Theorem lplnribN

Description: Property of a lattice plane expressed as the join of 3 atoms. (Contributed by NM, 30-Jul-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	islpln2a.l	|- .<_ = ( le ` K )
	islpln2a.j	|- .\/ = ( join ` K )
	islpln2a.a	|- A = ( Atoms ` K )
	islpln2a.p	|- P = ( LPlanes ` K )
	islpln2a.y	|- Y = ( ( Q .\/ R ) .\/ S )
Assertion	lplnribN	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. R .<_ ( Q .\/ S ) )

Proof

Step	Hyp	Ref	Expression
1		islpln2a.l	|- .<_ = ( le ` K )
2		islpln2a.j	|- .\/ = ( join ` K )
3		islpln2a.a	|- A = ( Atoms ` K )
4		islpln2a.p	|- P = ( LPlanes ` K )
5		islpln2a.y	|- Y = ( ( Q .\/ R ) .\/ S )
6	1 2 3	3noncolr1N	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> ( S =/= Q /\ -. R .<_ ( S .\/ Q ) ) )
7	6	simprd	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) -> -. R .<_ ( S .\/ Q ) )
8	7	3expia	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) -> -. R .<_ ( S .\/ Q ) ) )
9	1 2 3 4 5	islpln2ah	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P <-> ( Q =/= R /\ -. S .<_ ( Q .\/ R ) ) ) )
10	2 3	hlatjcom	|- ( ( K e. HL /\ Q e. A /\ S e. A ) -> ( Q .\/ S ) = ( S .\/ Q ) )
11	10	3adant3r2	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Q .\/ S ) = ( S .\/ Q ) )
12	11	breq2d	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( R .<_ ( Q .\/ S ) <-> R .<_ ( S .\/ Q ) ) )
13	12	notbid	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( -. R .<_ ( Q .\/ S ) <-> -. R .<_ ( S .\/ Q ) ) )
14	8 9 13	3imtr4d	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P -> -. R .<_ ( Q .\/ S ) ) )
15	14	3impia	|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. R .<_ ( Q .\/ S ) )