Theorem lvolnelpln

Description: No lattice volume is a lattice plane. (Contributed by NM, 19-Jun-2012)

	Ref	Expression
Hypotheses	lvolnelpln.p	|- P = ( LPlanes ` K )
	lvolnelpln.v	|- V = ( LVols ` K )
Assertion	lvolnelpln	|- ( ( K e. HL /\ X e. V ) -> -. X e. P )

Proof

Step	Hyp	Ref	Expression
1		lvolnelpln.p	|- P = ( LPlanes ` K )
2		lvolnelpln.v	|- V = ( LVols ` K )
3		hllat	|- ( K e. HL -> K e. Lat )
4		eqid	|- ( Base ` K ) = ( Base ` K )
5	4 2	lvolbase	|- ( X e. V -> X e. ( Base ` K ) )
6		eqid	|- ( le ` K ) = ( le ` K )
7	4 6	latref	|- ( ( K e. Lat /\ X e. ( Base ` K ) ) -> X ( le ` K ) X )
8	3 5 7	syl2an	|- ( ( K e. HL /\ X e. V ) -> X ( le ` K ) X )
9	6 1 2	lvolnlelpln	|- ( ( K e. HL /\ X e. V /\ X e. P ) -> -. X ( le ` K ) X )
10	9	3expia	|- ( ( K e. HL /\ X e. V ) -> ( X e. P -> -. X ( le ` K ) X ) )
11	8 10	mt2d	|- ( ( K e. HL /\ X e. V ) -> -. X e. P )