Theorem lhpat

Description: Create an atom under a co-atom. Part of proof of Lemma B in Crawley p. 112. (Contributed by NM, 23-May-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 )
Assertion	lhpat	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) 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		simp1l	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> K e. HL )
7		simp2l	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> P e. A )
8		simp3l	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> Q e. A )
9		simp1r	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W e. H )
10		eqid	|- ( Base ` K ) = ( Base ` K )
11	10 5	lhpbase	|- ( W e. H -> W e. ( Base ` K ) )
12	9 11	syl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W e. ( Base ` K ) )
13		simp3r	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> P =/= Q )
14		eqid	|- ( 1. ` K ) = ( 1. ` K )
15		eqid	|- (
16	14 15 5	lhp1cvr	|- ( ( K e. HL /\ W e. H ) -> W (
17	16	3ad2ant1	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> W (
18		simp2r	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> -. P .<_ W )
19	10 1 2 3 14 15 4	1cvrat	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ W e. ( Base ` K ) ) /\ ( P =/= Q /\ W ( ( ( P .\/ Q ) ./\ W ) e. A )
20	6 7 8 12 13 17 18 19	syl133anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ P =/= Q ) ) -> ( ( P .\/ Q ) ./\ W ) e. A )