . . 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 2589 . 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 698    = wceq 1342    e. wcel 2135   E.wrex 2443   class class class wbr 3979  (class class class)co 5839   CCcc 7745   RRcr 7746   _ici 7749    + caddc 7750    x. cmul 7752   -ucneg 8064   # cap 8473
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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-1cn 7840  ax-1re 7841  ax-icn 7842  ax-addcl 7843  ax-addrcl 7844  ax-mulcl 7845  ax-mulrcl 7846  ax-addcom 7847  ax-mulcom 7848  ax-addass 7849  ax-mulass 7850  ax-distr 7851  ax-i2m1 7852  ax-0lt1 7853  ax-1rid 7854  ax-0id 7855  ax-rnegex 7856  ax-precex 7857  ax-cnre 7858  ax-pre-ltirr 7859  ax-pre-lttrn 7861  ax-pre-apti 7862  ax-pre-ltadd 7863  ax-pre-mulgt0 7864
This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2726  df-sbc 2950  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-id 4268  df-xp 4607  df-rel 4608  df-cnv 4609  df-co 4610  df-dm 4611  df-iota 5150  df-fun 5187  df-fv 5193  df-riota 5795  df-ov 5842  df-oprab 5843  df-mpo 5844  df-pnf 7929  df-mnf 7930  df-ltxr 7932  df-sub 8065  df-neg 8066  df-reap 8467  df-ap 8474
This theorem is referenced by:  mulext1  8504  negap0  8522  div2subap  8727  cjap  10842  geosergap  11441
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