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Theorem apneg 8340
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 7730 . . 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 7730 . . . . . 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 508 . . . . . . . . 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 3912 . . . . . . . 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 509 . . . . . . . . 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 510 . . . . . . . . 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 505 . . . . . . . . . 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 506 . . . . . . . . . 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 8332 . . . . . . . . 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 1202 . . . . . . . 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 8110 . . . . . . . . . 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 8110 . . . . . . . . . 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 8110 . . . . . . . . . 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 8110 . . . . . . . . . 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 8332 . . . . . . . . . 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 1202 . . . . . . . . 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 7762 . . . . . . . . . . . 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 7683 . . . . . . . . . . . . . 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 7762 . . . . . . . . . . . . 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 7754 . . . . . . . . . . . 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 8054 . . . . . . . . . . 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 7925 . . . . . . . . . . 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 8142 . . . . . . . . . . . 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 5758 . . . . . . . . . . 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 2160 . . . . . . . . . 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 7762 . . . . . . . . . . . 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 7762 . . . . . . . . . . . . 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 7754 . . . . . . . . . . . 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 8054 . . . . . . . . . . 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 7925 . . . . . . . . . . 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 8142 . . . . . . . . . . . 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 5758 . . . . . . . . . . 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 2160 . . . . . . . . . 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 3912 . . . . . . . . 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 8326 . . . . . . . . . . 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 8326 . . . . . . . . . . 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 767 . . . . . . . . 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 2534 . . . . 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 2534 . 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 682    = wceq 1316    e. wcel 1465   E.wrex 2394   class class class wbr 3899  (class class class)co 5742   CCcc 7586   RRcr 7587   _ici 7590    + caddc 7591    x. cmul 7593   -ucneg 7902   # cap 8310
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-ltxr 7773  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311
This theorem is referenced by:  mulext1  8341  negap0  8359  div2subap  8563  cjap  10633  geosergap  11230
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