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 )