# Metamath Proof Explorer

## Theorem iscss2

Description: It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015)

Ref Expression
Hypotheses cssss.v
`|- V = ( Base ` W )`
cssss.c
`|- C = ( ClSubSp ` W )`
ocvcss.o
`|- ._|_ = ( ocv ` W )`
Assertion iscss2
`|- ( ( W e. PreHil /\ S C_ V ) -> ( S e. C <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )`

### Proof

Step Hyp Ref Expression
1 cssss.v
` |-  V = ( Base ` W )`
2 cssss.c
` |-  C = ( ClSubSp ` W )`
3 ocvcss.o
` |-  ._|_ = ( ocv ` W )`
4 3 2 iscss
` |-  ( W e. PreHil -> ( S e. C <-> S = ( ._|_ ` ( ._|_ ` S ) ) ) )`
` |-  ( ( W e. PreHil /\ S C_ V ) -> ( S e. C <-> S = ( ._|_ ` ( ._|_ ` S ) ) ) )`
` |-  ( ( W e. PreHil /\ S C_ V ) -> S C_ ( ._|_ ` ( ._|_ ` S ) ) )`
` |-  ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( S C_ ( ._|_ ` ( ._|_ ` S ) ) /\ ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )`
` |-  ( S C_ ( ._|_ ` ( ._|_ ` S ) ) -> ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )`
` |-  ( ( W e. PreHil /\ S C_ V ) -> ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )`
` |-  ( ( W e. PreHil /\ S C_ V ) -> ( S e. C <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )`