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Metamath Proof Explorer

Theorem 3atnelvolN

Description: The join of 3 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 3atnelvolN 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> -. ( ( P .\/ Q ) .\/ R ) 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 hllat 
 |-  ( K e. HL -> K e. Lat )
5 4 adantr 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> K e. Lat )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 1 2 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
8 7 3adant3r3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
9 simpr3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. A )
10 6 2 atbase 
 |-  ( R e. A -> R e. ( Base ` K ) )
11 9 10 syl 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> R e. ( Base ` K ) )
12 6 1 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ R e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
13 5 8 11 12 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) )
14 eqid 
 |-  ( le ` K ) = ( le ` K )
15 6 14 latref 
 |-  ( ( K e. Lat /\ ( ( P .\/ Q ) .\/ R ) e. ( Base ` K ) ) -> ( ( P .\/ Q ) .\/ R ) ( le ` K ) ( ( P .\/ Q ) .\/ R ) )
16 5 13 15 syl2anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( P .\/ Q ) .\/ R ) ( le ` K ) ( ( P .\/ Q ) .\/ R ) )
17 14 1 2 3 lvolnle3at 
 |-  ( ( ( K e. HL /\ ( ( P .\/ Q ) .\/ R ) e. V ) /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> -. ( ( P .\/ Q ) .\/ R ) ( le ` K ) ( ( P .\/ Q ) .\/ R ) )
18 17 an32s 
 |-  ( ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) /\ ( ( P .\/ Q ) .\/ R ) e. V ) -> -. ( ( P .\/ Q ) .\/ R ) ( le ` K ) ( ( P .\/ Q ) .\/ R ) )
19 18 ex 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> ( ( ( P .\/ Q ) .\/ R ) e. V -> -. ( ( P .\/ Q ) .\/ R ) ( le ` K ) ( ( P .\/ Q ) .\/ R ) ) )
20 16 19 mt2d 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) ) -> -. ( ( P .\/ Q ) .\/ R ) e. V )