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Definition df-cm 22162
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 22169 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
StepHypRef Expression
1 ccm 21516 . 2  class  C_H
2 vx . . . . . . 7  set  x
32cv 1622 . . . . . 6  class  x
4 cch 21509 . . . . . 6  class  CH
53, 4wcel 1684 . . . . 5  wff  x  e. 
CH
6 vy . . . . . . 7  set  y
76cv 1622 . . . . . 6  class  y
87, 4wcel 1684 . . . . 5  wff  y  e. 
CH
95, 8wa 358 . . . 4  wff  ( x  e.  CH  /\  y  e.  CH )
103, 7cin 3151 . . . . . 6  class  ( x  i^i  y )
11 cort 21510 . . . . . . . 8  class  _|_
127, 11cfv 5255 . . . . . . 7  class  ( _|_ `  y )
133, 12cin 3151 . . . . . 6  class  ( x  i^i  ( _|_ `  y
) )
14 chj 21513 . . . . . 6  class  vH
1510, 13, 14co 5858 . . . . 5  class  ( ( x  i^i  y )  vH  ( x  i^i  ( _|_ `  y
) ) )
163, 15wceq 1623 . . . 4  wff  x  =  ( ( x  i^i  y )  vH  (
x  i^i  ( _|_ `  y ) ) )
179, 16wa 358 . . 3  wff  ( ( x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) )
1817, 2, 6copab 4076 . 2  class  { <. x ,  y >.  |  ( ( x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
191, 18wceq 1623 1  wff  C_H  =  { <. x ,  y
>.  |  ( (
x  e.  CH  /\  y  e.  CH )  /\  x  =  (
( x  i^i  y
)  vH  ( x  i^i  ( _|_ `  y
) ) ) ) }
Colors of variables: wff set class
This definition is referenced by:  cmbr  22163
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