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Metamath Proof Explorer

Theorem cdleme00a

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 14-Jun-2012)

Ref Expression
Hypotheses cdleme0.l 
|- .<_ = ( le ` K )
cdleme0.j 
|- .\/ = ( join ` K )
cdleme0.m 
|- ./\ = ( meet ` K )
cdleme0.a 
|- A = ( Atoms ` K )
Assertion cdleme00a 
|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )

Proof

Step Hyp Ref Expression
1 cdleme0.l 
 |-  .<_ = ( le ` K )
2 cdleme0.j 
 |-  .\/ = ( join ` K )
3 cdleme0.m 
 |-  ./\ = ( meet ` K )
4 cdleme0.a 
 |-  A = ( Atoms ` K )
5 simp1 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> K e. HL )
6 simp23 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R e. A )
7 simp21 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> P e. A )
8 simp22 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> Q e. A )
9 simp3 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> -. R .<_ ( P .\/ Q ) )
10 1 2 4 atnlej1 
 |-  ( ( K e. HL /\ ( R e. A /\ P e. A /\ Q e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )
11 5 6 7 8 9 10 syl131anc 
 |-  ( ( K e. HL /\ ( P e. A /\ Q e. A /\ R e. A ) /\ -. R .<_ ( P .\/ Q ) ) -> R =/= P )