Metamath Proof Explorer

Theorem obsrcl

Description: Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015)

Ref Expression
Assertion obsrcl 
|- ( B e. ( OBasis ` W ) -> W e. PreHil )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Base ` W ) = ( Base ` W )
2 eqid 
 |-  ( .i ` W ) = ( .i ` W )
3 eqid 
 |-  ( Scalar ` W ) = ( Scalar ` W )
4 eqid 
 |-  ( 1r ` ( Scalar ` W ) ) = ( 1r ` ( Scalar ` W ) )
5 eqid 
 |-  ( 0g ` ( Scalar ` W ) ) = ( 0g ` ( Scalar ` W ) )
6 eqid 
 |-  ( ocv ` W ) = ( ocv ` W )
7 eqid 
 |-  ( 0g ` W ) = ( 0g ` W )
8 1 2 3 4 5 6 7 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 ) ) ) /\ ( ( ocv ` W ) ` B ) = { ( 0g ` W ) } ) ) )
9 8 simp1bi 
 |-  ( B e. ( OBasis ` W ) -> W e. PreHil )