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Metamath Proof Explorer

Theorem 2llnma2

Description: Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012)

Ref Expression
Hypotheses 2llnm.l 
|- .<_ = ( le ` K )
2llnm.j 
|- .\/ = ( join ` K )
2llnm.m 
|- ./\ = ( meet ` K )
2llnm.a 
|- A = ( Atoms ` K )
Assertion 2llnma2 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )

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 ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> K e. HL )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> P e. A )
7 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> R e. A )
8 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> Q e. A )
9 1 2 4 4atlem0ae 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> -. Q .<_ ( P .\/ R ) )
10 1 2 3 4 2llnma1 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A /\ Q e. A ) /\ -. Q .<_ ( P .\/ R ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )
11 5 6 7 8 9 10 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ ( P =/= Q /\ -. R .<_ ( P .\/ Q ) ) ) -> ( ( R .\/ P ) ./\ ( R .\/ Q ) ) = R )