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Metamath Proof Explorer

Theorem pol1N

Description: The polarity of the whole projective subspace is the empty space. Remark in Holland95 p. 223. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses polssat.a 
|- A = ( Atoms ` K )
polssat.p 
|- ._|_ = ( _|_P ` K )
Assertion pol1N 
|- ( K e. HL -> ( ._|_ ` A ) = (/) )

Proof

Step Hyp Ref Expression
1 polssat.a 
 |-  A = ( Atoms ` K )
2 polssat.p 
 |-  ._|_ = ( _|_P ` K )
3 ssid 
 |-  A C_ A
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 /\ A C_ A ) -> ( ._|_ ` A ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) )
8 3 7 mpan2 
 |-  ( K e. HL -> ( ._|_ ` A ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) )
9 hlop 
 |-  ( K e. HL -> K e. OP )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 1 atbase 
 |-  ( p e. A -> p e. ( Base ` K ) )
12 eqid 
 |-  ( le ` K ) = ( le ` K )
13 eqid 
 |-  ( 1. ` K ) = ( 1. ` K )
14 10 12 13 ople1 
 |-  ( ( K e. OP /\ p e. ( Base ` K ) ) -> p ( le ` K ) ( 1. ` K ) )
15 9 11 14 syl2an 
 |-  ( ( K e. HL /\ p e. A ) -> p ( le ` K ) ( 1. ` K ) )
16 15 ralrimiva 
 |-  ( K e. HL -> A. p e. A p ( le ` K ) ( 1. ` K ) )
17 rabid2 
 |-  ( A = { p e. A | p ( le ` K ) ( 1. ` K ) } <-> A. p e. A p ( le ` K ) ( 1. ` K ) )
18 16 17 sylibr 
 |-  ( K e. HL -> A = { p e. A | p ( le ` K ) ( 1. ` K ) } )
19 18 fveq2d 
 |-  ( K e. HL -> ( ( lub ` K ) ` A ) = ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) )
20 hlomcmat 
 |-  ( K e. HL -> ( K e. OML /\ K e. CLat /\ K e. AtLat ) )
21 10 13 op1cl 
 |-  ( K e. OP -> ( 1. ` K ) e. ( Base ` K ) )
22 9 21 syl 
 |-  ( K e. HL -> ( 1. ` K ) e. ( Base ` K ) )
23 10 12 4 1 atlatmstc 
 |-  ( ( ( K e. OML /\ K e. CLat /\ K e. AtLat ) /\ ( 1. ` K ) e. ( Base ` K ) ) -> ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) = ( 1. ` K ) )
24 20 22 23 syl2anc 
 |-  ( K e. HL -> ( ( lub ` K ) ` { p e. A | p ( le ` K ) ( 1. ` K ) } ) = ( 1. ` K ) )
25 19 24 eqtr2d 
 |-  ( K e. HL -> ( 1. ` K ) = ( ( lub ` K ) ` A ) )
26 25 fveq2d 
 |-  ( K e. HL -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) )
27 eqid 
 |-  ( 0. ` K ) = ( 0. ` K )
28 27 13 5 opoc1 
 |-  ( K e. OP -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( 0. ` K ) )
29 9 28 syl 
 |-  ( K e. HL -> ( ( oc ` K ) ` ( 1. ` K ) ) = ( 0. ` K ) )
30 26 29 eqtr3d 
 |-  ( K e. HL -> ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) = ( 0. ` K ) )
31 30 fveq2d 
 |-  ( K e. HL -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` A ) ) ) = ( ( pmap ` K ) ` ( 0. ` K ) ) )
32 hlatl 
 |-  ( K e. HL -> K e. AtLat )
33 27 6 pmap0 
 |-  ( K e. AtLat -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
34 32 33 syl 
 |-  ( K e. HL -> ( ( pmap ` K ) ` ( 0. ` K ) ) = (/) )
35 8 31 34 3eqtrd 
 |-  ( K e. HL -> ( ._|_ ` A ) = (/) )