Metamath Proof Explorer

Theorem obsss

Description: An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015)

Ref Expression
Hypothesis obsss.v 
|- V = ( Base ` W )
Assertion obsss 
|- ( B e. ( OBasis ` W ) -> B C_ V )

Proof

Step Hyp Ref Expression
1 obsss.v 
 |-  V = ( 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_ V /\ ( 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 simp2bi 
 |-  ( B e. ( OBasis ` W ) -> B C_ V )