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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempaddasslem8 39001 Lemma for paddass 39012. (Contributed by NM, 8-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝑧 ∈ 𝑍) ∧ 𝑠 ≀ (π‘₯ ∨ 𝑦) ∧ 𝑝 ≀ (𝑠 ∨ 𝑧))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem9 39002 Lemma for paddass 39012. Combine paddasslem7 39000 and paddasslem8 39001. (Contributed by NM, 9-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝑧 ∈ 𝑍) ∧ (Β¬ π‘Ÿ ≀ (π‘₯ ∨ 𝑦) ∧ π‘Ÿ ≀ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≀ (π‘₯ ∨ 𝑦) ∧ 𝑠 ≀ (𝑝 ∨ 𝑧)))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem10 39003 Lemma for paddass 39012. Use paddasslem4 38997 to eliminate 𝑠 from paddasslem9 39002. (Contributed by NM, 9-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ 𝑝 β‰  𝑧 ∧ π‘₯ β‰  𝑦) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝑧 ∈ 𝑍) ∧ (Β¬ π‘Ÿ ≀ (π‘₯ ∨ 𝑦) ∧ 𝑝 ≀ (π‘₯ ∨ π‘Ÿ) ∧ π‘Ÿ ≀ (𝑦 ∨ 𝑧)))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem11 39004 Lemma for paddass 39012. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) ∧ 𝑧 ∈ 𝑍) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem12 39005 Lemma for paddass 39012. The case when π‘₯ = 𝑦. (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘₯ = 𝑦) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴)) ∧ ((𝑦 ∈ π‘Œ ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≀ (π‘₯ ∨ π‘Ÿ) ∧ π‘Ÿ ≀ (𝑦 ∨ 𝑧)))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem13 39006 Lemma for paddass 39012. The case when π‘Ÿ ≀ (π‘₯ ∨ 𝑦). (Unlike the proof in Maeda and Maeda, we don't need π‘₯ β‰  𝑦.) (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ 𝑝 β‰  𝑧) ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ) ∧ (π‘Ÿ ≀ (π‘₯ ∨ 𝑦) ∧ 𝑝 ≀ (π‘₯ ∨ π‘Ÿ)))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem14 39007 Lemma for paddass 39012. Remove 𝑝 β‰  𝑧, π‘₯ β‰  𝑦, and Β¬ π‘Ÿ ≀ (π‘₯ ∨ 𝑦) from antecedent of paddasslem10 39003, using paddasslem11 39004, paddasslem12 39005, and paddasslem13 39006. (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ π‘Ÿ ∈ 𝐴)) ∧ ((π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ π‘Œ ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≀ (π‘₯ ∨ π‘Ÿ) ∧ π‘Ÿ ≀ (𝑦 ∨ 𝑧)))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem15 39008 Lemma for paddass 39012. Use elpaddn0 38974 to eliminate 𝑦 and 𝑧 from paddasslem14 39007. (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ (π‘Œ β‰  βˆ… ∧ 𝑍 β‰  βˆ…)) ∧ (𝑝 ∈ 𝐴 ∧ (π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ (π‘Œ + 𝑍)) ∧ 𝑝 ≀ (π‘₯ ∨ π‘Ÿ))) β†’ 𝑝 ∈ ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem16 39009 Lemma for paddass 39012. Use elpaddn0 38974 to eliminate π‘₯ and π‘Ÿ from paddasslem15 39008. (Contributed by NM, 11-Jan-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ ((𝑋 β‰  βˆ… ∧ (π‘Œ + 𝑍) β‰  βˆ…) ∧ (π‘Œ β‰  βˆ… ∧ 𝑍 β‰  βˆ…))) β†’ (𝑋 + (π‘Œ + 𝑍)) βŠ† ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem17 39010 Lemma for paddass 39012. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴) ∧ Β¬ ((𝑋 β‰  βˆ… ∧ (π‘Œ + 𝑍) β‰  βˆ…) ∧ (π‘Œ β‰  βˆ… ∧ 𝑍 β‰  βˆ…))) β†’ (𝑋 + (π‘Œ + 𝑍)) βŠ† ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddasslem18 39011 Lemma for paddass 39012. Combine paddasslem16 39009 and paddasslem17 39010. (Contributed by NM, 12-Jan-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + 𝑍)) βŠ† ((𝑋 + π‘Œ) + 𝑍))
 
Theorempaddass 39012 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 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + 𝑍) = (𝑋 + (π‘Œ + 𝑍)))
 
Theorempadd12N 39013 Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 + (π‘Œ + 𝑍)) = (π‘Œ + (𝑋 + 𝑍)))
 
Theorempadd4N 39014 Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 βŠ† 𝐴 ∧ π‘Š βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) + (𝑍 + π‘Š)) = ((𝑋 + 𝑍) + (π‘Œ + π‘Š)))
 
Theorempaddidm 39015 Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.)
𝑆 = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 + 𝑋) = 𝑋)
 
TheorempaddclN 39016 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 = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (𝑋 + π‘Œ) ∈ 𝑆)
 
Theorempaddssw1 39017 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 βŠ† 𝑍 ∧ π‘Œ βŠ† 𝑍) β†’ (𝑋 + π‘Œ) βŠ† (𝑍 + 𝑍)))
 
Theorempaddssw2 39018 Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.)
𝐴 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ ((𝑋 + π‘Œ) βŠ† 𝑍 β†’ (𝑋 βŠ† 𝑍 ∧ π‘Œ βŠ† 𝑍)))
 
Theorempaddss 39019 Subset law for projective subspace sum. (unss 4183 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ ((𝑋 βŠ† 𝑍 ∧ π‘Œ βŠ† 𝑍) ↔ (𝑋 + π‘Œ) βŠ† 𝑍))
 
Theorempmodlem1 39020* Lemma for pmod1i 39022. (Contributed by NM, 9-Mar-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) ∧ (𝑍 ∈ 𝑆 ∧ 𝑋 βŠ† 𝑍 ∧ 𝑝 ∈ 𝑍) ∧ (π‘ž ∈ 𝑋 ∧ π‘Ÿ ∈ π‘Œ ∧ 𝑝 ≀ (π‘ž ∨ π‘Ÿ))) β†’ 𝑝 ∈ (𝑋 + (π‘Œ ∩ 𝑍)))
 
Theorempmodlem2 39021 Lemma for pmod1i 39022. (Contributed by NM, 9-Mar-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 βŠ† 𝑍) β†’ ((𝑋 + π‘Œ) ∩ 𝑍) βŠ† (𝑋 + (π‘Œ ∩ 𝑍)))
 
Theorempmod1i 39022 The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 ∈ 𝑆)) β†’ (𝑋 βŠ† 𝑍 β†’ ((𝑋 + π‘Œ) ∩ 𝑍) = (𝑋 + (π‘Œ ∩ 𝑍))))
 
Theorempmod2iN 39023 Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑍 βŠ† 𝑋 β†’ ((𝑋 ∩ π‘Œ) + 𝑍) = (𝑋 ∩ (π‘Œ + 𝑍))))
 
TheorempmodN 39024 The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ π‘Œ βŠ† 𝐴 ∧ 𝑍 βŠ† 𝐴)) β†’ (𝑋 ∩ (π‘Œ + (𝑋 ∩ 𝑍))) = ((𝑋 ∩ π‘Œ) + (𝑋 ∩ 𝑍)))
 
Theorempmodl42N 39025 Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝑆 = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ π‘Š ∈ 𝑆)) β†’ (((𝑋 + π‘Œ) + 𝑍) ∩ ((𝑋 + π‘Œ) + π‘Š)) = ((𝑋 + π‘Œ) + ((𝑋 + 𝑍) ∩ (π‘Œ + π‘Š))))
 
Theorempmapjoin 39026 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ ((π‘€β€˜π‘‹) + (π‘€β€˜π‘Œ)) βŠ† (π‘€β€˜(𝑋 ∨ π‘Œ)))
 
Theorempmapjat1 39027 The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (π‘€β€˜(𝑋 ∨ 𝑄)) = ((π‘€β€˜π‘‹) + (π‘€β€˜π‘„)))
 
Theorempmapjat2 39028 The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴) β†’ (π‘€β€˜(𝑄 ∨ 𝑋)) = ((π‘€β€˜π‘„) + (π‘€β€˜π‘‹)))
 
Theorempmapjlln1 39029 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.)
𝐡 = (Baseβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ (π‘€β€˜(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((π‘€β€˜π‘‹) + (π‘€β€˜(𝑄 ∨ 𝑅))))
 
Theoremhlmod1i 39030 A version of the modular law pmod1i 39022 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   πΉ = (pmapβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡)) β†’ ((𝑋 ≀ 𝑍 ∧ (πΉβ€˜(𝑋 ∨ π‘Œ)) = ((πΉβ€˜π‘‹) + (πΉβ€˜π‘Œ))) β†’ ((𝑋 ∨ π‘Œ) ∧ 𝑍) = (𝑋 ∨ (π‘Œ ∧ 𝑍))))
 
Theorematmod1i1 39031 Version of modular law pmod1i 39022 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑃 ≀ π‘Œ) β†’ (𝑃 ∨ (𝑋 ∧ π‘Œ)) = ((𝑃 ∨ 𝑋) ∧ π‘Œ))
 
Theorematmod1i1m 39032 Version of modular law pmod1i 39022 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) ∧ (𝑋 ∧ 𝑃) ≀ 𝑍) β†’ ((𝑋 ∧ 𝑃) ∨ (π‘Œ ∧ 𝑍)) = (((𝑋 ∧ 𝑃) ∨ π‘Œ) ∧ 𝑍))
 
Theorematmod1i2 39033 Version of modular law pmod1i 39022 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 ∨ (𝑃 ∧ π‘Œ)) = ((𝑋 ∨ 𝑃) ∧ π‘Œ))
 
Theoremllnmod1i2 39034 Version of modular law pmod1i 39022 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ π‘Œ)) = ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ π‘Œ))
 
Theorematmod2i1 39035 Version of modular law pmod2iN 39023 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑃 ≀ 𝑋) β†’ ((𝑋 ∧ π‘Œ) ∨ 𝑃) = (𝑋 ∧ (π‘Œ ∨ 𝑃)))
 
Theorematmod2i2 39036 Version of modular law pmod2iN 39023 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ π‘Œ ≀ 𝑋) β†’ ((𝑋 ∧ 𝑃) ∨ π‘Œ) = (𝑋 ∧ (𝑃 ∨ π‘Œ)))
 
Theoremllnmod2i2 39037 Version of modular law pmod1i 39022 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ π‘Œ ≀ 𝑋) β†’ ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∨ π‘Œ) = (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ π‘Œ)))
 
Theorematmod3i1 39038 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑃 ≀ 𝑋) β†’ (𝑃 ∨ (𝑋 ∧ π‘Œ)) = (𝑋 ∧ (𝑃 ∨ π‘Œ)))
 
Theorematmod3i2 39039 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ (𝑋 ∨ (π‘Œ ∧ 𝑃)) = (π‘Œ ∧ (𝑋 ∨ 𝑃)))
 
Theorematmod4i1 39040 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑃 ≀ π‘Œ) β†’ ((𝑋 ∧ π‘Œ) ∨ 𝑃) = ((𝑋 ∨ 𝑃) ∧ π‘Œ))
 
Theorematmod4i2 39041 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.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) ∧ 𝑋 ≀ π‘Œ) β†’ ((𝑃 ∧ π‘Œ) ∨ 𝑋) = ((𝑃 ∨ 𝑋) ∧ π‘Œ))
 
Theoremllnexchb2lem 39042 Lemma for llnexchb2 39043. (Contributed by NM, 17-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘ = (LLinesβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ 𝑋) ∧ (𝑋 ∧ π‘Œ) ∈ 𝐴) β†’ ((𝑋 ∧ π‘Œ) ≀ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ π‘Œ) = (𝑋 ∧ (𝑃 ∨ 𝑄))))
 
Theoremllnexchb2 39043 Line exchange property (compare cvlatexchb2 38508 for atoms). (Contributed by NM, 17-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘ = (LLinesβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ π‘Œ) ∈ 𝐴 ∧ 𝑋 β‰  𝑍)) β†’ ((𝑋 ∧ π‘Œ) ≀ 𝑍 ↔ (𝑋 ∧ π‘Œ) = (𝑋 ∧ 𝑍)))
 
Theoremllnexch2N 39044 Line exchange property (compare cvlatexch2 38510 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘ = (LLinesβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ π‘Œ ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ π‘Œ) ∈ 𝐴 ∧ 𝑋 β‰  𝑍)) β†’ ((𝑋 ∧ π‘Œ) ≀ 𝑍 β†’ (𝑋 ∧ 𝑍) ≀ π‘Œ))
 
Theoremdalawlem1 39045 Lemma for dalaw 39060. Special case of dath2 38911, where 𝐢 is replaced by ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 38911. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘‚ = (LPlanesβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) ∧ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ ((𝑆 ∨ 𝑇) ∨ π‘ˆ) ∈ 𝑂) ∧ ((Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ 𝑃)) ∧ (Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑆 ∨ 𝑇) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑇 ∨ π‘ˆ) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (π‘ˆ ∨ 𝑆)) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ))) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem2 39046 Lemma for dalaw 39060. Utility lemma that breaks ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) into a join of two pieces. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ ((((𝑃 ∨ 𝑄) ∨ 𝑇) ∧ 𝑆) ∨ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇)))
 
Theoremdalawlem3 39047 Lemma for dalaw 39060. First piece of dalawlem5 39049. (Contributed by NM, 4-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑄 ∨ 𝑇) ∨ 𝑃) ∧ 𝑆) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem4 39048 Lemma for dalaw 39060. Second piece of dalawlem5 39049. (Contributed by NM, 4-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑆) ∨ 𝑄) ∧ 𝑇) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem5 39049 Lemma for dalaw 39060. Special case to eliminate the requirement Β¬ (𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) in dalawlem1 39045. (Contributed by NM, 4-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem6 39050 Lemma for dalaw 39060. First piece of dalawlem8 39052. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑇) ∧ 𝑆) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem7 39051 Lemma for dalaw 39060. Second piece of dalawlem8 39052. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem8 39052 Lemma for dalaw 39060. Special case to eliminate the requirement Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) in dalawlem1 39045. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem9 39053 Lemma for dalaw 39060. Special case to eliminate the requirement Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ 𝑃) in dalawlem1 39045. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ 𝑃) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem10 39054 Lemma for dalaw 39060. Combine dalawlem5 39049, dalawlem8 39052, and dalawlem9 . (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ Β¬ (Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ 𝑃)) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem11 39055 Lemma for dalaw 39060. First part of dalawlem13 39057. (Contributed by NM, 17-Sep-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑃 ≀ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem12 39056 Lemma for dalaw 39060. Second part of dalawlem13 39057. (Contributed by NM, 17-Sep-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem13 39057 Lemma for dalaw 39060. Special case to eliminate the requirement ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 in dalawlem1 39045. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘‚ = (LPlanesβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ Β¬ ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem14 39058 Lemma for dalaw 39060. Combine dalawlem10 39054 and dalawlem13 39057. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘‚ = (LPlanesβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ Β¬ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ (Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑄 ∨ 𝑅) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ 𝑃))) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalawlem15 39059 Lemma for dalaw 39060. Swap variable triples 𝑃𝑄𝑅 and π‘†π‘‡π‘ˆ in dalawlem14 39058, to obtain the elimination of the remaining conditions in dalawlem1 39045. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘‚ = (LPlanesβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ Β¬ (((𝑆 ∨ 𝑇) ∨ π‘ˆ) ∈ 𝑂 ∧ (Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑆 ∨ 𝑇) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑇 ∨ π‘ˆ) ∧ Β¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (π‘ˆ ∨ 𝑆))) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆))))
 
Theoremdalaw 39060 Desargues's law, derived from Desargues's theorem dath 38910 and with no conditions on the atoms. If triples βŸ¨π‘ƒ, 𝑄, π‘…βŸ© and βŸ¨π‘†, 𝑇, π‘ˆβŸ© are centrally perspective, i.e., ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ π‘ˆ ∈ 𝐴)) β†’ (((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≀ (𝑅 ∨ π‘ˆ) β†’ ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≀ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ π‘ˆ)) ∨ ((𝑅 ∨ 𝑃) ∧ (π‘ˆ ∨ 𝑆)))))
 
SyntaxcpclN 39061 Extend class notation with projective subspace closure.
class PCl
 
Definitiondf-pclN 39062* 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 39109.) (Contributed by NM, 7-Sep-2013.)
PCl = (π‘˜ ∈ V ↦ (π‘₯ ∈ 𝒫 (Atomsβ€˜π‘˜) ↦ ∩ {𝑦 ∈ (PSubSpβ€˜π‘˜) ∣ π‘₯ βŠ† 𝑦}))
 
TheorempclfvalN 39063* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ π‘ˆ = (π‘₯ ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ π‘₯ βŠ† 𝑦}))
 
TheorempclvalN 39064* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 βŠ† 𝑦})
 
TheorempclclN 39065 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) ∈ 𝑆)
 
TheoremelpclN 39066* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    &   π‘„ ∈ V    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ (𝑄 ∈ (π‘ˆβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑆 (𝑋 βŠ† 𝑦 β†’ 𝑄 ∈ 𝑦)))
 
TheoremelpcliN 39067 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   (((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ ∈ 𝑆) ∧ 𝑄 ∈ (π‘ˆβ€˜π‘‹)) β†’ 𝑄 ∈ π‘Œ)
 
TheorempclssN 39068 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† π‘Œ ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) βŠ† (π‘ˆβ€˜π‘Œ))
 
TheorempclssidN 39069 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† (π‘ˆβ€˜π‘‹))
 
TheorempclidN 39070 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) β†’ (π‘ˆβ€˜π‘‹) = 𝑋)
 
TheorempclbtwnN 39071 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 = (PSubSpβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑋 βŠ† (π‘ˆβ€˜π‘Œ))) β†’ 𝑋 = (π‘ˆβ€˜π‘Œ))
 
TheorempclunN 39072 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 = (Atomsβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝑉 ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (π‘ˆβ€˜(𝑋 + π‘Œ)))
 
Theorempclun2N 39073 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 = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ π‘Œ ∈ 𝑆) β†’ (π‘ˆβ€˜(𝑋 βˆͺ π‘Œ)) = (𝑋 + π‘Œ))
 
TheorempclfinN 39074* 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 39124. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) = βˆͺ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(π‘ˆβ€˜π‘¦))
 
TheorempclcmpatN 39075* 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 = (Atomsβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ AtLat ∧ 𝑋 βŠ† 𝐴 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘‹)) β†’ βˆƒπ‘¦ ∈ Fin (𝑦 βŠ† 𝑋 ∧ 𝑃 ∈ (π‘ˆβ€˜π‘¦)))
 
SyntaxcpolN 39076 Extend class notation with polarity of projective subspace $m$.
class βŠ₯𝑃
 
Definitiondf-polarityN 39077* 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 Atomsβ€˜π‘™ ensures it is defined when π‘š = βˆ…. (Contributed by NM, 23-Oct-2011.)
βŠ₯𝑃 = (𝑙 ∈ V ↦ (π‘š ∈ 𝒫 (Atomsβ€˜π‘™) ↦ ((Atomsβ€˜π‘™) ∩ ∩ 𝑝 ∈ π‘š ((pmapβ€˜π‘™)β€˜((ocβ€˜π‘™)β€˜π‘)))))
 
TheorempolfvalN 39078* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
βŠ₯ = (ocβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &   π‘ƒ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐡 β†’ 𝑃 = (π‘š ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ π‘š (π‘€β€˜( βŠ₯ β€˜π‘)))))
 
TheorempolvalN 39079* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
βŠ₯ = (ocβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &   π‘ƒ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (π‘€β€˜( βŠ₯ β€˜π‘))))
 
Theorempolval2N 39080 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.)
π‘ˆ = (lubβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &   π‘ƒ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ƒβ€˜π‘‹) = (π‘€β€˜( βŠ₯ β€˜(π‘ˆβ€˜π‘‹))))
 
TheorempolsubN 39081 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) ∈ 𝑆)
 
TheorempolssatN 39082 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜π‘‹) βŠ† 𝐴)
 
Theorempol0N 39083 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐡 β†’ ( βŠ₯ β€˜βˆ…) = 𝐴)
 
Theorempol1N 39084 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 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ ( βŠ₯ β€˜π΄) = βˆ…)
 
Theorem2pol0N 39085 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ ( βŠ₯ β€˜( βŠ₯ β€˜βˆ…)) = βˆ…)
 
TheorempolpmapN 39086 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    βŠ₯ = (ocβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &   π‘ƒ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘ƒβ€˜(π‘€β€˜π‘‹)) = (π‘€β€˜( βŠ₯ β€˜π‘‹)))
 
Theorem2polpmapN 39087 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜(π‘€β€˜π‘‹))) = (π‘€β€˜π‘‹))
 
Theorem2polvalN 39088 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
π‘ˆ = (lubβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) = (π‘€β€˜(π‘ˆβ€˜π‘‹)))
 
Theorem2polssN 39089 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ 𝑋 βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
 
Theorem3polN 39090 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ ( βŠ₯ β€˜( βŠ₯ β€˜( βŠ₯ β€˜π‘†))) = ( βŠ₯ β€˜π‘†))
 
Theorempolcon3N 39091 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† π‘Œ) β†’ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹))
 
Theorem2polcon4bN 39092 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘Œ)) ↔ ( βŠ₯ β€˜π‘Œ) βŠ† ( βŠ₯ β€˜π‘‹)))
 
Theorempolcon2N 39093 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ π‘Œ βŠ† 𝐴 ∧ 𝑋 βŠ† ( βŠ₯ β€˜π‘Œ)) β†’ π‘Œ βŠ† ( βŠ₯ β€˜π‘‹))
 
Theorempolcon2bN 39094 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴 ∧ π‘Œ βŠ† 𝐴) β†’ (𝑋 βŠ† ( βŠ₯ β€˜π‘Œ) ↔ π‘Œ βŠ† ( βŠ₯ β€˜π‘‹)))
 
Theorempclss2polN 39095 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 βŠ† 𝐴) β†’ (π‘ˆβ€˜π‘‹) βŠ† ( βŠ₯ β€˜( βŠ₯ β€˜π‘‹)))
 
Theorempcl0N 39096 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
π‘ˆ = (PClβ€˜πΎ)    β‡’   (𝐾 ∈ HL β†’ (π‘ˆβ€˜βˆ…) = βˆ…)
 
Theorempcl0bN 39097 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘ˆ = (PClβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑃 βŠ† 𝐴) β†’ ((π‘ˆβ€˜π‘ƒ) = βˆ… ↔ 𝑃 = βˆ…))
 
TheorempmaplubN 39098 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &   π‘ˆ = (lubβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐡) β†’ (π‘ˆβ€˜(π‘€β€˜π‘‹)) = 𝑋)
 
TheoremsspmaplubN 39099 A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
π‘ˆ = (lubβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ 𝑆 βŠ† (π‘€β€˜(π‘ˆβ€˜π‘†)))
 
Theorem2pmaplubN 39100 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
π‘ˆ = (lubβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π‘€ = (pmapβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ 𝑆 βŠ† 𝐴) β†’ (π‘€β€˜(π‘ˆβ€˜(π‘€β€˜(π‘ˆβ€˜π‘†)))) = (π‘€β€˜(π‘ˆβ€˜π‘†)))
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