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Metamath Proof Explorer

Theorem dochcl

Description: Closure of subspace orthocomplement for DVecH vector space. (Contributed by NM, 9-Mar-2014)

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

Proof

Step Hyp Ref Expression
1 dochcl.h 
 |-  H = ( LHyp ` K )
2 dochcl.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 dochcl.u 
 |-  U = ( ( DVecH ` K ) ` W )
4 dochcl.v 
 |-  V = ( Base ` U )
5 dochcl.o 
 |-  ._|_ = ( ( ocH ` K ) ` W )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 eqid 
 |-  ( glb ` K ) = ( glb ` K )
8 eqid 
 |-  ( oc ` K ) = ( oc ` K )
9 6 7 8 1 2 3 4 5 dochval 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) = ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) ) )
10 hlop 
 |-  ( K e. HL -> K e. OP )
11 10 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> K e. OP )
12 hlclat 
 |-  ( K e. HL -> K e. CLat )
13 12 ad2antrr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> K e. CLat )
14 ssrab2 
 |-  { y e. ( Base ` K ) | X C_ ( I ` y ) } C_ ( Base ` K )
15 6 7 clatglbcl 
 |-  ( ( K e. CLat /\ { y e. ( Base ` K ) | X C_ ( I ` y ) } C_ ( Base ` K ) ) -> ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) e. ( Base ` K ) )
16 13 14 15 sylancl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) e. ( Base ` K ) )
17 6 8 opoccl 
 |-  ( ( K e. OP /\ ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) e. ( Base ` K ) )
18 11 16 17 syl2anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) e. ( Base ` K ) )
19 6 1 2 dihcl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) e. ( Base ` K ) ) -> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) ) e. ran I )
20 18 19 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( I ` ( ( oc ` K ) ` ( ( glb ` K ) ` { y e. ( Base ` K ) | X C_ ( I ` y ) } ) ) ) e. ran I )
21 9 20 eqeltrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ V ) -> ( ._|_ ` X ) e. ran I )