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Theorem apneg 8064
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 7463 . . 3  |-  ( B  e.  CC  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w ) ) )
21adantl 271 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  E. z  e.  RR  E. w  e.  RR  B  =  ( z  +  ( _i  x.  w
) ) )
3 cnre 7463 . . . . . 6  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
43ad3antrrr 476 . . . . 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 108 . . . . . . . . 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 501 . . . . . . . . 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 3850 . . . . . . . 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 502 . . . . . . . . 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 503 . . . . . . . . 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 498 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  -> 
z  e.  RR )
1110ad3antrrr 476 . . . . . . . . 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 499 . . . . . . . . . 10  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( z  e.  RR  /\  w  e.  RR ) )  ->  w  e.  RR )
1312ad3antrrr 476 . . . . . . . . 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 8056 . . . . . . . . 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 1175 . . . . . . . 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 7837 . . . . . . . . . 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 7837 . . . . . . . . . 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 7837 . . . . . . . . . 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 7837 . . . . . . . . . 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 8056 . . . . . . . . . 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 1175 . . . . . . . . 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 7495 . . . . . . . . . . . 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 7419 . . . . . . . . . . . . . 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 7495 . . . . . . . . . . . . 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 7487 . . . . . . . . . . . 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 7785 . . . . . . . . . . 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 7656 . . . . . . . . . . 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 7869 . . . . . . . . . . . 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 5650 . . . . . . . . . . 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 2130 . . . . . . . . . 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 7495 . . . . . . . . . . . 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 7495 . . . . . . . . . . . . 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 7487 . . . . . . . . . . . 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 7785 . . . . . . . . . . 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 7656 . . . . . . . . . . 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 7869 . . . . . . . . . . . 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 5650 . . . . . . . . . . 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 2130 . . . . . . . . . 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 3850 . . . . . . . . 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 8050 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  z  e.  RR )  ->  ( x #  z  <->  -u x #  -u z ) )
428, 11, 41syl2anc 403 . . . . . . . . . 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 8050 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  w  e.  RR )  ->  ( y #  w  <->  -u y #  -u w ) )
449, 13, 43syl2anc 403 . . . . . . . . . 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 742 . . . . . . . . 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 219 . . . . . . . 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 212 . . . . . . 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 113 . . . . . 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 2496 . . . . 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 113 . . 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 2496 . 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 102    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   E.wrex 2360   class class class wbr 3837  (class class class)co 5634   CCcc 7327   RRcr 7328   _ici 7331    + caddc 7332    x. cmul 7334   -ucneg 7633   # cap 8034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-ltxr 7506  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035
This theorem is referenced by:  mulext1  8065  negap0  8082  div2subap  8274  cjap  10305  geosergap  10861
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