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Theorem ocval 24828
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 24546 . . 3  |-  ~H  e.  _V
21elpw2 4557 . 2  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 raleq 3016 . . . 4  |-  ( z  =  H  ->  ( A. y  e.  z 
( x  .ih  y
)  =  0  <->  A. y  e.  H  (
x  .ih  y )  =  0 ) )
43rabbidv 3063 . . 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 24800 . . 3  |-  _|_  =  ( z  e.  ~P ~H  |->  { x  e. 
~H  |  A. y  e.  z  ( x  .ih  y )  =  0 } )
61rabex 4544 . . 3  |-  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 }  e.  _V
74, 5, 6fvmpt 5876 . 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 1370    e. wcel 1758   A.wral 2795   {crab 2799    C_ wss 3429   ~Pcpw 3961   ` cfv 5519  (class class class)co 6193   0cc0 9386   ~Hchil 24466    .ih csp 24469   _|_cort 24477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-hilex 24546
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-iota 5482  df-fun 5521  df-fv 5527  df-oc 24800
This theorem is referenced by:  ocel  24829  ocsh  24831  occon  24835  chocvali  24847
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