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Definition df-ap 8570
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 8667 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8593), symmetry (apsym 8594), and cotransitivity (apcotr 8595). Apartness implies negated equality, as seen at apne 8611, 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 8610).

(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 8569 . 2  class #
2 vx . . . . . . . . . . 11  setvar  x
32cv 1363 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . . 12  setvar  r
54cv 1363 . . . . . . . . . . 11  class  r
6 ci 7844 . . . . . . . . . . . 12  class  _i
7 vs . . . . . . . . . . . . 13  setvar  s
87cv 1363 . . . . . . . . . . . 12  class  s
9 cmul 7847 . . . . . . . . . . . 12  class  x.
106, 8, 9co 5897 . . . . . . . . . . 11  class  ( _i  x.  s )
11 caddc 7845 . . . . . . . . . . 11  class  +
125, 10, 11co 5897 . . . . . . . . . 10  class  ( r  +  ( _i  x.  s ) )
133, 12wceq 1364 . . . . . . . . 9  wff  x  =  ( r  +  ( _i  x.  s ) )
14 vy . . . . . . . . . . 11  setvar  y
1514cv 1363 . . . . . . . . . 10  class  y
16 vt . . . . . . . . . . . 12  setvar  t
1716cv 1363 . . . . . . . . . . 11  class  t
18 vu . . . . . . . . . . . . 13  setvar  u
1918cv 1363 . . . . . . . . . . . 12  class  u
206, 19, 9co 5897 . . . . . . . . . . 11  class  ( _i  x.  u )
2117, 20, 11co 5897 . . . . . . . . . 10  class  ( t  +  ( _i  x.  u ) )
2215, 21wceq 1364 . . . . . . . . 9  wff  y  =  ( t  +  ( _i  x.  u ) )
2313, 22wa 104 . . . . . . . 8  wff  ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )
24 creap 8562 . . . . . . . . . 10  class #
255, 17, 24wbr 4018 . . . . . . . . 9  wff  r #  t
268, 19, 24wbr 4018 . . . . . . . . 9  wff  s #  u
2725, 26wo 709 . . . . . . . 8  wff  ( r #  t  \/  s #  u )
2823, 27wa 104 . . . . . . 7  wff  ( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )
29 cr 7841 . . . . . . 7  class  RR
3028, 18, 29wrex 2469 . . . . . 6  wff  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3130, 16, 29wrex 2469 . . . . 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 2469 . . . 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 2469 . . 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 4078 . 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 1364 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  8575  apreim  8591  aprcl  8634  aptap  8638
  Copyright terms: Public domain W3C validator