This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )