Metamath Proof Explorer


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 = ( ⊥𝑃𝐾 )
psubclset.c 𝐶 = ( PSubCl ‘ 𝐾 )
Assertion psubcliN ( ( 𝐾𝐷𝑋𝐶 ) → ( 𝑋𝐴 ∧ ( ‘ ( 𝑋 ) ) = 𝑋 ) )

Proof

Step Hyp Ref Expression
1 psubclset.a 𝐴 = ( Atoms ‘ 𝐾 )
2 psubclset.p = ( ⊥𝑃𝐾 )
3 psubclset.c 𝐶 = ( PSubCl ‘ 𝐾 )
4 1 2 3 ispsubclN ( 𝐾𝐷 → ( 𝑋𝐶 ↔ ( 𝑋𝐴 ∧ ( ‘ ( 𝑋 ) ) = 𝑋 ) ) )
5 4 biimpa ( ( 𝐾𝐷𝑋𝐶 ) → ( 𝑋𝐴 ∧ ( ‘ ( 𝑋 ) ) = 𝑋 ) )