Definition df-ap 8210

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 8300 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8233), symmetry (apsym 8234), and cotransitivity (apcotr 8235). Apartness implies negated equality, as seen at apne 8251, 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 8250).

(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 8209	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1298	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1298	. . . . . . . . . . 11 class r
6		ci 7502	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1298	. . . . . . . . . . . 12 class s
9		cmul 7505	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 5706	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 7503	. . . . . . . . . . 11 class +
12	5, 10, 11	co 5706	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1299	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1298	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1298	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1298	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 5706	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 5706	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1299	. . . . . . . . 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 8202	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 3875	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 3875	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 670	. . . . . . . 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 7499	. . . . . . 7 class RR
30	28, 18, 29	wrex 2376	. . . . . 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 2376	. . . . 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 2376	. . . 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 2376	. . 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 3928	. 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 1299	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  8215  apreim  8231