Theorem dochlss

Description: A subspace orthocomplement is a subspace of the DVecH vector space. (Contributed by NM, 22-Jul-2014)

	Ref	Expression
Hypotheses	dochlss.h	|- H = ( LHyp ` K )
	dochlss.u	|- U = ( ( DVecH ` K ) ` W )
	dochlss.v	|- V = ( Base ` U )
	dochlss.s	|- S = ( LSubSp ` U )
	dochlss.o	|- ._|_ = ( ( ocH ` K ) ` W )
Assertion	dochlss	|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. S )

Proof

Step	Hyp	Ref	Expression
1		dochlss.h	|- H = ( LHyp ` K )
2		dochlss.u	|- U = ( ( DVecH ` K ) ` W )
3		dochlss.v	|- V = ( Base ` U )
4		dochlss.s	|- S = ( LSubSp ` U )
5		dochlss.o	|- ._|_ = ( ( ocH ` K ) ` W )
6		eqid	|- ( ( DIsoH ` K ) ` W ) = ( ( DIsoH ` K ) ` W )
7	1 6 2 3 5	dochcl	|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. ran ( ( DIsoH ` K ) ` W ) )
8	1 2 6 4	dihrnlss	|- ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) e. ran ( ( DIsoH ` K ) ` W ) ) -> ( ._|_ ` X ) e. S )
9	7 8	syldan	|- ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. S )