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Theorem chocvali 22803
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 22736 . 2  |-  A  C_  ~H
3 ocval 22784 . 2  |-  ( A 
C_  ~H  ->  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 } )
42, 3ax-mp 8 1  |-  ( _|_ `  A )  =  {
x  e.  ~H  |  A. y  e.  A  ( x  .ih  y )  =  0 }
Colors of variables: wff set class
Syntax hints:    = wceq 1653    e. wcel 1726   A.wral 2707   {crab 2711    C_ wss 3322   ` cfv 5456  (class class class)co 6083   0cc0 8992   ~Hchil 22424    .ih csp 22427   CHcch 22434   _|_cort 22435
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405  ax-hilex 22504
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-sh 22711  df-ch 22726  df-oc 22756
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