Metamath Proof Explorer


Theorem polsubclN

Description: A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses polsubcl.a
|- A = ( Atoms ` K )
polsubcl.p
|- ._|_ = ( _|_P ` K )
polsubcl.c
|- C = ( PSubCl ` K )
Assertion polsubclN
|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. C )

Proof

Step Hyp Ref Expression
1 polsubcl.a
 |-  A = ( Atoms ` K )
2 polsubcl.p
 |-  ._|_ = ( _|_P ` K )
3 polsubcl.c
 |-  C = ( PSubCl ` K )
4 eqid
 |-  ( lub ` K ) = ( lub ` K )
5 eqid
 |-  ( oc ` K ) = ( oc ` K )
6 eqid
 |-  ( pmap ` K ) = ( pmap ` K )
7 4 5 1 6 2 polval2N
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) )
8 hlop
 |-  ( K e. HL -> K e. OP )
9 8 adantr
 |-  ( ( K e. HL /\ X C_ A ) -> K e. OP )
10 hlclat
 |-  ( K e. HL -> K e. CLat )
11 eqid
 |-  ( Base ` K ) = ( Base ` K )
12 11 1 atssbase
 |-  A C_ ( Base ` K )
13 sstr
 |-  ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) )
14 12 13 mpan2
 |-  ( X C_ A -> X C_ ( Base ` K ) )
15 11 4 clatlubcl
 |-  ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
16 10 14 15 syl2an
 |-  ( ( K e. HL /\ X C_ A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
17 11 5 opoccl
 |-  ( ( K e. OP /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
18 9 16 17 syl2anc
 |-  ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
19 11 6 3 pmapsubclN
 |-  ( ( K e. HL /\ ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. C )
20 18 19 syldan
 |-  ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. C )
21 7 20 eqeltrd
 |-  ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. C )