Metamath Proof Explorer

Theorem 2polatN

Description: Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2polat.a 
|- A = ( Atoms ` K )
2polat.p 
|- P = ( _|_P ` K )
Assertion 2polatN 
|- ( ( K e. HL /\ Q e. A ) -> ( P ` ( P ` { Q } ) ) = { Q } )

Proof

Step Hyp Ref Expression
1 2polat.a 
 |-  A = ( Atoms ` K )
2 2polat.p 
 |-  P = ( _|_P ` K )
3 hlol 
 |-  ( K e. HL -> K e. OL )
4 eqid 
 |-  ( oc ` K ) = ( oc ` K )
5 eqid 
 |-  ( pmap ` K ) = ( pmap ` K )
6 4 1 5 2 polatN 
 |-  ( ( K e. OL /\ Q e. A ) -> ( P ` { Q } ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) )
7 3 6 sylan 
 |-  ( ( K e. HL /\ Q e. A ) -> ( P ` { Q } ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) )
8 7 fveq2d 
 |-  ( ( K e. HL /\ Q e. A ) -> ( P ` ( P ` { Q } ) ) = ( P ` ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) ) )
9 hlop 
 |-  ( K e. HL -> K e. OP )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 1 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
12 10 4 opoccl 
 |-  ( ( K e. OP /\ Q e. ( Base ` K ) ) -> ( ( oc ` K ) ` Q ) e. ( Base ` K ) )
13 9 11 12 syl2an 
 |-  ( ( K e. HL /\ Q e. A ) -> ( ( oc ` K ) ` Q ) e. ( Base ` K ) )
14 10 4 5 2 polpmapN 
 |-  ( ( K e. HL /\ ( ( oc ` K ) ` Q ) e. ( Base ` K ) ) -> ( P ` ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) ) )
15 13 14 syldan 
 |-  ( ( K e. HL /\ Q e. A ) -> ( P ` ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) ) )
16 10 4 opococ 
 |-  ( ( K e. OP /\ Q e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) = Q )
17 9 11 16 syl2an 
 |-  ( ( K e. HL /\ Q e. A ) -> ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) = Q )
18 17 fveq2d 
 |-  ( ( K e. HL /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) ) = ( ( pmap ` K ) ` Q ) )
19 1 5 pmapat 
 |-  ( ( K e. HL /\ Q e. A ) -> ( ( pmap ` K ) ` Q ) = { Q } )
20 18 19 eqtrd 
 |-  ( ( K e. HL /\ Q e. A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( oc ` K ) ` Q ) ) ) = { Q } )
21 15 20 eqtrd 
 |-  ( ( K e. HL /\ Q e. A ) -> ( P ` ( ( pmap ` K ) ` ( ( oc ` K ) ` Q ) ) ) = { Q } )
22 8 21 eqtrd 
 |-  ( ( K e. HL /\ Q e. A ) -> ( P ` ( P ` { Q } ) ) = { Q } )