Definition df-dmd 29140

Description: Define the dual modular pair relation (on the Hilbert lattice). Definition 1.1 of [MaedaMaeda] p. 1, who use the notation (x,y)M* for "the ordered pair <x,y> is a dual modular pair." See dmdbr 29158 for membership relation. (Contributed by NM, 27-Apr-2006.) (New usage is discouraged.)

Assertion

Ref	Expression
df-dmd	|- MH* = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) ) ) }

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-dmd

Step	Hyp	Ref	Expression
1		cdmd 27824	. 2 class MH*
2		vx	. . . . . . 7 setvar x
3	2	cv 1482	. . . . . 6 class x
4		cch 27786	. . . . . 6 class CH
5	3, 4	wcel 1990	. . . . 5 wff x e. CH
6		vy	. . . . . . 7 setvar y
7	6	cv 1482	. . . . . 6 class y
8	7, 4	wcel 1990	. . . . 5 wff y e. CH
9	5, 8	wa 384	. . . 4 wff ( x e. CH /\ y e. CH )
10		vz	. . . . . . . 8 setvar z
11	10	cv 1482	. . . . . . 7 class z
12	7, 11	wss 3574	. . . . . 6 wff y C_ z
13	11, 3	cin 3573	. . . . . . . 8 class ( z i^i x )
14		chj 27790	. . . . . . . 8 class vH
15	13, 7, 14	co 6650	. . . . . . 7 class ( ( z i^i x ) vH y )
16	3, 7, 14	co 6650	. . . . . . . 8 class ( x vH y )
17	11, 16	cin 3573	. . . . . . 7 class ( z i^i ( x vH y ) )
18	15, 17	wceq 1483	. . . . . 6 wff ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) )
19	12, 18	wi 4	. . . . 5 wff ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) )
20	19, 10, 4	wral 2912	. . . 4 wff A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) )
21	9, 20	wa 384	. . 3 wff ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) ) )
22	21, 2, 6	copab 4712	. 2 class { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) ) ) }
23	1, 22	wceq 1483	1 wff MH* = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ A. z e. CH ( y C_ z -> ( ( z i^i x ) vH y ) = ( z i^i ( x vH y ) ) ) ) }

This definition is referenced by:  dmdbr  29158