Definition df-ap 8761

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 8858 which says that a number apart from zero has a reciprocal).

The defining characteristics of an apartness are irreflexivity (apirr 8784), symmetry (apsym 8785), and cotransitivity (apcotr 8786). Apartness implies negated equality, as seen at apne 8802, 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 8801).

(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 8760	. 2 class #
2		vx	. . . . . . . . . . 11 setvar x
3	2	cv 1396	. . . . . . . . . 10 class x
4		vr	. . . . . . . . . . . 12 setvar r
5	4	cv 1396	. . . . . . . . . . 11 class r
6		ci 8033	. . . . . . . . . . . 12 class _i
7		vs	. . . . . . . . . . . . 13 setvar s
8	7	cv 1396	. . . . . . . . . . . 12 class s
9		cmul 8036	. . . . . . . . . . . 12 class x.
10	6, 8, 9	co 6017	. . . . . . . . . . 11 class ( _i x. s )
11		caddc 8034	. . . . . . . . . . 11 class +
12	5, 10, 11	co 6017	. . . . . . . . . 10 class ( r + ( _i x. s ) )
13	3, 12	wceq 1397	. . . . . . . . 9 wff x = ( r + ( _i x. s ) )
14		vy	. . . . . . . . . . 11 setvar y
15	14	cv 1396	. . . . . . . . . 10 class y
16		vt	. . . . . . . . . . . 12 setvar t
17	16	cv 1396	. . . . . . . . . . 11 class t
18		vu	. . . . . . . . . . . . 13 setvar u
19	18	cv 1396	. . . . . . . . . . . 12 class u
20	6, 19, 9	co 6017	. . . . . . . . . . 11 class ( _i x. u )
21	17, 20, 11	co 6017	. . . . . . . . . 10 class ( t + ( _i x. u ) )
22	15, 21	wceq 1397	. . . . . . . . 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 8753	. . . . . . . . . 10 class #ℝ
25	5, 17, 24	wbr 4088	. . . . . . . . 9 wff r #ℝ t
26	8, 19, 24	wbr 4088	. . . . . . . . 9 wff s #ℝ u
27	25, 26	wo 715	. . . . . . . 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 8030	. . . . . . 7 class RR
30	28, 18, 29	wrex 2511	. . . . . 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 2511	. . . . 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 2511	. . . 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 2511	. . 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 4149	. 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 1397	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  8766  apreim  8782  aprcl  8825  aptap  8829