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 ) |
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 ) |