Definition df-cm 28442

Description: Define the commutes relation (on the Hilbert lattice). Definition of commutes in [Kalmbach] p. 20, who uses the notation xCy for "x commutes with y." See cmbri 28449 for membership relation. (Contributed by NM, 14-Jun-2004.) (New usage is discouraged.)

Assertion

Ref	Expression
df-cm	|- C_H = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) ) ) }

Distinct variable group:   x, y

Detailed syntax breakdown of Definition df-cm

Step	Hyp	Ref	Expression
1		ccm 27793	. 2 class C_H
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	3, 7	cin 3573	. . . . . 6 class ( x i^i y )
11		cort 27787	. . . . . . . 8 class _|_
12	7, 11	cfv 5888	. . . . . . 7 class ( _|_ ` y )
13	3, 12	cin 3573	. . . . . 6 class ( x i^i ( _|_ ` y ) )
14		chj 27790	. . . . . 6 class vH
15	10, 13, 14	co 6650	. . . . 5 class ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) )
16	3, 15	wceq 1483	. . . 4 wff x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) )
17	9, 16	wa 384	. . 3 wff ( ( x e. CH /\ y e. CH ) /\ x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) ) )
18	17, 2, 6	copab 4712	. 2 class { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) ) ) }
19	1, 18	wceq 1483	1 wff C_H = { <. x , y >. | ( ( x e. CH /\ y e. CH ) /\ x = ( ( x i^i y ) vH ( x i^i ( _|_ ` y ) ) ) ) }

This definition is referenced by:  cmbr  28443