Theorem cdlemeulpq

Description: Part of proof of Lemma E in Crawley p. 113. (Contributed by NM, 5-Dec-2012)

	Ref	Expression
Hypotheses	cdleme0.l	|- .<_ = ( le ` K )
	cdleme0.j	|- .\/ = ( join ` K )
	cdleme0.m	|- ./\ = ( meet ` K )
	cdleme0.a	|- A = ( Atoms ` K )
	cdleme0.h	|- H = ( LHyp ` K )
	cdleme0.u	|- U = ( ( P .\/ Q ) ./\ W )
Assertion	cdlemeulpq	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )

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		cdleme0.h	|- H = ( LHyp ` K )
6		cdleme0.u	|- U = ( ( P .\/ Q ) ./\ W )
7		simpll	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> K e. HL )
8	7	hllatd	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> K e. Lat )
9		simprl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> P e. A )
10		simprr	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> Q e. A )
11		eqid	|- ( Base ` K ) = ( Base ` K )
12	11 2 4	hlatjcl	|- ( ( K e. HL /\ P e. A /\ Q e. A ) -> ( P .\/ Q ) e. ( Base ` K ) )
13	7 9 10 12	syl3anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> ( P .\/ Q ) e. ( Base ` K ) )
14	11 5	lhpbase	|- ( W e. H -> W e. ( Base ` K ) )
15	14	ad2antlr	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> W e. ( Base ` K ) )
16	11 1 3	latmle1	|- ( ( K e. Lat /\ ( P .\/ Q ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
17	8 13 15 16	syl3anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> ( ( P .\/ Q ) ./\ W ) .<_ ( P .\/ Q ) )
18	6 17	eqbrtrid	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ Q e. A ) ) -> U .<_ ( P .\/ Q ) )