Definition df-plq0 7494

Description: Define addition on nonnegative fractions. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 2-Nov-2019.)

Assertion

Ref	Expression
df-plq0	|- +Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) ) }

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

Detailed syntax breakdown of Definition df-plq0

Step	Hyp	Ref	Expression
1		cplq0 7356	. 2 class +Q0
2		vx	. . . . . . 7 setvar x
3	2	cv 1363	. . . . . 6 class x
4		cnq0 7354	. . . . . 6 class Q0
5	3, 4	wcel 2167	. . . . 5 wff x e. Q0
6		vy	. . . . . . 7 setvar y
7	6	cv 1363	. . . . . 6 class y
8	7, 4	wcel 2167	. . . . 5 wff y e. Q0
9	5, 8	wa 104	. . . 4 wff ( x e. Q0 /\ y e. Q0 )
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 3625	. . . . . . . . . . . 12 class <. w , v >.
15		ceq0 7353	. . . . . . . . . . . 12 class ~Q0
16	14, 15	cec 6590	. . . . . . . . . . 11 class [ <. w , v >. ] ~Q0
17	3, 16	wceq 1364	. . . . . . . . . 10 wff x = [ <. w , v >. ] ~Q0
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 3625	. . . . . . . . . . . 12 class <. u , f >.
23	22, 15	cec 6590	. . . . . . . . . . 11 class [ <. u , f >. ] ~Q0
24	7, 23	wceq 1364	. . . . . . . . . 10 wff y = [ <. u , f >. ] ~Q0
25	17, 24	wa 104	. . . . . . . . 9 wff ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 )
26		vz	. . . . . . . . . . 11 setvar z
27	26	cv 1363	. . . . . . . . . 10 class z
28		comu 6472	. . . . . . . . . . . . . 14 class .o
29	11, 21, 28	co 5922	. . . . . . . . . . . . 13 class ( w .o f )
30	13, 19, 28	co 5922	. . . . . . . . . . . . 13 class ( v .o u )
31		coa 6471	. . . . . . . . . . . . 13 class +o
32	29, 30, 31	co 5922	. . . . . . . . . . . 12 class ( ( w .o f ) +o ( v .o u ) )
33	13, 21, 28	co 5922	. . . . . . . . . . . 12 class ( v .o f )
34	32, 33	cop 3625	. . . . . . . . . . 11 class <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >.
35	34, 15	cec 6590	. . . . . . . . . 10 class [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0
36	27, 35	wceq 1364	. . . . . . . . 9 wff z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0
37	25, 36	wa 104	. . . . . . . 8 wff ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 )
38	37, 20	wex 1506	. . . . . . 7 wff E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 )
39	38, 18	wex 1506	. . . . . 6 wff E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 )
40	39, 12	wex 1506	. . . . 5 wff E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 )
41	40, 10	wex 1506	. . . 4 wff E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 )
42	9, 41	wa 104	. . 3 wff ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) )
43	42, 2, 6, 26	coprab 5923	. 2 class { <. <. x , y >. , z >. | ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) ) }
44	1, 43	wceq 1364	1 wff +Q0 = { <. <. x , y >. , z >. | ( ( x e. Q0 /\ y e. Q0 ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~Q0 /\ y = [ <. u , f >. ] ~Q0 ) /\ z = [ <. ( ( w .o f ) +o ( v .o u ) ) , ( v .o f ) >. ] ~Q0 ) ) }

This definition is referenced by:  dfplq0qs  7497