Definition df-plp 3882

Description: Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 4034, 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 ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)})}

Distinct variable group(s):   x,y,z,w,v,u

Detailed syntax breakdown of Definition df-plp

Step	Hyp	Ref	Expression
1		cpp 3781	. 2 class +P
2		vx	. . . . . . 7 set x
3	2	cv 1089	. . . . . 6 class x
4		cnp 3779	. . . . . 6 class P
5	3, 4	wcel 1092	. . . . 5 wff x ∈ P
6		vy	. . . . . . 7 set y
7	6	cv 1089	. . . . . 6 class y
8	7, 4	wcel 1092	. . . . 5 wff y ∈ P
9	5, 8	wa 196	. . . 4 wff (x ∈ P ∧ y ∈ P)
10		vz	. . . . . 6 set z
11	10	cv 1089	. . . . 5 class z
12		vw	. . . . . . . . . 10 set w
13	12	cv 1089	. . . . . . . . 9 class w
14		vv	. . . . . . . . . . 11 set v
15	14	cv 1089	. . . . . . . . . 10 class v
16		vu	. . . . . . . . . . 11 set u
17	16	cv 1089	. . . . . . . . . 10 class u
18		cplq 3775	. . . . . . . . . 10 class +Q
19	15, 17, 18	co 3001	. . . . . . . . 9 class (v +Q u)
20	13, 19	wceq 1091	. . . . . . . 8 wff w = (v +Q u)
21	20, 16, 7	wrex 1202	. . . . . . 7 wff ∃u ∈ y w = (v +Q u)
22	21, 14, 3	wrex 1202	. . . . . 6 wff ∃v ∈ x ∃u ∈ y w = (v +Q u)
23	22, 12	cab 1090	. . . . 5 class {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)}
24	11, 23	wceq 1091	. . . 4 wff z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)}
25	9, 24	wa 196	. . 3 wff ((x ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)})
26	25, 2, 6, 10	copab2 3002	. 2 class {⟨⟨x, y⟩, z⟩∣((x ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)})}
27	1, 26	wceq 1091	1 wff +P = {⟨⟨x, y⟩, z⟩∣((x ∈ P ∧ y ∈ P) ∧ z = {w∣∃v ∈ x ∃u ∈ y w = (v +Q u)})}

This definition is referenced by:  plpv 3907  dmplp 3909  addclprlem2 3913  addclpr 3914  addasspr 3918  distrlem1pr 3921  distrlem2pr 3922  distrlem5pr 3925  ltaddpr 3934  ltexprlem6 3941  ltexprlem7 3942

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