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Metamath Proof Explorer

Theorem linepmap

Description: A line described with a projective map. (Contributed by NM, 3-Feb-2012)

Ref Expression
Hypotheses isline2.j 
|- .\/ = ( join ` K )
isline2.a 
|- A = ( Atoms ` K )
isline2.n 
|- N = ( Lines ` K )
isline2.m 
|- M = ( pmap ` K )
Assertion linepmap 
|- ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( M ` ( P .\/ Q ) ) e. N )

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 simpl1 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> K e. Lat )
6 simpl2 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P e. A )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 2 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
9 6 8 syl 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> P e. ( Base ` K ) )
10 simpl3 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> Q e. A )
11 7 2 atbase 
 |-  ( Q e. A -> Q e. ( Base ` K ) )
12 10 11 syl 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> Q e. ( Base ` K ) )
13 7 1 latjcl 
 |-  ( ( K e. Lat /\ P e. ( Base ` K ) /\ Q e. ( Base ` K ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
14 5 9 12 13 syl3anc 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( P .\/ Q ) e. ( Base ` K ) )
15 eqid 
 |-  ( le ` K ) = ( le ` K )
16 7 15 2 4 pmapval 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) ) -> ( M ` ( P .\/ Q ) ) = { r e. A | r ( le ` K ) ( P .\/ Q ) } )
17 5 14 16 syl2anc 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( M ` ( P .\/ Q ) ) = { r e. A | r ( le ` K ) ( P .\/ Q ) } )
18 eqid 
 |-  { r e. A | r ( le ` K ) ( P .\/ Q ) } = { r e. A | r ( le ` K ) ( P .\/ Q ) }
19 15 1 2 3 islinei 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ ( P =/= Q /\ { r e. A | r ( le ` K ) ( P .\/ Q ) } = { r e. A | r ( le ` K ) ( P .\/ Q ) } ) ) -> { r e. A | r ( le ` K ) ( P .\/ Q ) } e. N )
20 18 19 mpanr2 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> { r e. A | r ( le ` K ) ( P .\/ Q ) } e. N )
21 17 20 eqeltrd 
 |-  ( ( ( K e. Lat /\ P e. A /\ Q e. A ) /\ P =/= Q ) -> ( M ` ( P .\/ Q ) ) e. N )