Definition df-ap 8821

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 8918 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8844), symmetry (apsym 8845), and cotransitivity (apcotr 8846). Apartness implies negated equality, as seen at apne 8862, 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 8861).

(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 8820	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1397	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1397	. . . . . . . . . . 11 class r
6		ci 8094	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1397	. . . . . . . . . . . 12 class s
9		cmul 8097	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 6028	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 8095	. . . . . . . . . . 11 class +
12	5, 10, 11	co 6028	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1398	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1397	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1397	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1397	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 6028	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 6028	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1398	. . . . . . . . 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 8813	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4093	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 4093	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 716	. . . . . . . 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 8091	. . . . . . 7 class RR
30	28, 18, 29	wrex 2512	. . . . . 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 2512	. . . . 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 2512	. . . 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 2512	. . 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 4154	. 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 1398	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  8826  apreim  8842  aprcl  8885  aptap  8889