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Theorem List for Metamath Proof Explorer - 36801-36900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlncmp 36801 If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝑀 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))
 
Theorem2lnat 36802 Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝐹 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)
 
Theorem2atm2atN 36803 Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑅 𝑃) (𝑅 𝑄)) ≠ 0 )
 
Theorem2llnma1b 36804 Generalization of 2llnma1 36805. (Contributed by NM, 26-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)
 
Theorem2llnma1 36805 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)
 
Theorem2llnma3r 36806 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑅) ≠ (𝑄 𝑅)) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)
 
Theorem2llnma2 36807 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)
 
Theorem2llnma2rN 36808 Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)
 
20.24.13  Construction of a vector space from a Hilbert lattice
 
Theoremcdlema1N 36809 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)    &   𝐹 = (pmap‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) = (𝑋 𝑌))
 
Theoremcdlema2N 36810 A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋) = 0 )
 
Theoremcdlemblem 36811 Lemma for cdlemb 36812. (Contributed by NM, 8-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    < = (lt‘𝐾)    &    = (meet‘𝐾)    &   𝑉 = ((𝑃 𝑄) 𝑋)       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
 
Theoremcdlemb 36812* Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))
 
Syntaxcpadd 36813 Extend class notation with projective subspace sum.
class +𝑃
 
Definitiondf-padd 36814* Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
+𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
 
Theorempaddfval 36815* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (𝐾𝐵+ = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))
 
Theorempaddval 36816* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)}))
 
Theoremelpadd 36817* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑆 ∈ (𝑋 + 𝑌) ↔ ((𝑆𝑋𝑆𝑌) ∨ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟)))))
 
Theoremelpaddn0 36818* Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟))))
 
Theorempaddvaln0N 36819* Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)})
 
Theoremelpaddri 36820 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑄𝑋𝑅𝑌) ∧ (𝑆𝐴𝑆 (𝑄 𝑅))) → 𝑆 ∈ (𝑋 + 𝑌))
 
TheoremelpaddatriN 36821 Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑅𝑋𝑆𝐴𝑆 (𝑅 𝑄))) → 𝑆 ∈ (𝑋 + {𝑄}))
 
Theoremelpaddat 36822* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆𝐴 ∧ ∃𝑝𝑋 𝑆 (𝑝 𝑄))))
 
TheoremelpaddatiN 36823* Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝𝑋 𝑅 (𝑝 𝑄))
 
Theoremelpadd2at 36824 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑄𝐴𝑅𝐴) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ (𝑆𝐴𝑆 (𝑄 𝑅))))
 
Theoremelpadd2at2 36825 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 36826 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 36827 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 36828 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 36829 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑋 + ∅) = 𝑋)
 
Theorempadd02 36830 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (∅ + 𝑋) = 𝑋)
 
Theorempaddcom 36831 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
 
Theorempaddssat 36832 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)
 
Theoremsspadd1 36833 A projective subspace sum is a superset of its first summand. (ssun1 4147 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
 
Theoremsspadd2 36834 A projective subspace sum is a superset of its second summand. (ssun2 4148 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))
 
Theorempaddss1 36835 Subset law for projective subspace sum. (unss1 4154 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍)))
 
Theorempaddss2 36836 Subset law for projective subspace sum. (unss2 4156 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))
 
Theorempaddss12 36837 Subset law for projective subspace sum. (unss12 4157 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑊𝐴) → ((𝑋𝑌𝑍𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊)))
 
Theorempaddasslem1 36838 Lemma for paddass 36856. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑥𝐴𝑟𝐴𝑦𝐴) ∧ 𝑥𝑦) ∧ ¬ 𝑟 (𝑥 𝑦)) → ¬ 𝑥 (𝑟 𝑦))
 
Theorempaddasslem2 36839 Lemma for paddass 36856. (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 36840* Lemma for paddass 36856. Restate projective space axiom ps-2 36496. (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 36841* Lemma for paddass 36856. Combine paddasslem1 36838, paddasslem2 36839, and paddasslem3 36840. (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 36842 Lemma for paddass 36856. 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 36843 Lemma for paddass 36856. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ (𝑝𝐴𝑠𝐴) ∧ 𝑧𝐴) ∧ (𝑠𝑧𝑠 (𝑝 𝑧))) → 𝑝 (𝑠 𝑧))
 
Theorempaddasslem7 36844 Lemma for paddass 36856. Combine paddasslem5 36842 and paddasslem6 36843. (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 36845 Lemma for paddass 36856. (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 36846 Lemma for paddass 36856. Combine paddasslem7 36844 and paddasslem8 36845. (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 36847 Lemma for paddass 36856. Use paddasslem4 36841 to eliminate 𝑠 from paddasslem9 36846. (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 36848 Lemma for paddass 36856. 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 36849 Lemma for paddass 36856. 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 36850 Lemma for paddass 36856. 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 36851 Lemma for paddass 36856. Remove 𝑝𝑧, 𝑥𝑦, and ¬ 𝑟 (𝑥 𝑦) from antecedent of paddasslem10 36847, using paddasslem11 36848, paddasslem12 36849, and paddasslem13 36850. (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 36852 Lemma for paddass 36856. Use elpaddn0 36818 to eliminate 𝑦 and 𝑧 from paddasslem14 36851. (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 36853 Lemma for paddass 36856. Use elpaddn0 36818 to eliminate 𝑥 and 𝑟 from paddasslem15 36852. (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 36854 Lemma for paddass 36856. 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 36855 Lemma for paddass 36856. Combine paddasslem16 36853 and paddasslem17 36854. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐴𝑌𝐴𝑍𝐴)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍))
 
Theorempaddass 36856 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 36857 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 36858 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 36859 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 36860 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 36861 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 36862 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 36863 Subset law for projective subspace sum. (unss 4159 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵 ∧ (𝑋𝐴𝑌𝐴𝑍𝑆)) → ((𝑋𝑍𝑌𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍))
 
Theorempmodlem1 36864* Lemma for pmod1i 36866. (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 36865 Lemma for pmod1i 36866. (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 36866 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 36867 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 36868 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 36869 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 36870 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 36871 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 36872 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 36873 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 36874 A version of the modular law pmod1i 36866 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 36875 Version of modular law pmod1i 36866 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 36876 Version of modular law pmod1i 36866 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 36877 Version of modular law pmod1i 36866 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 36878 Version of modular law pmod1i 36866 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 36879 Version of modular law pmod2iN 36867 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 36880 Version of modular law pmod2iN 36867 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 36881 Version of modular law pmod1i 36866 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 36882 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 36883 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 36884 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 36885 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 36886 Lemma for llnexchb2 36887. (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 36887 Line exchange property (compare cvlatexchb2 36353 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 36888 Line exchange property (compare cvlatexch2 36355 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 36889 Lemma for dalaw 36904. Special case of dath2 36755, where 𝐶 is replaced by ((𝑃 𝑆) (𝑄 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 36755. (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 36890 Lemma for dalaw 36904. 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 36891 Lemma for dalaw 36904. First piece of dalawlem5 36893. (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 36892 Lemma for dalaw 36904. Second piece of dalawlem5 36893. (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 36893 Lemma for dalaw 36904. Special case to eliminate the requirement ¬ (𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) in dalawlem1 36889. (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 36894 Lemma for dalaw 36904. First piece of dalawlem8 36896. (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 36895 Lemma for dalaw 36904. Second piece of dalawlem8 36896. (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 36896 Lemma for dalaw 36904. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) in dalawlem1 36889. (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 36897 Lemma for dalaw 36904. Special case to eliminate the requirement ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃) in dalawlem1 36889. (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 36898 Lemma for dalaw 36904. Combine dalawlem5 36893, dalawlem8 36896, 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 36899 Lemma for dalaw 36904. First part of dalawlem13 36901. (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 36900 Lemma for dalaw 36904. Second part of dalawlem13 36901. (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))))
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 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 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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