Metamath Proof Explorer


Theorem psubcli2N

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

Ref Expression
Hypotheses psubcli2.p = ( ⊥𝑃𝐾 )
psubcli2.c 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion psubcli2N ( ( 𝐾𝐷𝑋𝐶 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )

Proof

Step Hyp Ref Expression
1 psubcli2.p = ( ⊥𝑃𝐾 )
2 psubcli2.c 𝐶 = ( PSubCl ‘ 𝐾 )
3 eqid ( Atoms ‘ 𝐾 ) = ( Atoms ‘ 𝐾 )
4 3 1 2 ispsubclN ( 𝐾𝐷 → ( 𝑋𝐶 ↔ ( 𝑋 ⊆ ( Atoms ‘ 𝐾 ) ∧ ( ‘ ( 𝑋 ) ) = 𝑋 ) ) )
5 4 simplbda ( ( 𝐾𝐷𝑋𝐶 ) → ( ‘ ( 𝑋 ) ) = 𝑋 )