Definition df-ap 7959

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 8044 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 7982), symmetry (apsym 7983), and cotransitivity (apcotr 7984). Apartness implies negated equality, as seen at apne 8000, 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 7999).

(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 7958	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1284	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1284	. . . . . . . . . . 11 class r
6		ci 7255	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1284	. . . . . . . . . . . 12 class s
9		cmul 7258	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5591	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7256	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5591	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1285	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1284	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1284	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1284	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5591	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5591	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1285	. . . . . . . . 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 7951	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3811	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 3811	. . . . . . . . 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 7252	. . . . . . 7 class RR
30	28, 18, 29	wrex 2354	. . . . . 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 2354	. . . . 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 2354	. . . 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 2354	. . 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 3864	. 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 1285	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  7964  apreim  7980