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Metamath Proof Explorer

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 )