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 )