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 #ℝ t \/ s #ℝ u ) ) }

Distinct variable group:   s, r, t, u, x, y

Detailed syntax breakdown of Definition df-ap

Step	Hyp	Ref	Expression
1		cap 8569	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1363	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1363	. . . . . . . . . . 11 class r
6		ci 7844	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1363	. . . . . . . . . . . 12 class s
9		cmul 7847	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5897	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7845	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5897	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1364	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1363	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1363	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1363	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5897	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5897	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1364	. . . . . . . . 9 wff y = ( t + ( _i x. u ) )
23	13, 22	wa 104	. . . . . . . 8 wff ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) )
24		creap 8562	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4018	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 4018	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 709	. . . . . . . 8 wff ( r #ℝ t \/ s #ℝ u )
28	23, 27	wa 104	. . . . . . 7 wff ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
29		cr 7841	. . . . . . 7 class RR
30	28, 18, 29	wrex 2469	. . . . . 6 wff E. u e. RR ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
31	30, 16, 29	wrex 2469	. . . . 5 wff E. t e. RR E. u e. RR ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
32	31, 7, 29	wrex 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 #ℝ t \/ s #ℝ u ) )
33	32, 4, 29	wrex 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 #ℝ t \/ s #ℝ u ) )
34	33, 2, 14	copab 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 #ℝ t \/ s #ℝ u ) ) }
35	1, 34	wceq 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 #ℝ t \/ s #ℝ u ) ) }

This definition is referenced by:  apreap  8575  apreim  8591  aprcl  8634  aptap  8638