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Theorem List for Metamath Proof Explorer - 36001-36100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorempaddass 36001 Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))

Theorempadd12N 36002 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Theorempadd4N 36003 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴) ∧ (𝑍𝐴𝑊𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Theorempaddidm 36004 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝑆) → (𝑋 + 𝑋) = 𝑋)

TheorempaddclN 36005 The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑋 + 𝑌) ∈ 𝑆)

Theorempaddssw1 36006 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋𝑍𝑌𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍)))

Theorempaddssw2 36007 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋𝑍𝑌𝑍)))

Theorempaddss 36008 Subset law for projective subspace sum. (unss 4010 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → ((𝑋𝑍𝑌𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍))

Theorempmodlem1 36009* Lemma for pmod1i 36011. (Contributed by NM, 9-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑍𝑆𝑋𝑍𝑝𝑍) ∧ (𝑞𝑋𝑟𝑌𝑝 (𝑞 𝑟))) → 𝑝 ∈ (𝑋 + (𝑌𝑍)))

Theorempmodlem2 36010 Lemma for pmod1i 36011. (Contributed by NM, 9-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝑆) ∧ 𝑋𝑍) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌𝑍)))

Theorempmod1i 36011 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → (𝑋𝑍 → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 + (𝑌𝑍))))

Theorempmod2iN 36012 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑆𝑌𝐴𝑍𝐴)) → (𝑍𝑋 → ((𝑋𝑌) + 𝑍) = (𝑋 ∩ (𝑌 + 𝑍))))

TheorempmodN 36013 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑆𝑌𝐴𝑍𝐴)) → (𝑋 ∩ (𝑌 + (𝑋𝑍))) = ((𝑋𝑌) + (𝑋𝑍)))

Theorempmodl42N 36014 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) ∧ (𝑍𝑆𝑊𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊))))

Theorempmapjoin 36015 The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) + (𝑀𝑌)) ⊆ (𝑀‘(𝑋 𝑌)))

Theorempmapjat1 36016 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (𝑀‘(𝑋 𝑄)) = ((𝑀𝑋) + (𝑀𝑄)))

Theorempmapjat2 36017 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑄𝐴) → (𝑀‘(𝑄 𝑋)) = ((𝑀𝑄) + (𝑀𝑋)))

Theorempmapjlln1 36018 The projective map of the join of a lattice element and a lattice line (expressed as the join 𝑄 𝑅 of two atoms). (Contributed by NM, 16-Sep-2012.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑄𝐴𝑅𝐴)) → (𝑀‘(𝑋 (𝑄 𝑅))) = ((𝑀𝑋) + (𝑀‘(𝑄 𝑅))))

Theoremhlmod1i 36019 A version of the modular law pmod1i 36011 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑌𝐵𝑍𝐵)) → ((𝑋 𝑍 ∧ (𝐹‘(𝑋 𝑌)) = ((𝐹𝑋) + (𝐹𝑌))) → ((𝑋 𝑌) 𝑍) = (𝑋 (𝑌 𝑍))))

Theorematmod1i1 36020 Version of modular law pmod1i 36011 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑌) → (𝑃 (𝑋 𝑌)) = ((𝑃 𝑋) 𝑌))

Theorematmod1i1m 36021 Version of modular law pmod1i 36011 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴) ∧ (𝑋𝐵𝑌𝐵𝑍𝐵) ∧ (𝑋 𝑃) 𝑍) → ((𝑋 𝑃) (𝑌 𝑍)) = (((𝑋 𝑃) 𝑌) 𝑍))

Theorematmod1i2 36022 Version of modular law pmod1i 36011 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 (𝑃 𝑌)) = ((𝑋 𝑃) 𝑌))

Theoremllnmod1i2 36023 Version of modular law pmod1i 36011 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑋 𝑌) → (𝑋 ((𝑃 𝑄) 𝑌)) = ((𝑋 (𝑃 𝑄)) 𝑌))

Theorematmod2i1 36024 Version of modular law pmod2iN 36012 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → ((𝑋 𝑌) 𝑃) = (𝑋 (𝑌 𝑃)))

Theorematmod2i2 36025 Version of modular law pmod2iN 36012 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑌 𝑋) → ((𝑋 𝑃) 𝑌) = (𝑋 (𝑃 𝑌)))

Theoremllnmod2i2 36026 Version of modular law pmod1i 36011 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑌 𝑋) → ((𝑋 (𝑃 𝑄)) 𝑌) = (𝑋 ((𝑃 𝑄) 𝑌)))

Theorematmod3i1 36027 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑋) → (𝑃 (𝑋 𝑌)) = (𝑋 (𝑃 𝑌)))

Theorematmod3i2 36028 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 (𝑌 𝑃)) = (𝑌 (𝑋 𝑃)))

Theorematmod4i1 36029 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑃 𝑌) → ((𝑋 𝑌) 𝑃) = ((𝑋 𝑃) 𝑌))

Theorematmod4i2 36030 Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → ((𝑃 𝑌) 𝑋) = ((𝑃 𝑋) 𝑌))

Theoremllnexchb2lem 36031 Lemma for llnexchb2 36032. (Contributed by NM, 17-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (LLines‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝑁𝑌𝑁) ∧ (𝑃𝐴𝑄𝐴 ∧ ¬ 𝑃 𝑋) ∧ (𝑋 𝑌) ∈ 𝐴) → ((𝑋 𝑌) (𝑃 𝑄) ↔ (𝑋 𝑌) = (𝑋 (𝑃 𝑄))))

Theoremllnexchb2 36032 Line exchange property (compare cvlatexchb2 35498 for atoms). (Contributed by NM, 17-Nov-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (LLines‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑍𝑁) ∧ ((𝑋 𝑌) ∈ 𝐴𝑋𝑍)) → ((𝑋 𝑌) 𝑍 ↔ (𝑋 𝑌) = (𝑋 𝑍)))

Theoremllnexch2N 36033 Line exchange property (compare cvlatexch2 35500 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (LLines‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝑁𝑌𝑁𝑍𝑁) ∧ ((𝑋 𝑌) ∈ 𝐴𝑋𝑍)) → ((𝑋 𝑌) 𝑍 → (𝑋 𝑍) 𝑌))

Theoremdalawlem1 36034 Lemma for dalaw 36049. Special case of dath2 35900, where 𝐶 is replaced by ((𝑃 𝑆) (𝑄 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 35900. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑆 𝑇) 𝑈) ∈ 𝑂) ∧ ((¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈))) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem2 36035 Lemma for dalaw 36049. Utility lemma that breaks ((𝑃 𝑄) (𝑆 𝑇)) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) ((((𝑃 𝑄) 𝑇) 𝑆) (((𝑃 𝑄) 𝑆) 𝑇)))

Theoremdalawlem3 36036 Lemma for dalaw 36049. First piece of dalawlem5 36038. (Contributed by NM, 4-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑄 𝑇) 𝑃) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem4 36037 Lemma for dalaw 36049. Second piece of dalawlem5 36038. (Contributed by NM, 4-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) 𝑄) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem5 36038 Lemma for dalaw 36049. Special case to eliminate the requirement ¬ (𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) in dalawlem1 36034. (Contributed by NM, 4-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem6 36039 Lemma for dalaw 36049. First piece of dalawlem8 36041. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑇) 𝑆) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem7 36040 Lemma for dalaw 36049. Second piece of dalawlem8 36041. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑄) 𝑆) 𝑇) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem8 36041 Lemma for dalaw 36049. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) in dalawlem1 36034. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem9 36042 Lemma for dalaw 36049. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) in dalawlem1 36034. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem10 36043 Lemma for dalaw 36049. Combine dalawlem5 36038, dalawlem8 36041, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃)) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem11 36044 Lemma for dalaw 36049. First part of dalawlem13 36046. (Contributed by NM, 17-Sep-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃 (𝑄 𝑅) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem12 36045 Lemma for dalaw 36049. Second part of dalawlem13 36046. (Contributed by NM, 17-Sep-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem13 36046 Lemma for dalaw 36049. Special case to eliminate the requirement ((𝑃 𝑄) 𝑅) ∈ 𝑂 in dalawlem1 36034. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem14 36047 Lemma for dalaw 36049. Combine dalawlem10 36043 and dalawlem13 36046. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalawlem15 36048 Lemma for dalaw 36049. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 36047, to obtain the elimination of the remaining conditions in dalawlem1 36034. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))

Theoremdalaw 36049 Desargues's law, derived from Desargues's theorem dath 35899 and with no conditions on the atoms. If triples 𝑃, 𝑄, 𝑅 and 𝑆, 𝑇, 𝑈 are centrally perspective, i.e. ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))

SyntaxcpclN 36050 Extend class notation with projective subspace closure.
class PCl

Definitiondf-pclN 36051* Projective subspace closure, which is the smallest projective subspace containing an arbitrary set of atoms. The subspace closure of the union of a set of projective subspaces is their supremum in PSubSp. Related to an analogous definition of closure used in Lemma 3.1.4 of [PtakPulmannova] p. 68. (Note that this closure is not necessarily one of the closed projective subspaces PSubCl of df-psubclN 36098.) (Contributed by NM, 7-Sep-2013.)
PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))

TheorempclfvalN 36052* 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 36053* 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 36054 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 36055* 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 36056 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 36057 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 36058 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 36059 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 36060 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 36061 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 36062 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 36063* 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 36113. (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 36064* 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 36065 Extend class notation with polarity of projective subspace \$m\$.
class 𝑃

Definitiondf-polarityN 36066* 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 36067* 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 36068* 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 36069 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 36070 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 36071 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 36072 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 36073 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 36074 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
= (⊥𝑃𝐾)       (𝐾 ∈ HL → ( ‘( ‘∅)) = ∅)

TheorempolpmapN 36075 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 36076 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 36077 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 36078 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 36079 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 36080 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 36081 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 36082 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 36083 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 36084 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 36085 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 36086 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 36087 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 36088 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 36089 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 36090 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 6452.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = ( ‘(𝑆𝑇)))

Theorempoldmj1N 36091 De Morgan's law for polarity of projective sum. (oldmj1 35384 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = (( 𝑆) ∩ ( 𝑇)))

Theorempmapj2N 36092 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 36093 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 36094 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 36095 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 36096 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 36097 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl

Definitiondf-psubclN 36098* 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 36099* 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 36100 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)))

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