Metamath Proof Explorer

Theorem 2llnma1

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses 2llnm.l 
|- .<_ = ( le ` K )
2llnm.j 
|- .\/ = ( join ` K )
2llnm.m 
|- ./\ = ( meet ` K )
2llnm.a 
|- A = ( Atoms ` K )
Assertion 2llnma1 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )

Proof

Step Hyp Ref Expression
1 2llnm.l 
 |-  .<_ = ( le ` K )
2 2llnm.j 
 |-  .\/ = ( join ` K )
3 2llnm.m 
 |-  ./\ = ( meet ` K )
4 2llnm.a 
 |-  A = ( Atoms ` K )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
7 eqid 
 |-  ( Base ` K ) = ( Base ` K )
8 7 4 atbase 
 |-  ( P e. A -> P e. ( Base ` K ) )
9 6 8 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. ( Base ` K ) )
10 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
11 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
12 simp3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) )
13 2 4 hlatjcom 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) = ( Q .\/ P ) )
14 5 6 10 13 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
15 14 breq2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( R .<_ ( P .\/ Q ) <-> R .<_ ( Q .\/ P ) ) )
16 12 15 mtbid 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( Q .\/ P ) )
17 7 1 2 3 4 2llnma1b 
 |-  ( ( K e. HL /\ ( P e. ( Base ` K ) /\ Q e. A /\ R e. A ) /\ -. R .<_ ( Q .\/ P ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )
18 5 9 10 11 16 17 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> ( ( Q .\/ P ) ./\ ( Q .\/ R ) ) = Q )