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Theorem ocval 26012
Description: Value of orthogonal complement of a subset of Hilbert space. (Contributed by NM, 7-Aug-2000.) (Revised by Mario Carneiro, 23-Dec-2013.) (New usage is discouraged.)
Assertion
Ref Expression
ocval  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
Distinct variable group:    x, y, H

Proof of Theorem ocval
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 ax-hilex 25730 . . 3  |-  ~H  e.  _V
21elpw2 4617 . 2  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 raleq 3063 . . . 4  |-  ( z  =  H  ->  ( A. y  e.  z 
( x  .ih  y
)  =  0  <->  A. y  e.  H  (
x  .ih  y )  =  0 ) )
43rabbidv 3110 . . 3  |-  ( z  =  H  ->  { x  e.  ~H  |  A. y  e.  z  ( x  .ih  y )  =  0 }  =  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
5 df-oc 25984 . . 3  |-  _|_  =  ( z  e.  ~P ~H  |->  { x  e. 
~H  |  A. y  e.  z  ( x  .ih  y )  =  0 } )
61rabex 4604 . . 3  |-  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 }  e.  _V
74, 5, 6fvmpt 5957 . 2  |-  ( H  e.  ~P ~H  ->  ( _|_ `  H )  =  { x  e. 
~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
82, 7sylbir 213 1  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2817   {crab 2821    C_ wss 3481   ~Pcpw 4016   ` cfv 5594  (class class class)co 6295   0cc0 9504   ~Hchil 25650    .ih csp 25653   _|_cort 25661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-hilex 25730
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-oc 25984
This theorem is referenced by:  ocel  26013  ocsh  26015  occon  26019  chocvali  26031
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