Definition df-ap 8601

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 8698 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8624), symmetry (apsym 8625), and cotransitivity (apcotr 8626). Apartness implies negated equality, as seen at apne 8642, 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 8641).

(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 8600	. 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 7874	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1363	. . . . . . . . . . . 12 class s
9		cmul 7877	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5918	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7875	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5918	. . . . . . . . . 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 5918	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5918	. . . . . . . . . 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 8593	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4029	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 4029	. . . . . . . . 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 7871	. . . . . . 7 class RR
30	28, 18, 29	wrex 2473	. . . . . 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 2473	. . . . 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 2473	. . . 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 2473	. . 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 4089	. 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  8606  apreim  8622  aprcl  8665  aptap  8669