Theorem dochssv

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

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

Proof

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