Theorem lhpat4N

Description: Property of an atom under a co-atom. (Contributed by NM, 24-Nov-2013) (New usage is discouraged.)

	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	lhpat4N	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )

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		simp1	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( K e. HL /\ W e. H ) )
7		simp2	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( P e. A /\ -. P .<_ W ) )
8		simp3l	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U e. A )
9		eqid	|- ( Base ` K ) = ( Base ` K )
10	9 4	atbase	|- ( U e. A -> U e. ( Base ` K ) )
11	8 10	syl	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U e. ( Base ` K ) )
12		simp3r	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> U .<_ W )
13	9 1 2 3 4 5	lhple	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. ( Base ` K ) /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )
14	6 7 11 12 13	syl112anc	|- ( ( ( K e. HL /\ W e. H ) /\ ( P e. A /\ -. P .<_ W ) /\ ( U e. A /\ U .<_ W ) ) -> ( ( P .\/ U ) ./\ W ) = U )