Definition df-ap 8344

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 8439 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8367), symmetry (apsym 8368), and cotransitivity (apcotr 8369). Apartness implies negated equality, as seen at apne 8385, 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 8384).

(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 8343	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1330	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1330	. . . . . . . . . . 11 class r
6		ci 7622	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1330	. . . . . . . . . . . 12 class s
9		cmul 7625	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5774	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7623	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5774	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1331	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1330	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1330	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1330	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5774	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5774	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1331	. . . . . . . . 9 wff y = ( t + ( _i x. u ) )
23	13, 22	wa 103	. . . . . . . 8 wff ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) )
24		creap 8336	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3929	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 3929	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 697	. . . . . . . 8 wff ( r #ℝ t \/ s #ℝ u )
28	23, 27	wa 103	. . . . . . 7 wff ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
29		cr 7619	. . . . . . 7 class RR
30	28, 18, 29	wrex 2417	. . . . . 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 2417	. . . . 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 2417	. . . 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 2417	. . 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 3988	. 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 1331	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  8349  apreim  8365  aprcl  8408