HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  df-oc Structured version   Visualization version   GIF version

Definition df-oc 30505
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 30533 and chocvali 30552 for its value. Textbooks usually denote this unary operation with the symbol โŠฅ as a small superscript, although Mittelstaedt uses the symbol as a prefix operation. Here we define a function (prefix operation) โŠฅ rather than introducing a new syntactic 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 โŠฅ = (๐‘ฅ โˆˆ ๐’ซ โ„‹ โ†ฆ {๐‘ฆ โˆˆ โ„‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ฆ ยทih ๐‘ง) = 0})
Distinct variable group:   ๐‘ฅ,๐‘ฆ,๐‘ง

Detailed syntax breakdown of Definition df-oc
StepHypRef Expression
1 cort 30183 . 2 class โŠฅ
2 vx . . 3 setvar ๐‘ฅ
3 chba 30172 . . . 4 class โ„‹
43cpw 4603 . . 3 class ๐’ซ โ„‹
5 vy . . . . . . . 8 setvar ๐‘ฆ
65cv 1541 . . . . . . 7 class ๐‘ฆ
7 vz . . . . . . . 8 setvar ๐‘ง
87cv 1541 . . . . . . 7 class ๐‘ง
9 csp 30175 . . . . . . 7 class ยทih
106, 8, 9co 7409 . . . . . 6 class (๐‘ฆ ยทih ๐‘ง)
11 cc0 11110 . . . . . 6 class 0
1210, 11wceq 1542 . . . . 5 wff (๐‘ฆ ยทih ๐‘ง) = 0
132cv 1541 . . . . 5 class ๐‘ฅ
1412, 7, 13wral 3062 . . . 4 wff โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ฆ ยทih ๐‘ง) = 0
1514, 5, 3crab 3433 . . 3 class {๐‘ฆ โˆˆ โ„‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ฆ ยทih ๐‘ง) = 0}
162, 4, 15cmpt 5232 . 2 class (๐‘ฅ โˆˆ ๐’ซ โ„‹ โ†ฆ {๐‘ฆ โˆˆ โ„‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ฆ ยทih ๐‘ง) = 0})
171, 16wceq 1542 1 wff โŠฅ = (๐‘ฅ โˆˆ ๐’ซ โ„‹ โ†ฆ {๐‘ฆ โˆˆ โ„‹ โˆฃ โˆ€๐‘ง โˆˆ ๐‘ฅ (๐‘ฆ ยทih ๐‘ง) = 0})
Colors of variables: wff setvar class
This definition is referenced by:  ocval  30533
  Copyright terms: Public domain W3C validator