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Theorem chocvali 21803
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 21736 . 2  |-  A  C_  ~H
3 ocval 21784 . 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 2519    C_ wss 3094   ` cfv 4638  (class class class)co 5757   0cc0 8670   ~Hchil 21424    .ih csp 21427   CHcch 21434   _|_cort 21435
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 4081  ax-nul 4089  ax-pr 4152  ax-hilex 21504
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 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3769  df-br 3964  df-opab 4018  df-mpt 4019  df-id 4246  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fv 4654  df-ov 5760  df-sh 21711  df-ch 21726  df-oc 21756
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