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Theorem obsocv 16955
Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
Hypotheses
Ref Expression
obsocv.z  |-  .0.  =  ( 0g `  W )
obsocv.o  |-  ._|_  =  ( ocv `  W )
Assertion
Ref Expression
obsocv  |-  ( B  e.  (OBasis `  W
)  ->  (  ._|_  `  B )  =  {  .0.  } )

Proof of Theorem obsocv
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2438 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2438 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2438 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2438 . . . 4  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
5 eqid 2438 . . . 4  |-  ( 0g
`  (Scalar `  W )
)  =  ( 0g
`  (Scalar `  W )
)
6 obsocv.o . . . 4  |-  ._|_  =  ( ocv `  W )
7 obsocv.z . . . 4  |-  .0.  =  ( 0g `  W )
81, 2, 3, 4, 5, 6, 7isobs 16949 . . 3  |-  ( B  e.  (OBasis `  W
)  <->  ( W  e. 
PreHil  /\  B  C_  ( Base `  W )  /\  ( A. x  e.  B  A. y  e.  B  ( x ( .i
`  W ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  /\  (  ._|_  `  B
)  =  {  .0.  } ) ) )
98simp3bi 975 . 2  |-  ( B  e.  (OBasis `  W
)  ->  ( A. x  e.  B  A. y  e.  B  (
x ( .i `  W ) y )  =  if ( x  =  y ,  ( 1r `  (Scalar `  W ) ) ,  ( 0g `  (Scalar `  W ) ) )  /\  (  ._|_  `  B
)  =  {  .0.  } ) )
109simprd 451 1  |-  ( B  e.  (OBasis `  W
)  ->  (  ._|_  `  B )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   A.wral 2707    C_ wss 3322   ifcif 3741   {csn 3816   ` cfv 5456  (class class class)co 6083   Basecbs 13471  Scalarcsca 13534   .icip 13536   0gc0g 13725   1rcur 15664   PreHilcphl 16857   ocvcocv 16889  OBasiscobs 16931
This theorem is referenced by:  obs2ocv  16956  obs2ss  16958
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-obs 16934
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