Definition df-ap 8655

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 8752 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8678), symmetry (apsym 8679), and cotransitivity (apcotr 8680). Apartness implies negated equality, as seen at apne 8696, 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 8695).

(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 8654	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1372	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1372	. . . . . . . . . . 11 class r
6		ci 7927	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1372	. . . . . . . . . . . 12 class s
9		cmul 7930	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5944	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7928	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5944	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1373	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1372	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1372	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1372	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5944	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5944	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1373	. . . . . . . . 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 8647	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4044	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 4044	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 710	. . . . . . . 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 7924	. . . . . . 7 class RR
30	28, 18, 29	wrex 2485	. . . . . 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 2485	. . . . 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 2485	. . . 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 2485	. . 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 4104	. 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 1373	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  8660  apreim  8676  aprcl  8719  aptap  8723