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Theorem obsocv 16626
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 2283 . . . 4  |-  ( Base `  W )  =  (
Base `  W )
2 eqid 2283 . . . 4  |-  ( .i
`  W )  =  ( .i `  W
)
3 eqid 2283 . . . 4  |-  (Scalar `  W )  =  (Scalar `  W )
4 eqid 2283 . . . 4  |-  ( 1r
`  (Scalar `  W )
)  =  ( 1r
`  (Scalar `  W )
)
5 eqid 2283 . . . 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 16620 . . 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 972 . 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 449 1  |-  ( B  e.  (OBasis `  W
)  ->  (  ._|_  `  B )  =  {  .0.  } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   ifcif 3565   {csn 3640   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .icip 13213   0gc0g 13400   1rcur 15339   PreHilcphl 16528   ocvcocv 16560  OBasiscobs 16602
This theorem is referenced by:  obs2ocv  16627  obs2ss  16629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-obs 16605
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