Theorem lhpat2

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 21-Nov-2012)

	Ref	Expression
Hypotheses	lhpat.l	|- .<_ = ( le ` K )
	lhpat.j	|- .\/ = ( join ` K )
	lhpat.m	|- ./\ = ( meet ` K )
	lhpat.a	|- A = ( Atoms ` K )
	lhpat.h	|- H = ( LHyp ` K )
	lhpat2.r	|- R = ( ( P .\/ Q ) ./\ W )
Assertion	lhpat2	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> R e. A )

Proof

Step	Hyp	Ref	Expression
1		lhpat.l	|- .<_ = ( le ` K )
2		lhpat.j	|- .\/ = ( join ` K )
3		lhpat.m	|- ./\ = ( meet ` K )
4		lhpat.a	|- A = ( Atoms ` K )
5		lhpat.h	|- H = ( LHyp ` K )
6		lhpat2.r	|- R = ( ( P .\/ Q ) ./\ W )
7	1 2 3 4 5	lhpat	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )
8	6 7	eqeltrid	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> R e. A )