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Definition df-oc 24590
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 24618 and chocvali 24637 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 24267 . 2  class  _|_
2 vx . . 3  setvar  x
3 chil 24256 . . . 4  class  ~H
43cpw 3857 . . 3  class  ~P ~H
5 vy . . . . . . . 8  setvar  y
65cv 1363 . . . . . . 7  class  y
7 vz . . . . . . . 8  setvar  z
87cv 1363 . . . . . . 7  class  z
9 csp 24259 . . . . . . 7  class  .ih
106, 8, 9co 6090 . . . . . 6  class  ( y 
.ih  z )
11 cc0 9278 . . . . . 6  class  0
1210, 11wceq 1364 . . . . 5  wff  ( y 
.ih  z )  =  0
132cv 1363 . . . . 5  class  x
1412, 7, 13wral 2713 . . . 4  wff  A. z  e.  x  ( y  .ih  z )  =  0
1514, 5, 3crab 2717 . . 3  class  { y  e.  ~H  |  A. z  e.  x  (
y  .ih  z )  =  0 }
162, 4, 15cmpt 4347 . 2  class  ( x  e.  ~P ~H  |->  { y  e.  ~H  |  A. z  e.  x  ( y  .ih  z
)  =  0 } )
171, 16wceq 1364 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  24618
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