Definition df-mqqs 7434

Description: Define multiplication on 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 9-2.4 of [Gleason] p. 119. (Contributed by NM, 24-Aug-1995.)

Assertion

Ref	Expression
df-mqqs	|- .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) }

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

Detailed syntax breakdown of Definition df-mqqs

Step	Hyp	Ref	Expression
1		cmq 7367	. 2 class .Q
2		vx	. . . . . . 7 setvar x
3	2	cv 1363	. . . . . 6 class x
4		cnq 7364	. . . . . 6 class Q.
5	3, 4	wcel 2167	. . . . 5 wff x e. Q.
6		vy	. . . . . . 7 setvar y
7	6	cv 1363	. . . . . 6 class y
8	7, 4	wcel 2167	. . . . 5 wff y e. Q.
9	5, 8	wa 104	. . . 4 wff ( x e. Q. /\ y e. Q. )
10		vw	. . . . . . . . . . . . . 14 setvar w
11	10	cv 1363	. . . . . . . . . . . . 13 class w
12		vv	. . . . . . . . . . . . . 14 setvar v
13	12	cv 1363	. . . . . . . . . . . . 13 class v
14	11, 13	cop 3626	. . . . . . . . . . . 12 class <. w , v >.
15		ceq 7363	. . . . . . . . . . . 12 class ~Q
16	14, 15	cec 6599	. . . . . . . . . . 11 class [ <. w , v >. ] ~Q
17	3, 16	wceq 1364	. . . . . . . . . 10 wff x = [ <. w , v >. ] ~Q
18		vu	. . . . . . . . . . . . . 14 setvar u
19	18	cv 1363	. . . . . . . . . . . . 13 class u
20		vf	. . . . . . . . . . . . . 14 setvar f
21	20	cv 1363	. . . . . . . . . . . . 13 class f
22	19, 21	cop 3626	. . . . . . . . . . . 12 class <. u , f >.
23	22, 15	cec 6599	. . . . . . . . . . 11 class [ <. u , f >. ] ~Q
24	7, 23	wceq 1364	. . . . . . . . . 10 wff y = [ <. u , f >. ] ~Q
25	17, 24	wa 104	. . . . . . . . 9 wff ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q )
26		vz	. . . . . . . . . . 11 setvar z
27	26	cv 1363	. . . . . . . . . 10 class z
28		cmpq 7361	. . . . . . . . . . . 12 class .pQ
29	14, 22, 28	co 5925	. . . . . . . . . . 11 class ( <. w , v >. .pQ <. u , f >. )
30	29, 15	cec 6599	. . . . . . . . . 10 class [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q
31	27, 30	wceq 1364	. . . . . . . . 9 wff z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q
32	25, 31	wa 104	. . . . . . . 8 wff ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q )
33	32, 20	wex 1506	. . . . . . 7 wff E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q )
34	33, 18	wex 1506	. . . . . 6 wff E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q )
35	34, 12	wex 1506	. . . . 5 wff E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q )
36	35, 10	wex 1506	. . . 4 wff E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q )
37	9, 36	wa 104	. . 3 wff ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) )
38	37, 2, 6, 26	coprab 5926	. 2 class { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) }
39	1, 38	wceq 1364	1 wff .Q = { <. <. x , y >. , z >. | ( ( x e. Q. /\ y e. Q. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q /\ y = [ <. u , f >. ] ~Q ) /\ z = [ ( <. w , v >. .pQ <. u , f >. ) ] ~Q ) ) }

This definition is referenced by:  mulpipqqs  7457  dmmulpq  7464