Definition df-ap 8488

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 8583 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8511), symmetry (apsym 8512), and cotransitivity (apcotr 8513). Apartness implies negated equality, as seen at apne 8529, 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 8528).

(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 8487	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1347	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1347	. . . . . . . . . . 11 class r
6		ci 7763	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1347	. . . . . . . . . . . 12 class s
9		cmul 7766	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5850	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7764	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5850	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1348	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1347	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1347	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1347	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5850	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5850	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1348	. . . . . . . . 9 wff y = ( t + ( _i x. u ) )
23	13, 22	wa 103	. . . . . . . 8 wff ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) )
24		creap 8480	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3987	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 3987	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 703	. . . . . . . 8 wff ( r #ℝ t \/ s #ℝ u )
28	23, 27	wa 103	. . . . . . 7 wff ( ( x = ( r + ( _i x. s ) ) /\ y = ( t + ( _i x. u ) ) ) /\ ( r #ℝ t \/ s #ℝ u ) )
29		cr 7760	. . . . . . 7 class RR
30	28, 18, 29	wrex 2449	. . . . . 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 2449	. . . . 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 2449	. . . 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 2449	. . 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 4047	. 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 1348	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  8493  apreim  8509  aprcl  8552