This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem 2atnelvolN

Description: The join of two atoms is not a lattice volume. (Contributed by NM, 17-Jul-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 3atnelvol.j 
|- .\/ = ( join ` K )
3atnelvol.a 
|- A = ( Atoms ` K )
3atnelvol.v 
|- V = ( LVols ` K )
Assertion 2atnelvolN 
|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> -. ( P .\/ Q ) e. V )

Proof

Step Hyp Ref Expression
1 3atnelvol.j 
 |-  .\/ = ( join ` K )
2 3atnelvol.a 
 |-  A = ( Atoms ` K )
3 3atnelvol.v 
 |-  V = ( LVols ` K )
4 1 2 hlatjidm 
 |-  ( ( K e. HL /\ P e. A ) -> ( P .\/ P ) = P )
5 4 3adant3 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ P ) = P )
6 5 oveq1d 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( ( P .\/ P ) .\/ Q ) = ( P .\/ Q ) )
7 simp1 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> K e. HL )
8 simp2 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> P e. A )
9 simp3 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> Q e. A )
10 1 2 3 3atnelvolN 
 |-  ( ( K e. HL /\ ( P e. A /\ P e. A /\ Q e. A ) ) -> -. ( ( P .\/ P ) .\/ Q ) e. V )
11 7 8 8 9 10 syl13anc 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> -. ( ( P .\/ P ) .\/ Q ) e. V )
12 6 11 eqneltrrd 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> -. ( P .\/ Q ) e. V )