Definition df-mp 9806

Description: Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9942, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)

Assertion

Ref	Expression
df-mp	|- .P. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v .Q u ) } )

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

Detailed syntax breakdown of Definition df-mp

Step	Hyp	Ref	Expression
1		cmp 9684	. 2 class .P.
2		vx	. . 3 setvar x
3		vy	. . 3 setvar y
4		cnp 9681	. . 3 class P.
5		vw	. . . . . . . 8 setvar w
6	5	cv 1482	. . . . . . 7 class w
7		vv	. . . . . . . . 9 setvar v
8	7	cv 1482	. . . . . . . 8 class v
9		vu	. . . . . . . . 9 setvar u
10	9	cv 1482	. . . . . . . 8 class u
11		cmq 9678	. . . . . . . 8 class .Q
12	8, 10, 11	co 6650	. . . . . . 7 class ( v .Q u )
13	6, 12	wceq 1483	. . . . . 6 wff w = ( v .Q u )
14	3	cv 1482	. . . . . 6 class y
15	13, 9, 14	wrex 2913	. . . . 5 wff E. u e. y w = ( v .Q u )
16	2	cv 1482	. . . . 5 class x
17	15, 7, 16	wrex 2913	. . . 4 wff E. v e. x E. u e. y w = ( v .Q u )
18	17, 5	cab 2608	. . 3 class { w | E. v e. x E. u e. y w = ( v .Q u ) }
19	2, 3, 4, 4, 18	cmpt2 6652	. 2 class ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v .Q u ) } )
20	1, 19	wceq 1483	1 wff .P. = ( x e. P. , y e. P. |-> { w | E. v e. x E. u e. y w = ( v .Q u ) } )

This definition is referenced by:  mpv  9833  dmmp  9835  mulclprlem  9841  mulclpr  9842  mulasspr  9846  distrlem1pr  9847  distrlem4pr  9848  distrlem5pr  9849  1idpr  9851  reclem3pr  9871  reclem4pr  9872