# 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 )`