Metamath Proof Explorer

Theorem linepsubclN

Description: A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012) (New usage is discouraged.)

Ref Expression
Hypotheses linepsubcl.n 
|- N = ( Lines ` K )
linepsubcl.c 
|- C = ( PSubCl ` K )
Assertion linepsubclN 
|- ( ( K e. HL /\ X e. N ) -> X e. C )

Proof

Step Hyp Ref Expression
1 linepsubcl.n 
 |-  N = ( Lines ` K )
2 linepsubcl.c 
 |-  C = ( PSubCl ` K )
3 hllat 
 |-  ( K e. HL -> K e. Lat )
4 eqid 
 |-  ( join ` K ) = ( join ` K )
5 eqid 
 |-  ( Atoms ` K ) = ( Atoms ` K )
6 eqid 
 |-  ( pmap ` K ) = ( pmap ` K )
7 4 5 1 6 isline2 
 |-  ( K e. Lat -> ( X e. N <-> E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) ) ) )
8 3 7 syl 
 |-  ( K e. HL -> ( X e. N <-> E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) ) ) )
9 3 adantr 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> K e. Lat )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 5 atbase 
 |-  ( p e. ( Atoms ` K ) -> p e. ( Base ` K ) )
12 11 ad2antrl 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> p e. ( Base ` K ) )
13 10 5 atbase 
 |-  ( q e. ( Atoms ` K ) -> q e. ( Base ` K ) )
14 13 ad2antll 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> q e. ( Base ` K ) )
15 10 4 latjcl 
 |-  ( ( K e. Lat /\ p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( p ( join ` K ) q ) e. ( Base ` K ) )
16 9 12 14 15 syl3anc 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> ( p ( join ` K ) q ) e. ( Base ` K ) )
17 10 6 2 pmapsubclN 
 |-  ( ( K e. HL /\ ( p ( join ` K ) q ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) e. C )
18 16 17 syldan 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) e. C )
19 eleq1a 
 |-  ( ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) e. C -> ( X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) -> X e. C ) )
20 18 19 syl 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> ( X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) -> X e. C ) )
21 20 adantld 
 |-  ( ( K e. HL /\ ( p e. ( Atoms ` K ) /\ q e. ( Atoms ` K ) ) ) -> ( ( p =/= q /\ X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) ) -> X e. C ) )
22 21 rexlimdvva 
 |-  ( K e. HL -> ( E. p e. ( Atoms ` K ) E. q e. ( Atoms ` K ) ( p =/= q /\ X = ( ( pmap ` K ) ` ( p ( join ` K ) q ) ) ) -> X e. C ) )
23 8 22 sylbid 
 |-  ( K e. HL -> ( X e. N -> X e. C ) )
24 23 imp 
 |-  ( ( K e. HL /\ X e. N ) -> X e. C )