Metamath Proof Explorer
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 ) ) |