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Metamath Proof Explorer

Theorem lplnnleat

Description: A lattice plane cannot majorize an atom. (Contributed by NM, 14-Jul-2012)

Ref Expression
Hypotheses lplnnleat.l 
|- .<_ = ( le ` K )
lplnnleat.a 
|- A = ( Atoms ` K )
lplnnleat.p 
|- P = ( LPlanes ` K )
Assertion lplnnleat 
|- ( ( K e. HL /\ X e. P /\ Q e. A ) -> -. X .<_ Q )

Proof

Step Hyp Ref Expression
1 lplnnleat.l 
 |-  .<_ = ( le ` K )
2 lplnnleat.a 
 |-  A = ( Atoms ` K )
3 lplnnleat.p 
 |-  P = ( LPlanes ` K )
4 simp1 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> K e. HL )
5 simp2 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> X e. P )
6 simp3 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> Q e. A )
7 eqid 
 |-  ( join ` K ) = ( join ` K )
8 1 7 2 3 lplnnle2at 
 |-  ( ( K e. HL /\ ( X e. P /\ Q e. A /\ Q e. A ) ) -> -. X .<_ ( Q ( join ` K ) Q ) )
9 4 5 6 6 8 syl13anc 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> -. X .<_ ( Q ( join ` K ) Q ) )
10 7 2 hlatjidm 
 |-  ( ( K e. HL /\ Q e. A ) -> ( Q ( join ` K ) Q ) = Q )
11 10 3adant2 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> ( Q ( join ` K ) Q ) = Q )
12 11 breq2d 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> ( X .<_ ( Q ( join ` K ) Q ) <-> X .<_ Q ) )
13 9 12 mtbid 
 |-  ( ( K e. HL /\ X e. P /\ Q e. A ) -> -. X .<_ Q )