Theorem psubcliN

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

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

Proof

Step	Hyp	Ref	Expression
1		psubclset.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
2		psubclset.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
3		psubclset.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
4	1 2 3	ispsubclN	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝐶 ↔ ( 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) )
5	4	biimpa	⊢ ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶 ) → ( 𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) )