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Metamath Proof Explorer

Theorem 2atnelpln

Description: The join of two atoms is not a lattice plane. (Contributed by NM, 16-Jul-2012)

Ref Expression
Hypotheses 2atnelpln.j 
|- .\/ = ( join ` K )
2atnelpln.a 
|- A = ( Atoms ` K )
2atnelpln.p 
|- P = ( LPlanes ` K )
Assertion 2atnelpln 
|- ( ( K e. HL /\ Q e. A /\ R e. A ) -> -. ( Q .\/ R ) e. P )

Proof

Step Hyp Ref Expression
1 2atnelpln.j 
 |-  .\/ = ( join ` K )
2 2atnelpln.a 
 |-  A = ( Atoms ` K )
3 2atnelpln.p 
 |-  P = ( LPlanes ` K )
4 hllat 
 |-  ( K e. HL -> K e. Lat )
5 4 3ad2ant1 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> K e. Lat )
6 eqid 
 |-  ( Base ` K ) = ( Base ` K )
7 6 1 2 hlatjcl 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) e. ( Base ` K ) )
8 eqid 
 |-  ( le ` K ) = ( le ` K )
9 6 8 latref 
 |-  ( ( K e. Lat /\ ( Q .\/ R ) e. ( Base ` K ) ) -> ( Q .\/ R ) ( le ` K ) ( Q .\/ R ) )
10 5 7 9 syl2anc 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( Q .\/ R ) ( le ` K ) ( Q .\/ R ) )
11 simpl1 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( Q .\/ R ) e. P ) -> K e. HL )
12 simpr 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( Q .\/ R ) e. P ) -> ( Q .\/ R ) e. P )
13 simpl2 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( Q .\/ R ) e. P ) -> Q e. A )
14 simpl3 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( Q .\/ R ) e. P ) -> R e. A )
15 8 1 2 3 lplnnle2at 
 |-  ( ( K e. HL /\ ( ( Q .\/ R ) e. P /\ Q e. A /\ R e. A ) ) -> -. ( Q .\/ R ) ( le ` K ) ( Q .\/ R ) )
16 11 12 13 14 15 syl13anc 
 |-  ( ( ( K e. HL /\ Q e. A /\ R e. A ) /\ ( Q .\/ R ) e. P ) -> -. ( Q .\/ R ) ( le ` K ) ( Q .\/ R ) )
17 16 ex 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> ( ( Q .\/ R ) e. P -> -. ( Q .\/ R ) ( le ` K ) ( Q .\/ R ) ) )
18 10 17 mt2d 
 |-  ( ( K e. HL /\ Q e. A /\ R e. A ) -> -. ( Q .\/ R ) e. P )