Theorem psubcli2N

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

	Ref	Expression
Hypotheses	psubcli2.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
	psubcli2.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion	psubcli2N	⊢ ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		psubcli2.p	⊢ ⊥ = ( ⊥𝑃 ‘ 𝐾 )
2		psubcli2.c	⊢ 𝐶 = ( PSubCl ‘ 𝐾 )
3		eqid	⊢ ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
4	3 1 2	ispsubclN	⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝐶 ↔ ( 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 ) ) )
5	4	simplbda	⊢ ( ( 𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑋 ) ) = 𝑋 )