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 ) ) ) )
5 4 adantr 
 |-  ( ( W e. PreHil /\ S C_ V ) -> ( S e. C <-> S = ( ._|_ ` ( ._|_ ` S ) ) ) )
6 1 3 ocvocv 
 |-  ( ( W e. PreHil /\ S C_ V ) -> S C_ ( ._|_ ` ( ._|_ ` S ) ) )
7 eqss 
 |-  ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( S C_ ( ._|_ ` ( ._|_ ` S ) ) /\ ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )
8 7 baib 
 |-  ( S C_ ( ._|_ ` ( ._|_ ` S ) ) -> ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )
9 6 8 syl 
 |-  ( ( W e. PreHil /\ S C_ V ) -> ( S = ( ._|_ ` ( ._|_ ` S ) ) <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )
10 5 9 bitrd 
 |-  ( ( W e. PreHil /\ S C_ V ) -> ( S e. C <-> ( ._|_ ` ( ._|_ ` S ) ) C_ S ) )