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Metamath Proof Explorer

Theorem lplnri1

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

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

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 eqid 
 |-  ( le ` K ) = ( le ` K )
6 5 1 2 3 4 islpln2ah 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) ) -> ( Y e. P <-> ( Q =/= R /\ -. S ( le ` K ) ( Q .\/ R ) ) ) )
7 6 biimp3a 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> ( Q =/= R /\ -. S ( le ` K ) ( Q .\/ R ) ) )
8 7 simpld 
 |-  ( ( K e. HL /\ ( Q e. A /\ R e. A /\ S e. A ) /\ Y e. P ) -> Q =/= R )