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Metamath Proof Explorer

Theorem lplnri3N

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 lplnri1.j 
|- .\/ = ( join ` K )
lplnri1.a 
|- A = ( Atoms ` K )
lplnri1.p 
|- P = ( LPlanes ` K )
lplnri1.y 
|- Y = ( ( Q .\/ R ) .\/ S )
Assertion lplnri3N 
|- ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> R =/= S )

Proof

Step Hyp Ref Expression
1 lplnri1.j 
 |-  .\/ = ( join ` K )
2 lplnri1.a 
 |-  A = ( Atoms ` K )
3 lplnri1.p 
 |-  P = ( LPlanes ` K )
4 lplnri1.y 
 |-  Y = ( ( Q .\/ R ) .\/ S )
5 simp1 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> K e. HL )
6 simp22 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> R e. A )
7 simp21 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> Q e. A )
8 simp23 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> S e. A )
9 eqid 
 |-  ( le ` K ) = ( le ` K )
10 9 1 2 3 4 lplnribN 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> -. R ( le ` K ) ( Q .\/ S ) )
11 9 1 2 atnlej2 
 |-  ( ( K e. HL /\ ( R e. A /\ Q e. A /\ S e. A ) /\ -. R ( le ` K ) ( Q .\/ S ) ) -> R =/= S )
12 5 6 7 8 10 11 syl131anc 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> R =/= S )