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Definition df-ap 8000
Description: Define complex apartness. Definition 6.1 of [Geuvers], p. 17.

Two numbers are considered apart if it is possible to separate them. One common usage is that we can divide by a number if it is apart from zero (see for example recclap 8085 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8023), symmetry (apsym 8024), and cotransitivity (apcotr 8025). Apartness implies negated equality, as seen at apne 8041, and the converse would also follow if we assumed excluded middle.

In addition, apartness of complex numbers is tight, which means that two numbers which are not apart are equal (apti 8040).

(Contributed by Jim Kingdon, 26-Jan-2020.)

Assertion
Ref Expression
df-ap  |- #  =  { <. x ,  y >.  |  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) }
Distinct variable group:    s, r, t, u, x, y

Detailed syntax breakdown of Definition df-ap
StepHypRef Expression
1 cap 7999 . 2  class #
2 vx . . . . . . . . . . 11  setvar  x
32cv 1286 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . . 12  setvar  r
54cv 1286 . . . . . . . . . . 11  class  r
6 ci 7296 . . . . . . . . . . . 12  class  _i
7 vs . . . . . . . . . . . . 13  setvar  s
87cv 1286 . . . . . . . . . . . 12  class  s
9 cmul 7299 . . . . . . . . . . . 12  class  x.
106, 8, 9co 5613 . . . . . . . . . . 11  class  ( _i  x.  s )
11 caddc 7297 . . . . . . . . . . 11  class  +
125, 10, 11co 5613 . . . . . . . . . 10  class  ( r  +  ( _i  x.  s ) )
133, 12wceq 1287 . . . . . . . . 9  wff  x  =  ( r  +  ( _i  x.  s ) )
14 vy . . . . . . . . . . 11  setvar  y
1514cv 1286 . . . . . . . . . 10  class  y
16 vt . . . . . . . . . . . 12  setvar  t
1716cv 1286 . . . . . . . . . . 11  class  t
18 vu . . . . . . . . . . . . 13  setvar  u
1918cv 1286 . . . . . . . . . . . 12  class  u
206, 19, 9co 5613 . . . . . . . . . . 11  class  ( _i  x.  u )
2117, 20, 11co 5613 . . . . . . . . . 10  class  ( t  +  ( _i  x.  u ) )
2215, 21wceq 1287 . . . . . . . . 9  wff  y  =  ( t  +  ( _i  x.  u ) )
2313, 22wa 102 . . . . . . . 8  wff  ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )
24 creap 7992 . . . . . . . . . 10  class #
255, 17, 24wbr 3820 . . . . . . . . 9  wff  r #  t
268, 19, 24wbr 3820 . . . . . . . . 9  wff  s #  u
2725, 26wo 662 . . . . . . . 8  wff  ( r #  t  \/  s #  u )
2823, 27wa 102 . . . . . . 7  wff  ( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )
29 cr 7293 . . . . . . 7  class  RR
3028, 18, 29wrex 2356 . . . . . 6  wff  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3130, 16, 29wrex 2356 . . . . 5  wff  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3231, 7, 29wrex 2356 . . . 4  wff  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3332, 4, 29wrex 2356 . . 3  wff  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3433, 2, 14copab 3873 . 2  class  { <. x ,  y >.  |  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) }
351, 34wceq 1287 1  wff #  =  { <. x ,  y >.  |  E. r  e.  RR  E. s  e.  RR  E. t  e.  RR  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) ) }
Colors of variables: wff set class
This definition is referenced by:  apreap  8005  apreim  8021
  Copyright terms: Public domain W3C validator