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Definition df-oc 22742
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 22770 and chocvali 22789 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 22421 . 2  class  _|_
2 vx . . 3  set  x
3 chil 22410 . . . 4  class  ~H
43cpw 3791 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1651 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1651 . . . . . . 7  class  z
9 csp 22413 . . . . . . 7  class  .ih
106, 8, 9co 6072 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8979 . . . . . 6  class  0
1210, 11wceq 1652 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1651 . . . . 5  class  x
1412, 7, 13wral 2697 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2701 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4258 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1652 1  wff  _|_  =  ( x  e.  ~P ~H  |->  { y  e. 
~H  |  A. z  e.  x  ( y  .ih  z )  =  0 } )
Colors of variables: wff set class
This definition is referenced by:  ocval  22770
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