Definition df-plp 5088

Description: Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5240, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123.

Assertion

Ref	Expression
df-plp	|- +P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})}

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

Detailed syntax breakdown of Definition df-plp

Step	Hyp	Ref	Expression
1		cpp 4987	. 2 class +P.
2		vx	. . . . . . 7 set x
3	2	cv 955	. . . . . 6 class x
4		cnp 4985	. . . . . 6 class P.
5	3, 4	wcel 958	. . . . 5 wff x e. P.
6		vy	. . . . . . 7 set y
7	6	cv 955	. . . . . 6 class y
8	7, 4	wcel 958	. . . . 5 wff y e. P.
9	5, 8	wa 223	. . . 4 wff (x e. P. /\ y e. P.)
10		vz	. . . . . 6 set z
11	10	cv 955	. . . . 5 class z
12		vw	. . . . . . . . . 10 set w
13	12	cv 955	. . . . . . . . 9 class w
14		vv	. . . . . . . . . . 11 set v
15	14	cv 955	. . . . . . . . . 10 class v
16		vu	. . . . . . . . . . 11 set u
17	16	cv 955	. . . . . . . . . 10 class u
18		cplq 4981	. . . . . . . . . 10 class +Q
19	15, 17, 18	co 3963	. . . . . . . . 9 class (v +Q u)
20	13, 19	wceq 956	. . . . . . . 8 wff w = (v +Q u)
21	20, 16, 7	wrex 1646	. . . . . . 7 wff E.u e. y w = (v +Q u)
22	21, 14, 3	wrex 1646	. . . . . 6 wff E.v e. x E.u e. y w = (v +Q u)
23	22, 12	cab 1463	. . . . 5 class {w | E.v e. x E.u e. y w = (v +Q u)}
24	11, 23	wceq 956	. . . 4 wff z = {w | E.v e. x E.u e. y w = (v +Q u)}
25	9, 24	wa 223	. . 3 wff ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})
26	25, 2, 6, 10	copab2 3964	. 2 class {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})}
27	1, 26	wceq 956	1 wff +P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})}

This definition is referenced by:  plpv 5113  dmplp 5115  addclprlem2 5119  addclpr 5120  addasspr 5124  distrlem1pr 5127  distrlem2pr 5128  distrlem5pr 5131  ltaddpr 5140  ltexprlem6 5147  ltexprlem7 5148

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