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Theorem List for Metamath Proof Explorer - 34001-34100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremelpadd2at2 34001 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ (𝑄𝐴𝑅𝐴𝑆𝐴)) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ 𝑆 (𝑄 𝑅)))
 
TheorempaddunssN 34002 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))
 
Theoremelpadd0 34003 Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝑋𝑆𝑌)))
 
Theorempaddval0 34004 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = (𝑋𝑌))
 
Theorempadd01 34005 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑋 + ∅) = 𝑋)
 
Theorempadd02 34006 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (∅ + 𝑋) = 𝑋)
 
Theorempaddcom 34007 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theorempaddssat 34008 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
 
Theoremsspadd1 34009 A projective subspace sum is a superset of its first summand. (ssun1 3642 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
 
Theoremsspadd2 34010 A projective subspace sum is a superset of its second summand. (ssun2 3643 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))
 
Theorempaddss1 34011 Subset law for projective subspace sum. (unss1 3648 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍)))
 
Theorempaddss2 34012 Subset law for projective subspace sum. (unss2 3650 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))
 
Theorempaddss12 34013 Subset law for projective subspace sum. (unss12 3651 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑊𝐴) → ((𝑋𝑌𝑍𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊)))
 
Theorempaddasslem1 34014 Lemma for paddass 34032. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ 𝑥𝑦) ∧ ¬ 𝑟 (𝑥 𝑦)) → ¬ 𝑥 (𝑟 𝑦))
 
Theorempaddasslem2 34015 Lemma for paddass 34032. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧))) → 𝑧 (𝑟 𝑦))
 
Theorempaddasslem3 34016* Lemma for paddass 34032. Restate projective space axiom ps-2 33672. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ (𝑝𝐴𝑧𝐴)) → (((¬ 𝑥 (𝑟 𝑦) ∧ 𝑝𝑧) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑧 (𝑟 𝑦))) → ∃𝑠𝐴 (𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧))))
 
Theorempaddasslem4 34017* Lemma for paddass 34032. Combine paddasslem1 34014, paddasslem2 34015, and paddasslem3 34016. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝐴𝑟𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴) ∧ (𝑝𝑧𝑥𝑦 ∧ ¬ 𝑟 (𝑥 𝑦))) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧))) → ∃𝑠𝐴 (𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧)))
 
Theorempaddasslem5 34018 Lemma for paddass 34032. Show 𝑠𝑧 by contradiction. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑟𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧) ∧ 𝑠 (𝑥 𝑦))) → 𝑠𝑧)
 
Theorempaddasslem6 34019 Lemma for paddass 34032. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑠𝐴) ∧ 𝑧𝐴) ∧ (𝑠𝑧𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))
 
Theorempaddasslem7 34020 Lemma for paddass 34032. Combine paddasslem5 34018 and paddasslem6 34019. (Contributed by NM, 9-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑟𝐴𝑠𝐴) ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) ∧ ((¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧) ∧ 𝑠 (𝑥 𝑦)) ∧ 𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))
 
Theorempaddasslem8 34021 Lemma for paddass 34032. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑠𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ 𝑠 (𝑥 𝑦) ∧ 𝑝 (𝑠 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem9 34022 Lemma for paddass 34032. Combine paddasslem7 34020 and paddasslem8 34021. (Contributed by NM, 9-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑟 (𝑦 𝑧)) ∧ (𝑠𝐴𝑠 (𝑥 𝑦) ∧ 𝑠 (𝑝 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem10 34023 Lemma for paddass 34032. Use paddasslem4 34017 to eliminate 𝑠 from paddasslem9 34022. (Contributed by NM, 9-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝑧𝑥𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (¬ 𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem11 34024 Lemma for paddass 34032. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) ∧ 𝑧𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem12 34025 Lemma for paddass 34032. The case when 𝑥 = 𝑦. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem13 34026 Lemma for paddass 34032. The case when 𝑟 (𝑥 𝑦). (Unlike the proof in Maeda and Maeda, we don't need 𝑥𝑦.) (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((((𝐾 ∈ HL ∧ 𝑝𝑧) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌) ∧ (𝑟 (𝑥 𝑦) ∧ 𝑝 (𝑥 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem14 34027 Lemma for paddass 34032. Remove 𝑝𝑧, 𝑥𝑦, and ¬ 𝑟 (𝑥 𝑦) from antecedent of paddasslem10 34023, using paddasslem11 34024, paddasslem12 34025, and paddasslem13 34026. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑝𝐴𝑟𝐴)) ∧ ((𝑥𝑋𝑦𝑌𝑧𝑍) ∧ (𝑝 (𝑥 𝑟) ∧ 𝑟 (𝑦 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem15 34028 Lemma for paddass 34032. Use elpaddn0 33994 to eliminate 𝑦 and 𝑧 from paddasslem14 34027. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ∧ (𝑝𝐴 ∧ (𝑥𝑋𝑟 ∈ (𝑌 + 𝑍)) ∧ 𝑝 (𝑥 𝑟))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem16 34029 Lemma for paddass 34032. Use elpaddn0 33994 to eliminate 𝑥 and 𝑟 from paddasslem15 34028. (Contributed by NM, 11-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem17 34030 Lemma for paddass 34032. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ ¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddasslem18 34031 Lemma for paddass 34032. Combine paddasslem16 34029 and paddasslem17 34030. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddass 34032 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 34033 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 34034 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 34035 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 34036 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 34037 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 34038 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 34039 Subset law for projective subspace sum. (unss 3653 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → ((𝑋𝑍𝑌𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍))
 
Theorempmodlem1 34040* Lemma for pmod1i 34042. (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 34041 Lemma for pmod1i 34042. (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 34042 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 34043 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 34044 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 34045 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 34046 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 34047 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 34048 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 34049 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 34050 A version of the modular law pmod1i 34042 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 34051 Version of modular law pmod1i 34042 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 34052 Version of modular law pmod1i 34042 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 34053 Version of modular law pmod1i 34042 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 34054 Version of modular law pmod1i 34042 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 34055 Version of modular law pmod2iN 34043 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 34056 Version of modular law pmod2iN 34043 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 34057 Version of modular law pmod1i 34042 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 34058 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 34059 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 34060 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 34061 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 34062 Lemma for llnexchb2 34063. (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 34063 Line exchange property (compare cvlatexchb2 33530 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 34064 Line exchange property (compare cvlatexch2 33532 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 34065 Lemma for dalaw 34080. Special case of dath2 33931, where 𝐶 is replaced by ((𝑃 𝑆) (𝑄 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 33931. (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 34066 Lemma for dalaw 34080. 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 34067 Lemma for dalaw 34080. First piece of dalawlem5 34069. (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 34068 Lemma for dalaw 34080. Second piece of dalawlem5 34069. (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 34069 Lemma for dalaw 34080. Special case to eliminate the requirement ¬ (𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) in dalawlem1 34065. (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 34070 Lemma for dalaw 34080. First piece of dalawlem8 34072. (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 34071 Lemma for dalaw 34080. Second piece of dalawlem8 34072. (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 34072 Lemma for dalaw 34080. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) in dalawlem1 34065. (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 34073 Lemma for dalaw 34080. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) in dalawlem1 34065. (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 34074 Lemma for dalaw 34080. Combine dalawlem5 34069, dalawlem8 34072, 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 34075 Lemma for dalaw 34080. First part of dalawlem13 34077. (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 34076 Lemma for dalaw 34080. Second part of dalawlem13 34077. (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 34077 Lemma for dalaw 34080. Special case to eliminate the requirement ((𝑃 𝑄) 𝑅) ∈ 𝑂 in dalawlem1 34065. (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 34078 Lemma for dalaw 34080. Combine dalawlem10 34074 and dalawlem13 34077. (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 34079 Lemma for dalaw 34080. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 34078, to obtain the elimination of the remaining conditions in dalawlem1 34065. (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 34080 Desargues' law, derived from Desargues' theorem dath 33930 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 34081 Extend class notation with projective subspace closure.
class PCl
 
Definitiondf-pclN 34082* 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 34129.) (Contributed by NM, 7-Sep-2013.)
PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
 
TheorempclfvalN 34083* 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 34084* 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 34085 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 34086* 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 34087 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 34088 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 34089 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 34090 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 34091 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 34092 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 34093 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 34094* 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 34144. (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 34095* 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 34096 Extend class notation with polarity of projective subspace $m$.
class 𝑃
 
Definitiondf-polarityN 34097* 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 34098* 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 34099* 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 34100 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))))
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
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