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Definition df-oc 21847
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 21875 and chocvali 21894 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 21526 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21515 . . . 4  class  ~H
43cpw 3638 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1631 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1631 . . . . . . 7  class  z
9 csp 21518 . . . . . . 7  class  .ih
106, 8, 9co 5874 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8753 . . . . . 6  class  0
1210, 11wceq 1632 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1631 . . . . 5  class  x
1412, 7, 13wral 2556 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2560 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4093 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1632 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  21875
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