Theorem psubclsubN

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	psubclsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
	psubclsub.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion	psubclsubN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ∈ 𝑆 )

Proof

Step	Hyp	Ref	Expression
1		psubclsub.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
2		psubclsub.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
3		eqid	⊢ ( ⊥𝑃 ‘ 𝐾 ) = ( ⊥𝑃 ‘ 𝐾 )
4	3 2	psubcli2N	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 )
5		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
6	5 3 2	psubcliN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ) = 𝑋 ) )
7	6	simpld	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ⊆ ( Atoms ‘ 𝐾 ) )
8	5 1 3	polsubN	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝑆 )
9	7 8	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝑆 )
10	5 1	psubssat	⊢ ( ( 𝐾 ∈ HL ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ∈ 𝑆 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) )
11	9 10	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) )
12	5 1 3	polsubN	⊢ ( ( 𝐾 ∈ HL ∧ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ⊆ ( Atoms ‘ 𝐾 ) ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ 𝑆 )
13	11 12	syldan	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → ( ( ⊥𝑃 ‘ 𝐾 ) ‘ ( ( ⊥𝑃 ‘ 𝐾 ) ‘ 𝑋 ) ) ∈ 𝑆 )
14	4 13	eqeltrrd	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ) → 𝑋 ∈ 𝑆 )