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Theorem ocval 22772
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 22492 . . 3  |-  ~H  e.  _V
21elpw2 4356 . 2  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 raleq 2896 . . . 4  |-  ( z  =  H  ->  ( A. y  e.  z 
( x  .ih  y
)  =  0  <->  A. y  e.  H  (
x  .ih  y )  =  0 ) )
43rabbidv 2940 . . 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 22744 . . 3  |-  _|_  =  ( z  e.  ~P ~H  |->  { x  e. 
~H  |  A. y  e.  z  ( x  .ih  y )  =  0 } )
61rabex 4346 . . 3  |-  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 }  e.  _V
74, 5, 6fvmpt 5798 . 2  |-  ( H  e.  ~P ~H  ->  ( _|_ `  H )  =  { x  e. 
~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
82, 7sylbir 205 1  |-  ( H 
C_  ~H  ->  ( _|_ `  H )  =  {
x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   A.wral 2697   {crab 2701    C_ wss 3312   ~Pcpw 3791   ` cfv 5446  (class class class)co 6073   0cc0 8980   ~Hchil 22412    .ih csp 22415   _|_cort 22423
This theorem is referenced by:  ocel  22773  ocsh  22775  occon  22779  chocvali  22791
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-hilex 22492
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-oc 22744
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