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Metamath Proof Explorer

Theorem isline2

Description: Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012)

Ref Expression
Hypotheses isline2.j 
|- .\/ = ( join ` K )
isline2.a 
|- A = ( Atoms ` K )
isline2.n 
|- N = ( Lines ` K )
isline2.m 
|- M = ( pmap ` K )
Assertion isline2 
|- ( K e. Lat -> ( X e. N <-> E. p e. A E. q e. A ( p =/= q /\ X = ( M ` ( p .\/ q ) ) ) ) )

Proof

Step Hyp Ref Expression
1 isline2.j 
 |-  .\/ = ( join ` K )
2 isline2.a 
 |-  A = ( Atoms ` K )
3 isline2.n 
 |-  N = ( Lines ` K )
4 isline2.m 
 |-  M = ( pmap ` K )
5 eqid 
 |-  ( le ` K ) = ( le ` K )
6 5 1 2 3 isline 
 |-  ( K e. Lat -> ( X e. N <-> E. p e. A E. q e. A ( p =/= q /\ X = { r e. A | r ( le ` K ) ( p .\/ q ) } ) ) )
7 simpl 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> K e. Lat )
8 eqid 
 |-  ( Base ` K ) = ( Base ` K )
9 8 2 atbase 
 |-  ( p e. A -> p e. ( Base ` K ) )
10 9 ad2antrl 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> p e. ( Base ` K ) )
11 8 2 atbase 
 |-  ( q e. A -> q e. ( Base ` K ) )
12 11 ad2antll 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> q e. ( Base ` K ) )
13 8 1 latjcl 
 |-  ( ( K e. Lat /\ p e. ( Base ` K ) /\ q e. ( Base ` K ) ) -> ( p .\/ q ) e. ( Base ` K ) )
14 7 10 12 13 syl3anc 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> ( p .\/ q ) e. ( Base ` K ) )
15 8 5 2 4 pmapval 
 |-  ( ( K e. Lat /\ ( p .\/ q ) e. ( Base ` K ) ) -> ( M ` ( p .\/ q ) ) = { r e. A | r ( le ` K ) ( p .\/ q ) } )
16 14 15 syldan 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> ( M ` ( p .\/ q ) ) = { r e. A | r ( le ` K ) ( p .\/ q ) } )
17 16 eqeq2d 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> ( X = ( M ` ( p .\/ q ) ) <-> X = { r e. A | r ( le ` K ) ( p .\/ q ) } ) )
18 17 anbi2d 
 |-  ( ( K e. Lat /\ ( p e. A /\ q e. A ) ) -> ( ( p =/= q /\ X = ( M ` ( p .\/ q ) ) ) <-> ( p =/= q /\ X = { r e. A | r ( le ` K ) ( p .\/ q ) } ) ) )
19 18 2rexbidva 
 |-  ( K e. Lat -> ( E. p e. A E. q e. A ( p =/= q /\ X = ( M ` ( p .\/ q ) ) ) <-> E. p e. A E. q e. A ( p =/= q /\ X = { r e. A | r ( le ` K ) ( p .\/ q ) } ) ) )
20 6 19 bitr4d 
 |-  ( K e. Lat -> ( X e. N <-> E. p e. A E. q e. A ( p =/= q /\ X = ( M ` ( p .\/ q ) ) ) ) )