Definition df-ap 8000

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 8085 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8023), symmetry (apsym 8024), and cotransitivity (apcotr 8025). Apartness implies negated equality, as seen at apne 8041, 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 8040).

(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 7999	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1286	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1286	. . . . . . . . . . 11 class r
6		ci 7296	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1286	. . . . . . . . . . . 12 class s
9		cmul 7299	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5613	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7297	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5613	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1287	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1286	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1286	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1286	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5613	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5613	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1287	. . . . . . . . 9 wff y = ( t + ( _i x. u ) )
23	13, 22	wa 102	. . . . . . . 8 wff ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) )
24		creap 7992	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3820	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 3820	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 662	. . . . . . . 8 wff ( r #ℝ t \/ s #ℝ u )
28	23, 27	wa 102	. . . . . . 7 wff ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
29		cr 7293	. . . . . . 7 class RR
30	28, 18, 29	wrex 2356	. . . . . 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 2356	. . . . 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 2356	. . . 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 2356	. . 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 3873	. 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 1287	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  8005  apreim  8021