Theorem polsubclN

Description: A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	polsubcl.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	polsubcl.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
	polsubcl.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion	polsubclN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		polsubcl.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		polsubcl.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3		polsubcl.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
4		eqid	⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
5		eqid	⊢ ( oc ‘ 𝐾 ) = ( oc ‘ 𝐾 )
6		eqid	⊢ ( pmap ‘ 𝐾 ) = ( pmap ‘ 𝐾 )
7	4 5 1 6 2	polval2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) = ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) )
8		hlop	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP )
9	8	adantr	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → 𝐾 ∈ OP )
10		hlclat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CLat )
11		eqid	⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12	11 1	atssbase	⊢ 𝐴 ⊆ ( Base ‘ 𝐾 )
13		sstr	⊢ ( ( 𝑋 ⊆ 𝐴 ∧ 𝐴 ⊆ ( Base ‘ 𝐾 ) ) → 𝑋 ⊆ ( Base ‘ 𝐾 ) )
14	12 13	mpan2	⊢ ( 𝑋 ⊆ 𝐴 → 𝑋 ⊆ ( Base ‘ 𝐾 ) )
15	11 4	clatlubcl	⊢ ( ( 𝐾 ∈ CLat ∧ 𝑋 ⊆ ( Base ‘ 𝐾 ) ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
16	10 14 15	syl2an	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) )
17	11 5	opoccl	⊢ ( ( 𝐾 ∈ OP ∧ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) )
18	9 16 17	syl2anc	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) )
19	11 6 3	pmapsubclN	⊢ ( ( 𝐾 ∈ HL ∧ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ∈ 𝐶 )
20	18 19	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ( pmap ‘ 𝐾 ) ‘ ( ( oc ‘ 𝐾 ) ‘ ( ( lub ‘ 𝐾 ) ‘ 𝑋 ) ) ) ∈ 𝐶 )
21	7 20	eqeltrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ) → ( ⊥ ‘ 𝑋 ) ∈ 𝐶 )