Definition df-plr 7812

Description: Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)

Assertion

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

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

Detailed syntax breakdown of Definition df-plr

Step	Hyp	Ref	Expression
1		cplr 7385	. 2 class +R
2		vx	. . . . . . 7 setvar x
3	2	cv 1363	. . . . . 6 class x
4		cnr 7381	. . . . . 6 class R.
5	3, 4	wcel 2167	. . . . 5 wff x e. R.
6		vy	. . . . . . 7 setvar y
7	6	cv 1363	. . . . . 6 class y
8	7, 4	wcel 2167	. . . . 5 wff y e. R.
9	5, 8	wa 104	. . . 4 wff ( x e. R. /\ y e. R. )
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		cer 7380	. . . . . . . . . . . 12 class ~R
16	14, 15	cec 6599	. . . . . . . . . . 11 class [ <. w , v >. ] ~R
17	3, 16	wceq 1364	. . . . . . . . . 10 wff x = [ <. w , v >. ] ~R
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 >. ] ~R
24	7, 23	wceq 1364	. . . . . . . . . 10 wff y = [ <. u , f >. ] ~R
25	17, 24	wa 104	. . . . . . . . 9 wff ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R )
26		vz	. . . . . . . . . . 11 setvar z
27	26	cv 1363	. . . . . . . . . 10 class z
28		cpp 7377	. . . . . . . . . . . . 13 class +P.
29	11, 19, 28	co 5925	. . . . . . . . . . . 12 class ( w +P. u )
30	13, 21, 28	co 5925	. . . . . . . . . . . 12 class ( v +P. f )
31	29, 30	cop 3626	. . . . . . . . . . 11 class <. ( w +P. u ) , ( v +P. f ) >.
32	31, 15	cec 6599	. . . . . . . . . 10 class [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R
33	27, 32	wceq 1364	. . . . . . . . 9 wff z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R
34	25, 33	wa 104	. . . . . . . 8 wff ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R )
35	34, 20	wex 1506	. . . . . . 7 wff E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R )
36	35, 18	wex 1506	. . . . . 6 wff E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R )
37	36, 12	wex 1506	. . . . 5 wff E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R )
38	37, 10	wex 1506	. . . 4 wff E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R )
39	9, 38	wa 104	. . 3 wff ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) )
40	39, 2, 6, 26	coprab 5926	. 2 class { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) }
41	1, 40	wceq 1364	1 wff +R = { <. <. x , y >. , z >. | ( ( x e. R. /\ y e. R. ) /\ E. w E. v E. u E. f ( ( x = [ <. w , v >. ] ~R /\ y = [ <. u , f >. ] ~R ) /\ z = [ <. ( w +P. u ) , ( v +P. f ) >. ] ~R ) ) }

This definition is referenced by:  addsrpr  7829