Metamath Proof Explorer

Theorem 4atlem0be

Description: Lemma for 4at . (Contributed by NM, 10-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem0be 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
5 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
6 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
7 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
8 simp3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) )
9 1 2 3 atnlej1 
 |-  ( ( K e. HL /\ ( R e. A /\ P e. A /\ Q e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
10 9 necomd 
 |-  ( ( K e. HL /\ ( R e. A /\ P e. A /\ Q e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R )
11 4 5 6 7 8 10 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P =/= R )