Definition df-enq 7414

Description: Define equivalence relation for positive fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-2.1 of [Gleason] p. 117. (Contributed by NM, 27-Aug-1995.)

Assertion

Ref	Expression
df-enq	|- ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }

Distinct variable group:   x, y, z, w, v, u

Detailed syntax breakdown of Definition df-enq

Step	Hyp	Ref	Expression
1		ceq 7346	. 2 class ~Q
2		vx	. . . . . . 7 setvar x
3	2	cv 1363	. . . . . 6 class x
4		cnpi 7339	. . . . . . 7 class N.
5	4, 4	cxp 4661	. . . . . 6 class ( N. X. N. )
6	3, 5	wcel 2167	. . . . 5 wff x e. ( N. X. N. )
7		vy	. . . . . . 7 setvar y
8	7	cv 1363	. . . . . 6 class y
9	8, 5	wcel 2167	. . . . 5 wff y e. ( N. X. N. )
10	6, 9	wa 104	. . . 4 wff ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) )
11		vz	. . . . . . . . . . . . 13 setvar z
12	11	cv 1363	. . . . . . . . . . . 12 class z
13		vw	. . . . . . . . . . . . 13 setvar w
14	13	cv 1363	. . . . . . . . . . . 12 class w
15	12, 14	cop 3625	. . . . . . . . . . 11 class <. z , w >.
16	3, 15	wceq 1364	. . . . . . . . . 10 wff x = <. z , w >.
17		vv	. . . . . . . . . . . . 13 setvar v
18	17	cv 1363	. . . . . . . . . . . 12 class v
19		vu	. . . . . . . . . . . . 13 setvar u
20	19	cv 1363	. . . . . . . . . . . 12 class u
21	18, 20	cop 3625	. . . . . . . . . . 11 class <. v , u >.
22	8, 21	wceq 1364	. . . . . . . . . 10 wff y = <. v , u >.
23	16, 22	wa 104	. . . . . . . . 9 wff ( x = <. z , w >. /\ y = <. v , u >. )
24		cmi 7341	. . . . . . . . . . 11 class .N
25	12, 20, 24	co 5922	. . . . . . . . . 10 class ( z .N u )
26	14, 18, 24	co 5922	. . . . . . . . . 10 class ( w .N v )
27	25, 26	wceq 1364	. . . . . . . . 9 wff ( z .N u ) = ( w .N v )
28	23, 27	wa 104	. . . . . . . 8 wff ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) )
29	28, 19	wex 1506	. . . . . . 7 wff E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) )
30	29, 17	wex 1506	. . . . . 6 wff E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) )
31	30, 13	wex 1506	. . . . 5 wff E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) )
32	31, 11	wex 1506	. . . 4 wff E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) )
33	10, 32	wa 104	. . 3 wff ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) )
34	33, 2, 7	copab 4093	. 2 class { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }
35	1, 34	wceq 1364	1 wff ~Q = { <. x , y >. | ( ( x e. ( N. X. N. ) /\ y e. ( N. X. N. ) ) /\ E. z E. w E. v E. u ( ( x = <. z , w >. /\ y = <. v , u >. ) /\ ( z .N u ) = ( w .N v ) ) ) }

This definition is referenced by:  enqbreq  7423  enqer  7425  enqex  7427  addpipqqs  7437  mulpipqqs  7440  enq0enq  7498