HSE Home Hilbert Space Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  HSE Home  >  Th. List  >  ocval Structured version   Unicode version

Theorem ocval 26918
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 26637 . . 3  |-  ~H  e.  _V
21elpw2 4584 . 2  |-  ( H  e.  ~P ~H  <->  H  C_  ~H )
3 raleq 3025 . . . 4  |-  ( z  =  H  ->  ( A. y  e.  z 
( x  .ih  y
)  =  0  <->  A. y  e.  H  (
x  .ih  y )  =  0 ) )
43rabbidv 3072 . . 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 26890 . . 3  |-  _|_  =  ( z  e.  ~P ~H  |->  { x  e. 
~H  |  A. y  e.  z  ( x  .ih  y )  =  0 } )
61rabex 4571 . . 3  |-  { x  e.  ~H  |  A. y  e.  H  ( x  .ih  y )  =  0 }  e.  _V
74, 5, 6fvmpt 5960 . 2  |-  ( H  e.  ~P ~H  ->  ( _|_ `  H )  =  { x  e. 
~H  |  A. y  e.  H  ( x  .ih  y )  =  0 } )
82, 7sylbir 216 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 1437    e. wcel 1868   A.wral 2775   {crab 2779    C_ wss 3436   ~Pcpw 3979   ` cfv 5597  (class class class)co 6301   0cc0 9539   ~Hchil 26557    .ih csp 26560   _|_cort 26568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pr 4656  ax-hilex 26637
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4764  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-iota 5561  df-fun 5599  df-fv 5605  df-oc 26890
This theorem is referenced by:  ocel  26919  ocsh  26921  occon  26925  chocvali  26937
  Copyright terms: Public domain W3C validator