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Definition df-oc 24660
Description: Define orthogonal complement of a subset (usually a subspace) of Hilbert space. The orthogonal complement is the set of all vectors orthogonal to all vectors in the subset. See ocval 24688 and chocvali 24707 for its value. Textbooks usually denote this unary operation with the symbol  _|_ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation)  _|_ rather than introducing a new syntactical form. This lets us take advantage of the theorems about functions that we already have proved under set theory. Definition of [Mittelstaedt] p. 9. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.)
Assertion
Ref Expression
df-oc  |-  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Distinct variable group:    x, y, z

Detailed syntax breakdown of Definition df-oc
StepHypRef Expression
1 cort 24337 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 24326 . . . 4  class  ~H
43cpw 3865 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1368 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1368 . . . . . . 7  class  z
9 csp 24329 . . . . . . 7  class  .ih
106, 8, 9co 6096 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9287 . . . . . 6  class  0
1210, 11wceq 1369 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1368 . . . . 5  class  x
1412, 7, 13wral 2720 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2724 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4355 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1369 1  wff  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Colors of variables: wff setvar class
This definition is referenced by:  ocval  24688
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