HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  chocvali Unicode version

Theorem chocvali 21824
Description: Value of the orthogonal complement of a Hilbert lattice element. The orthogonal complement of  A is the set of vectors that are orthogonal to all vectors in  A. (Contributed by NM, 8-Aug-2004.) (New usage is discouraged.)
Hypothesis
Ref Expression
chocval.1  |-  A  e. 
CH
Assertion
Ref Expression
chocvali  |-  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
Distinct variable group:    x, y, A

Proof of Theorem chocvali
StepHypRef Expression
1 chocval.1 . . 3  |-  A  e. 
CH
21chssii 21757 . 2  |-  A  C_  ~H
3 ocval 21805 . 2  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
42, 3ax-mp 10 1  |-  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
Colors of variables: wff set class
Syntax hints:    = wceq 1619    e. wcel 1621   A.wral 2516   {crab 2520    C_ wss 3113   ` cfv 4659  (class class class)co 5778   0cc0 8691   ~Hchil 21445    .ih csp 21448   CHcch 21455   _|_cort 21456
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4101  ax-nul 4109  ax-pr 4172  ax-hilex 21525
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-op 3609  df-uni 3788  df-br 3984  df-opab 4038  df-mpt 4039  df-id 4267  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fv 4675  df-ov 5781  df-sh 21732  df-ch 21747  df-oc 21777
  Copyright terms: Public domain W3C validator