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Definition df-oc 26035
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 26063 and chocvali 26082 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 25712 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 25701 . . . 4  class  ~H
43cpw 3993 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1380 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1380 . . . . . . 7  class  z
9 csp 25704 . . . . . . 7  class  .ih
106, 8, 9co 6277 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9490 . . . . . 6  class  0
1210, 11wceq 1381 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1380 . . . . 5  class  x
1412, 7, 13wral 2791 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2795 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4491 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1381 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  26063
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