Metamath Proof Explorer


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 )