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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 )