Theorem polsubN

Description: The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	polsubsp.a	|- A = ( Atoms ` K )
	polsubsp.s	|- S = ( PSubSp ` K )
	polsubsp.p	|- ._|_ = ( _|_P ` K )
Assertion	polsubN	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. S )

Proof

Step	Hyp	Ref	Expression
1		polsubsp.a	|- A = ( Atoms ` K )
2		polsubsp.s	|- S = ( PSubSp ` K )
3		polsubsp.p	|- ._|_ = ( _|_P ` K )
4		eqid	|- ( lub ` K ) = ( lub ` K )
5		eqid	|- ( oc ` K ) = ( oc ` K )
6		eqid	|- ( pmap ` K ) = ( pmap ` K )
7	4 5 1 6 3	polval2N	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) = ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) )
8		hllat	|- ( K e. HL -> K e. Lat )
9	8	adantr	|- ( ( K e. HL /\ X C_ A ) -> K e. Lat )
10		hlop	|- ( K e. HL -> K e. OP )
11	10	adantr	|- ( ( K e. HL /\ X C_ A ) -> K e. OP )
12		hlclat	|- ( K e. HL -> K e. CLat )
13		eqid	|- ( Base ` K ) = ( Base ` K )
14	13 1	atssbase	|- A C_ ( Base ` K )
15		sstr	|- ( ( X C_ A /\ A C_ ( Base ` K ) ) -> X C_ ( Base ` K ) )
16	14 15	mpan2	|- ( X C_ A -> X C_ ( Base ` K ) )
17	13 4	clatlubcl	|- ( ( K e. CLat /\ X C_ ( Base ` K ) ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
18	12 16 17	syl2an	|- ( ( K e. HL /\ X C_ A ) -> ( ( lub ` K ) ` X ) e. ( Base ` K ) )
19	13 5	opoccl	|- ( ( K e. OP /\ ( ( lub ` K ) ` X ) e. ( Base ` K ) ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
20	11 18 19	syl2anc	|- ( ( K e. HL /\ X C_ A ) -> ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) )
21	13 2 6	pmapsub	|- ( ( K e. Lat /\ ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) e. ( Base ` K ) ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. S )
22	9 20 21	syl2anc	|- ( ( K e. HL /\ X C_ A ) -> ( ( pmap ` K ) ` ( ( oc ` K ) ` ( ( lub ` K ) ` X ) ) ) e. S )
23	7 22	eqeltrd	|- ( ( K e. HL /\ X C_ A ) -> ( ._|_ ` X ) e. S )