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Theorem List for Metamath Proof Explorer - 35001-35100   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheorempclfvalN 35001* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))

TheorempclvalN 35002* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})

TheorempclclN 35003 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)

TheoremelpclN 35004* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)    &   𝑄 ∈ V       ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))

TheoremelpcliN 35005 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)

TheorempclssN 35006 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))

TheorempclssidN 35007 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))

TheorempclidN 35008 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)

TheorempclbtwnN 35009 A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))

TheorempclunN 35010 The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴𝑌𝐴) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))

Theorempclun2N 35011 The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))

TheorempclfinN 35012* The projective subspace closure of a set equals the union of the closures of its finite subsets. Analogous to Lemma 3.3.6 of [PtakPulmannova] p. 72. Compare the closed subspace version pclfinclN 35062. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑈𝑋) = 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦))

TheorempclcmpatN 35013* The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))

SyntaxcpolN 35014 Extend class notation with polarity of projective subspace \$m\$.
class 𝑃

Definitiondf-polarityN 35015* Define polarity of projective subspace, which is a kind of complement of the subspace. Item 2 in [Holland95] p. 222 bottom. For more generality, we define it for all subsets of atoms, not just projective subspaces. The intersection with Atoms‘𝑙 ensures it is defined when 𝑚 = ∅. (Contributed by NM, 23-Oct-2011.)
𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ 𝑝𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))

TheorempolfvalN 35016* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))

TheorempolvalN 35017* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))

Theorempolval2N 35018 Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))

TheorempolsubN 35019 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ 𝑆)

TheorempolssatN 35020 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)

Theorempol0N 35021 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       (𝐾𝐵 → ( ‘∅) = 𝐴)

Theorempol1N 35022 The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       (𝐾 ∈ HL → ( 𝐴) = ∅)

Theorem2pol0N 35023 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
= (⊥𝑃𝐾)       (𝐾 ∈ HL → ( ‘( ‘∅)) = ∅)

TheorempolpmapN 35024 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑃‘(𝑀𝑋)) = (𝑀‘( 𝑋)))

Theorem2polpmapN 35025 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → ( ‘( ‘(𝑀𝑋))) = (𝑀𝑋))

Theorem2polvalN 35026 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘( 𝑋)) = (𝑀‘(𝑈𝑋)))

Theorem2polssN 35027 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))

Theorem3polN 35028 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → ( ‘( ‘( 𝑆))) = ( 𝑆))

Theorempolcon3N 35029 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))

Theorem2polcon4bN 35030 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (( ‘( 𝑋)) ⊆ ( ‘( 𝑌)) ↔ ( 𝑌) ⊆ ( 𝑋)))

Theorempolcon2N 35031 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) → 𝑌 ⊆ ( 𝑋))

Theorempolcon2bN 35032 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 ⊆ ( 𝑌) ↔ 𝑌 ⊆ ( 𝑋)))

Theorempclss2polN 35033 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈𝑋) ⊆ ( ‘( 𝑋)))

Theorempcl0N 35034 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝑈 = (PCl‘𝐾)       (𝐾 ∈ HL → (𝑈‘∅) = ∅)

Theorempcl0bN 35035 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → ((𝑈𝑃) = ∅ ↔ 𝑃 = ∅))

TheorempmaplubN 35036 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘(𝑀𝑋)) = 𝑋)

TheoremsspmaplubN 35037 A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → 𝑆 ⊆ (𝑀‘(𝑈𝑆)))

Theorem2pmaplubN 35038 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈𝑆)))) = (𝑀‘(𝑈𝑆)))

TheorempaddunN 35039 The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 6193.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = ( ‘(𝑆𝑇)))

Theorempoldmj1N 35040 De Morgan's law for polarity of projective sum. (oldmj1 34334 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = (( 𝑆) ∩ ( 𝑇)))

Theorempmapj2N 35041 The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(𝑋 𝑌)) = ( ‘( ‘((𝑀𝑋) + (𝑀𝑌)))))

TheorempmapocjN 35042 The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)    &   𝑁 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘( ‘(𝑋 𝑌))) = (𝑁‘((𝐹𝑋) + (𝐹𝑌))))

TheorempolatN 35043 The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))

Theorem2polatN 35044 Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄})

TheorempnonsingN 35045 The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 ∩ (𝑃𝑋)) = ∅)

SyntaxcpscN 35046 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl

Definitiondf-psubclN 35047* Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})

TheorempsubclsetN 35048* The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})

TheoremispsubclN 35049 The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))

TheorempsubcliN 35050 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))

Theorempsubcli2N 35051 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
= (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)

TheorempsubclsubN 35052 A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)

TheorempsubclssatN 35053 A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)

TheorempmapidclN 35054 Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑀‘(𝑈𝑋)) = 𝑋)

Theorem0psubclN 35055 The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → ∅ ∈ 𝐶)

Theorem1psubclN 35056 The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → 𝐴𝐶)

TheorematpsubclN 35057 A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑄𝐴) → {𝑄} ∈ 𝐶)

TheorempmapsubclN 35058 A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐶)

Theoremispsubcl2N 35059* Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))

TheorempsubclinN 35060 The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) → (𝑋𝑌) ∈ 𝐶)

TheorempaddatclN 35061 The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶𝑄𝐴) → (𝑋 + {𝑄}) ∈ 𝐶)

TheorempclfinclN 35062 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 35012 and also pclcmpatN 35013. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)    &   𝑆 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑋 ∈ Fin) → (𝑈𝑋) ∈ 𝑆)

TheoremlinepsubclN 35063 A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝑁 = (Lines‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)

TheorempolsubclN 35064 A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ 𝐶)

Theorempoml4N 35065 Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → ((𝑋𝑌 ∧ ( ‘( 𝑌)) = 𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋))))

Theorempoml5N 35066 Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) → (( ‘(( 𝑋) ∩ ( 𝑌))) ∩ ( 𝑌)) = ( ‘( 𝑋)))

Theorempoml6N 35067 Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Theoremosumcllem1N 35068 Lemma for osumclN 35079. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → (𝑈𝑀) = 𝑀)

Theoremosumcllem2N 35069 Lemma for osumclN 35079. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → 𝑋 ⊆ (𝑈𝑀))

Theoremosumcllem3N 35070 Lemma for osumclN 35079. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       ((𝐾 ∈ HL ∧ 𝑌𝐶𝑋 ⊆ ( 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)

Theoremosumcllem4N 35071 Lemma for osumclN 35079. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) ∧ (𝑟𝑋𝑞𝑌)) → 𝑞𝑟)

Theoremosumcllem5N 35072 Lemma for osumclN 35079. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝐴 ∧ (𝑟𝑋𝑞𝑌𝑝 (𝑟 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem6N 35073 Lemma for osumclN 35079. Use atom exchange hlatexch1 34507 to swap 𝑝 and 𝑞. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑝𝐴) ∧ (𝑟𝑋𝑞𝑌𝑞 (𝑟 𝑝))) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem7N 35074* Lemma for osumclN 35079. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ 𝑞 ∈ (𝑌𝑀)) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem8N 35075 Lemma for osumclN 35079. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)

Theoremosumcllem9N 35076 Lemma for osumclN 35079. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)

Theoremosumcllem10N 35077 Lemma for osumclN 35079. Contradict osumcllem9N 35076. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝑋)

Theoremosumcllem11N 35078 Lemma for osumclN 35079. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
+ = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ‘( ‘(𝑋 + 𝑌))))

TheoremosumclN 35079 Closure of orthogonal sum. If 𝑋 and 𝑌 are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
+ = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋 ⊆ ( 𝑌)) → (𝑋 + 𝑌) ∈ 𝐶)

TheorempmapojoinN 35080 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 34964 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (oc‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 ( 𝑌)) → (𝑀‘(𝑋 𝑌)) = ((𝑀𝑋) + (𝑀𝑌)))

TheorempexmidN 35081 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 35065. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 35079. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Theorempexmidlem1N 35082 Lemma for pexmidN 35081. Holland's proof implicitly requires 𝑞𝑟, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋))) → 𝑞𝑟)

Theorempexmidlem2N 35083 Lemma for pexmidN 35081. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋) ∧ 𝑝 (𝑟 𝑞))) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem3N 35084 Lemma for pexmidN 35081. Use atom exchange hlatexch1 34507 to swap 𝑝 and 𝑞. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋)) ∧ 𝑞 (𝑟 𝑝)) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem4N 35085* Lemma for pexmidN 35081. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑞 ∈ (( 𝑋) ∩ 𝑀))) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem5N 35086 Lemma for pexmidN 35081. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → (( 𝑋) ∩ 𝑀) = ∅)

Theorempexmidlem6N 35087 Lemma for pexmidN 35081. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → 𝑀 = 𝑋)

Theorempexmidlem7N 35088 Lemma for pexmidN 35081. Contradict pexmidlem6N 35087. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → 𝑀𝑋)

Theorempexmidlem8N 35089 Lemma for pexmidN 35081. The contradiction of pexmidlem6N 35087 and pexmidlem7N 35088 shows that there can be no atom 𝑝 that is not in 𝑋 + ( 𝑋), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)

TheorempexmidALTN 35090 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 35065. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Theorempl42lem1N 35091 Lemma for pl42N 35095. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → (𝐹‘((((𝑋 𝑌) 𝑍) 𝑊) 𝑉)) = (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉))))

Theorempl42lem2N 35092 Lemma for pl42N 35095. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))

Theorempl42lem3N 35093 Lemma for pl42N 35095. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉)) ⊆ ((((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∩ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))))

Theorempl42lem4N 35094 Lemma for pl42N 35095. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → (𝐹‘((((𝑋 𝑌) 𝑍) 𝑊) 𝑉)) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉))))))

Theorempl42N 35095 Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → ((((𝑋 𝑌) 𝑍) 𝑊) 𝑉) ((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))

Syntaxclh 35096 Extend class notation with set of all co-atoms (lattice hyperplanes).
class LHyp

Syntaxclaut 35097 Extend class notation with set of all lattice automorphisms.
class LAut

SyntaxcwpointsN 35098 Extend class notation with W points.
class WAtoms

SyntaxcpautN 35099 Extend class notation with set of all projective automorphisms.
class PAut

Definitiondf-lhyp 35100* Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e. all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)
LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})

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