This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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 ⊢ ⊥ = ( ⊥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 ) )