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Definition df-oc 21833
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 21861 and chocvali 21880 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 21512 . 2  class  _|_
2 vx . . 3  set  x
3 chil 21501 . . . 4  class  ~H
43cpw 3627 . . 3  class  ~P ~H
5 vy . . . . . . . 8  set  y
65cv 1624 . . . . . . 7  class  y
7 vz . . . . . . . 8  set  z
87cv 1624 . . . . . . 7  class  z
9 csp 21504 . . . . . . 7  class  .ih
106, 8, 9co 5860 . . . . . 6  class  ( y 
.ih  z )
11 cc0 8739 . . . . . 6  class  0
1210, 11wceq 1625 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1624 . . . . 5  class  x
1412, 7, 13wral 2545 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2549 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4079 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1625 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  21861
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