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Theorem ocval 26325
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 26043 . . 3  |-  ~H  e.  _V
21elpw2 4620 . 2  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 raleq 3054 . . . 4  |-  ( z  =  H  ->  ( A. y  e.  z 
( x  .ih  y
)  =  0  <->  A. y  e.  H  (
x  .ih  y )  =  0 ) )
43rabbidv 3101 . . 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 26297 . . 3  |-  _|_  =  ( z  e.  ~P ~H  |->  { x  e. 
~H  |  A. y  e.  z  ( x  .ih  y )  =  0 } )
61rabex 4607 . . 3  |-  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 }  e.  _V
74, 5, 6fvmpt 5956 . 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 1395    e. wcel 1819   A.wral 2807   {crab 2811    C_ wss 3471   ~Pcpw 4015   ` cfv 5594  (class class class)co 6296   0cc0 9509   ~Hchil 25963    .ih csp 25966   _|_cort 25974
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-hilex 26043
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-oc 26297
This theorem is referenced by:  ocel  26326  ocsh  26328  occon  26332  chocvali  26344
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