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Definition df-ap 8210
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 8300 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8233), symmetry (apsym 8234), and cotransitivity (apcotr 8235). Apartness implies negated equality, as seen at apne 8251, 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 8250).

(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 8209 . 2  class #
2 vx . . . . . . . . . . 11  setvar  x
32cv 1298 . . . . . . . . . 10  class  x
4 vr . . . . . . . . . . . 12  setvar  r
54cv 1298 . . . . . . . . . . 11  class  r
6 ci 7502 . . . . . . . . . . . 12  class  _i
7 vs . . . . . . . . . . . . 13  setvar  s
87cv 1298 . . . . . . . . . . . 12  class  s
9 cmul 7505 . . . . . . . . . . . 12  class  x.
106, 8, 9co 5706 . . . . . . . . . . 11  class  ( _i  x.  s )
11 caddc 7503 . . . . . . . . . . 11  class  +
125, 10, 11co 5706 . . . . . . . . . 10  class  ( r  +  ( _i  x.  s ) )
133, 12wceq 1299 . . . . . . . . 9  wff  x  =  ( r  +  ( _i  x.  s ) )
14 vy . . . . . . . . . . 11  setvar  y
1514cv 1298 . . . . . . . . . 10  class  y
16 vt . . . . . . . . . . . 12  setvar  t
1716cv 1298 . . . . . . . . . . 11  class  t
18 vu . . . . . . . . . . . . 13  setvar  u
1918cv 1298 . . . . . . . . . . . 12  class  u
206, 19, 9co 5706 . . . . . . . . . . 11  class  ( _i  x.  u )
2117, 20, 11co 5706 . . . . . . . . . 10  class  ( t  +  ( _i  x.  u ) )
2215, 21wceq 1299 . . . . . . . . 9  wff  y  =  ( t  +  ( _i  x.  u ) )
2313, 22wa 103 . . . . . . . 8  wff  ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )
24 creap 8202 . . . . . . . . . 10  class #
255, 17, 24wbr 3875 . . . . . . . . 9  wff  r #  t
268, 19, 24wbr 3875 . . . . . . . . 9  wff  s #  u
2725, 26wo 670 . . . . . . . 8  wff  ( r #  t  \/  s #  u )
2823, 27wa 103 . . . . . . 7  wff  ( ( x  =  ( r  +  ( _i  x.  s ) )  /\  y  =  ( t  +  ( _i  x.  u ) ) )  /\  ( r #  t  \/  s #  u ) )
29 cr 7499 . . . . . . 7  class  RR
3028, 18, 29wrex 2376 . . . . . 6  wff  E. u  e.  RR  ( ( x  =  ( r  +  ( _i  x.  s
) )  /\  y  =  ( t  +  ( _i  x.  u
) ) )  /\  ( r #  t  \/  s #  u
) )
3130, 16, 29wrex 2376 . . . . 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 2376 . . . 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 2376 . . 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 3928 . 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 1299 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  8215  apreim  8231
  Copyright terms: Public domain W3C validator