Metamath Proof Explorer

Theorem obsocv

Description: An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015)

Ref Expression
Hypotheses obsocv.z 
|- .0. = ( 0g ` W )
obsocv.o 
|- ._|_ = ( ocv ` W )
Assertion obsocv 
|- ( B e. ( OBasis ` W ) -> ( ._|_ ` B ) = { .0. } )

Proof

Step Hyp Ref Expression
1 obsocv.z 
 |-  .0. = ( 0g ` W )
2 obsocv.o 
 |-  ._|_ = ( ocv ` W )
3 eqid 
 |-  ( Base ` W ) = ( Base ` W )
4 eqid 
 |-  ( .i ` W ) = ( .i ` W )
5 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
6 eqid 
 |-  ( 1r ` ( Scalar ` W ) ) = ( 1r ` ( Scalar ` W ) )
7 eqid 
 |-  ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
8 3 4 5 6 7 2 1 isobs 
 |-  ( 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. } ) ) )
9 8 simp3bi 
 |-  ( 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. } ) )
10 9 simprd 
 |-  ( B e. ( OBasis ` W ) -> ( ._|_ ` B ) = { .0. } )