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 )