Definition df-mp 5101

Description: Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 5252, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124.

Assertion

Ref	Expression
df-mp	|- .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-mp

Step	Hyp	Ref	Expression
1		cmp 5000	. 2 class .P.
2		vx	. . . . . . 7 set x
3	2	cv 957	. . . . . 6 class x
4		cnp 4997	. . . . . 6 class P.
5	3, 4	wcel 960	. . . . 5 wff x e. P.
6		vy	. . . . . . 7 set y
7	6	cv 957	. . . . . 6 class y
8	7, 4	wcel 960	. . . . 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 957	. . . . 5 class z
12		vw	. . . . . . . . . 10 set w
13	12	cv 957	. . . . . . . . 9 class w
14		vv	. . . . . . . . . . 11 set v
15	14	cv 957	. . . . . . . . . 10 class v
16		vu	. . . . . . . . . . 11 set u
17	16	cv 957	. . . . . . . . . 10 class u
18		cmq 4994	. . . . . . . . . 10 class .Q
19	15, 17, 18	co 3969	. . . . . . . . 9 class (v .Q u)
20	13, 19	wceq 958	. . . . . . . 8 wff w = (v .Q u)
21	20, 16, 7	wrex 1649	. . . . . . 7 wff E.u e. y w = (v .Q u)
22	21, 14, 3	wrex 1649	. . . . . 6 wff E.v e. x E.u e. y w = (v .Q u)
23	22, 12	cab 1466	. . . . 5 class {w | E.v e. x E.u e. y w = (v .Q u)}
24	11, 23	wceq 958	. . . 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 3970	. 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 958	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:  mpv 5126  dmmp 5128  mulclprlem 5133  mulclpr 5134  mulasspr 5138  distrlem1pr 5139  distrlem2pr 5140  distrlem5pr 5143  1idpr 5145  reclem3pr 5170  reclem4pr 5171

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