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Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheorempsubcliN 36101 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋))

Theorempsubcli2N 36102 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
= (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)

TheorempsubclsubN 36103 A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶) → 𝑋𝑆)

TheorempsubclssatN 36104 A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → 𝑋𝐴)

TheorempmapidclN 36105 Projective map of the LUB of a closed subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶) → (𝑀‘(𝑈𝑋)) = 𝑋)

Theorem0psubclN 36106 The empty set is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → ∅ ∈ 𝐶)

Theorem1psubclN 36107 The set of all atoms is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → 𝐴𝐶)

TheorematpsubclN 36108 A point (singleton of an atom) is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑄𝐴) → {𝑄} ∈ 𝐶)

TheorempmapsubclN 36109 A projective map value is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝐶)

Theoremispsubcl2N 36110* Alternate predicate for "is a closed projective subspace". Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾 ∈ HL → (𝑋𝐶 ↔ ∃𝑦𝐵 𝑋 = (𝑀𝑦)))

TheorempsubclinN 36111 The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) → (𝑋𝑌) ∈ 𝐶)

TheorempaddatclN 36112 The projective sum of a closed subspace and an atom is a closed projective subspace. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐶𝑄𝐴) → (𝑋 + {𝑄}) ∈ 𝐶)

TheorempclfinclN 36113 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 36063 and also pclcmpatN 36064. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)    &   𝑆 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑋 ∈ Fin) → (𝑈𝑋) ∈ 𝑆)

TheoremlinepsubclN 36114 A line is a closed projective subspace. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝑁 = (Lines‘𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑁) → 𝑋𝐶)

TheorempolsubclN 36115 A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ 𝐶)

Theorempoml4N 36116 Orthomodular law for projective lattices. Lemma 3.3(1) in [Holland95] p. 215. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → ((𝑋𝑌 ∧ ( ‘( 𝑌)) = 𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋))))

Theorempoml5N 36117 Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) → (( ‘(( 𝑋) ∩ ( 𝑌))) ∩ ( 𝑌)) = ( ‘( 𝑋)))

Theorempoml6N 36118 Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐶 = (PSubCl‘𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋𝑌) → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Theoremosumcllem1N 36119 Lemma for osumclN 36130. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → (𝑈𝑀) = 𝑀)

Theoremosumcllem2N 36120 Lemma for osumclN 36130. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝑈) → 𝑋 ⊆ (𝑈𝑀))

Theoremosumcllem3N 36121 Lemma for osumclN 36130. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       ((𝐾 ∈ HL ∧ 𝑌𝐶𝑋 ⊆ ( 𝑌)) → (( 𝑋) ∩ 𝑈) = 𝑌)

Theoremosumcllem4N 36122 Lemma for osumclN 36130. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) ∧ (𝑟𝑋𝑞𝑌)) → 𝑞𝑟)

Theoremosumcllem5N 36123 Lemma for osumclN 36130. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝐴 ∧ (𝑟𝑋𝑞𝑌𝑝 (𝑟 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem6N 36124 Lemma for osumclN 36130. Use atom exchange hlatexch1 35558 to swap 𝑝 and 𝑞. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑝𝐴) ∧ (𝑟𝑋𝑞𝑌𝑞 (𝑟 𝑝))) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem7N 36125* Lemma for osumclN 36130. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ 𝑞 ∈ (𝑌𝑀)) → 𝑝 ∈ (𝑋 + 𝑌))

Theoremosumcllem8N 36126 Lemma for osumclN 36130. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌𝑀) = ∅)

Theoremosumcllem9N 36127 Lemma for osumclN 36130. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋)

Theoremosumcllem10N 36128 Lemma for osumclN 36130. Contradict osumcllem9N 36127. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)    &   𝑀 = (𝑋 + {𝑝})    &   𝑈 = ( ‘( ‘(𝑋 + 𝑌)))       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) ∧ 𝑝𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀𝑋)

Theoremosumcllem11N 36129 Lemma for osumclN 36130. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
+ = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ (𝑋 ⊆ ( 𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ‘( ‘(𝑋 + 𝑌))))

TheoremosumclN 36130 Closure of orthogonal sum. If 𝑋 and 𝑌 are orthogonal closed projective subspaces, then their sum is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
+ = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐶𝑌𝐶) ∧ 𝑋 ⊆ ( 𝑌)) → (𝑋 + 𝑌) ∈ 𝐶)

TheorempmapojoinN 36131 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 36015 where only one direction holds. (Contributed by NM, 11-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (oc‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ 𝑋 ( 𝑌)) → (𝑀‘(𝑋 𝑌)) = ((𝑀𝑋) + (𝑀𝑌)))

TheorempexmidN 36132 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 36116. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 36130. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Theorempexmidlem1N 36133 Lemma for pexmidN 36132. Holland's proof implicitly requires 𝑞𝑟, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋))) → 𝑞𝑟)

Theorempexmidlem2N 36134 Lemma for pexmidN 36132. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋) ∧ 𝑝 (𝑟 𝑞))) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem3N 36135 Lemma for pexmidN 36132. Use atom exchange hlatexch1 35558 to swap 𝑝 and 𝑞. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑟𝑋𝑞 ∈ ( 𝑋)) ∧ 𝑞 (𝑟 𝑝)) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem4N 36136* Lemma for pexmidN 36132. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑞 ∈ (( 𝑋) ∩ 𝑀))) → 𝑝 ∈ (𝑋 + ( 𝑋)))

Theorempexmidlem5N 36137 Lemma for pexmidN 36132. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → (( 𝑋) ∩ 𝑀) = ∅)

Theorempexmidlem6N 36138 Lemma for pexmidN 36132. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → 𝑀 = 𝑋)

Theorempexmidlem7N 36139 Lemma for pexmidN 36132. Contradict pexmidlem6N 36138. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑀 = (𝑋 + {𝑝})       (((𝐾 ∈ HL ∧ 𝑋𝐴𝑝𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( 𝑋)))) → 𝑀𝑋)

Theorempexmidlem8N 36140 Lemma for pexmidN 36132. The contradiction of pexmidlem6N 36138 and pexmidlem7N 36139 shows that there can be no atom 𝑝 that is not in 𝑋 + ( 𝑋), which is therefore the whole atom space. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ (( ‘( 𝑋)) = 𝑋𝑋 ≠ ∅)) → (𝑋 + ( 𝑋)) = 𝐴)

TheorempexmidALTN 36141 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 36116. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. TODO: should we make this obsolete? (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)

Theorempl42lem1N 36142 Lemma for pl42N 36146. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → (𝐹‘((((𝑋 𝑌) 𝑍) 𝑊) 𝑉)) = (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉))))

Theorempl42lem2N 36143 Lemma for pl42N 36146. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((𝐹𝑋) + (𝐹𝑌)) + (((𝐹𝑋) + (𝐹𝑊)) ∩ ((𝐹𝑌) + (𝐹𝑉)))) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))

Theorempl42lem3N 36144 Lemma for pl42N 36146. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → (((((𝐹𝑋) + (𝐹𝑌)) ∩ (𝐹𝑍)) + (𝐹𝑊)) ∩ (𝐹𝑉)) ⊆ ((((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑊)) ∩ (((𝐹𝑋) + (𝐹𝑌)) + (𝐹𝑉))))

Theorempl42lem4N 36145 Lemma for pl42N 36146. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → (𝐹‘((((𝑋 𝑌) 𝑍) 𝑊) 𝑉)) ⊆ (𝐹‘((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉))))))

Theorempl42N 36146 Law holding in a Hilbert lattice that fails in orthomodular lattice L42 (Figure 7 in [MegPav2000] p. 2366). (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑍𝐵𝑊𝐵𝑉𝐵)) → ((𝑋 ( 𝑌) ∧ 𝑍 ( 𝑊)) → ((((𝑋 𝑌) 𝑍) 𝑊) 𝑉) ((𝑋 𝑌) ((𝑋 𝑊) (𝑌 𝑉)))))

Syntaxclh 36147 Extend class notation with set of all co-atoms (lattice hyperplanes).
class LHyp

Syntaxclaut 36148 Extend class notation with set of all lattice automorphisms.
class LAut

SyntaxcwpointsN 36149 Extend class notation with W points.
class WAtoms

SyntaxcpautN 36150 Extend class notation with set of all projective automorphisms.
class PAut

Definitiondf-lhyp 36151* Define the set of lattice hyperplanes, which are all lattice elements covered by 1 (i.e., all co-atoms). We call them "hyperplanes" instead of "co-atoms" in analogy with projective geometry hyperplanes. (Contributed by NM, 11-May-2012.)
LHyp = (𝑘 ∈ V ↦ {𝑥 ∈ (Base‘𝑘) ∣ 𝑥( ⋖ ‘𝑘)(1.‘𝑘)})

Definitiondf-laut 36152* Define set of lattice autoisomorphisms. (Contributed by NM, 11-May-2012.)
LAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(Base‘𝑘)–1-1-onto→(Base‘𝑘) ∧ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 ↔ (𝑓𝑥)(le‘𝑘)(𝑓𝑦)))})

Definitiondf-watsN 36153* Define W-atoms corresponding to an arbitrary "fiducial (i.e. reference) atom" 𝑑. These are all atoms not in the polarity of {𝑑}), which is the hyperplane determined by 𝑑. Definition of set W in [Crawley] p. 111. (Contributed by NM, 26-Jan-2012.)
WAtoms = (𝑘 ∈ V ↦ (𝑑 ∈ (Atoms‘𝑘) ↦ ((Atoms‘𝑘) ∖ ((⊥𝑃𝑘)‘{𝑑}))))

Definitiondf-pautN 36154* Define set of all projective automorphisms. This is the intended definition of automorphism in [Crawley] p. 112. (Contributed by NM, 26-Jan-2012.)
PAut = (𝑘 ∈ V ↦ {𝑓 ∣ (𝑓:(PSubSp‘𝑘)–1-1-onto→(PSubSp‘𝑘) ∧ ∀𝑥 ∈ (PSubSp‘𝑘)∀𝑦 ∈ (PSubSp‘𝑘)(𝑥𝑦 ↔ (𝑓𝑥) ⊆ (𝑓𝑦)))})

TheoremwatfvalN 36155* The W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)       (𝐾𝐵𝑊 = (𝑑𝐴 ↦ (𝐴 ∖ ((⊥𝑃𝐾)‘{𝑑}))))

TheoremwatvalN 36156 Value of the W atoms function. (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝑊𝐷) = (𝐴 ∖ ((⊥𝑃𝐾)‘{𝐷})))

TheoremiswatN 36157 The predicate "is a W atom" (corresponding to fiducial atom 𝐷). (Contributed by NM, 26-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝑃 ∈ (𝑊𝐷) ↔ (𝑃𝐴 ∧ ¬ 𝑃 ∈ ((⊥𝑃𝐾)‘{𝐷}))))

Theoremlhpset 36158* The set of co-atoms (lattice hyperplanes). (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝐴𝐻 = {𝑤𝐵𝑤𝐶 1 })

Theoremislhp 36159 The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 11-May-2012.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝐴 → (𝑊𝐻 ↔ (𝑊𝐵𝑊𝐶 1 )))

Theoremislhp2 36160 The predicate "is a co-atom (lattice hyperplane)". (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &    1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾𝐴𝑊𝐵) → (𝑊𝐻𝑊𝐶 1 ))

Theoremlhpbase 36161 A co-atom is a member of the lattice base set (i.e., a lattice element). (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝑊𝐻𝑊𝐵)

Theoremlhp1cvr 36162 The lattice unit covers a co-atom (lattice hyperplane). (Contributed by NM, 18-May-2012.)
1 = (1.‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾𝐴𝑊𝐻) → 𝑊𝐶 1 )

Theoremlhplt 36163 An atom under a co-atom is strictly less than it. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
= (le‘𝐾)    &    < = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑃 𝑊)) → 𝑃 < 𝑊)

Theoremlhp2lt 36164 The join of two atoms under a co-atom is strictly less than it. (Contributed by NM, 8-Jul-2013.)
= (le‘𝐾)    &    < = (lt‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑃 𝑊) ∧ (𝑄𝐴𝑄 𝑊)) → (𝑃 𝑄) < 𝑊)

Theoremlhpexlt 36165* There exists an atom less than a co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
< = (lt‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 𝑝 < 𝑊)

Theoremlhp0lt 36166 A co-atom is greater than zero. TODO: is this needed? (Contributed by NM, 1-Jun-2012.)
< = (lt‘𝐾)    &    0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 < 𝑊)

Theoremlhpn0 36167 A co-atom is nonzero. TODO: is this needed? (Contributed by NM, 26-Apr-2013.)
0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑊0 )

Theoremlhpexle 36168* There exists an atom under a co-atom. (Contributed by NM, 26-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 𝑝 𝑊)

Theoremlhpexnle 36169* There exists an atom not under a co-atom. (Contributed by NM, 12-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 ¬ 𝑝 𝑊)

Theoremlhpexle1lem 36170* Lemma for lhpexle1 36171 and others that eliminates restrictions on 𝑋. (Contributed by NM, 24-Jul-2013.)
(𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓))    &   ((𝜑 ∧ (𝑋𝐴𝑋 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))       (𝜑 → ∃𝑝𝐴 (𝑝 𝑊𝜓𝑝𝑋))

Theoremlhpexle1 36171* There exists an atom under a co-atom different from any given element. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋))

Theoremlhpexle2lem 36172* Lemma for lhpexle2 36173. (Contributed by NM, 19-Jun-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐴𝑋 𝑊) ∧ (𝑌𝐴𝑌 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋𝑝𝑌))

Theoremlhpexle2 36173* There exists atom under a co-atom different from any two other elements. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊𝑝𝑋𝑝𝑌))

Theoremlhpexle3lem 36174* There exists atom under a co-atom different from any three other atoms. TODO: study if adant*, simp* usage can be improved. (Contributed by NM, 9-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐴𝑌𝐴𝑍𝐴) ∧ (𝑋 𝑊𝑌 𝑊𝑍 𝑊)) → ∃𝑝𝐴 (𝑝 𝑊 ∧ (𝑝𝑋𝑝𝑌𝑝𝑍)))

Theoremlhpexle3 36175* There exists atom under a co-atom different from any three other elements. (Contributed by NM, 24-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴 (𝑝 𝑊 ∧ (𝑝𝑋𝑝𝑌𝑝𝑍)))

Theoremlhpex2leN 36176* There exist at least two different atoms under a co-atom. This allows us to create a line under the co-atom. TODO: is this needed? (Contributed by NM, 1-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∃𝑝𝐴𝑞𝐴 (𝑝 𝑊𝑞 𝑊𝑝𝑞))

Theoremlhpoc 36177 The orthocomplement of a co-atom (lattice hyperplane) is an atom. (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝑊𝐻 ↔ ( 𝑊) ∈ 𝐴))

Theoremlhpoc2N 36178 The orthocomplement of an atom is a co-atom (lattice hyperplane). (Contributed by NM, 20-Jun-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐵) → (𝑊𝐴 ↔ ( 𝑊) ∈ 𝐻))

Theoremlhpocnle 36179 The orthocomplement of a co-atom is not under it. (Contributed by NM, 22-May-2012.)
= (le‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ¬ ( 𝑊) 𝑊)

Theoremlhpocat 36180 The orthocomplement of a co-atom is an atom. (Contributed by NM, 9-Feb-2013.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( 𝑊) ∈ 𝐴)

Theoremlhpocnel 36181 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (( 𝑊) ∈ 𝐴 ∧ ¬ ( 𝑊) 𝑊))

Theoremlhpocnel2 36182 The orthocomplement of a co-atom is an atom not under it. Provides a convenient construction when we need the existence of any object with this property. (Contributed by NM, 20-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))

Theoremlhpjat1 36183 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 18-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑊 𝑃) = 1 )

Theoremlhpjat2 36184 The join of a co-atom (hyperplane) and an atom not under it is the lattice unit. (Contributed by NM, 4-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 1 )

Theoremlhpj1 36185 The join of a co-atom (hyperplane) and an element not under it is the lattice unit. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    1 = (1.‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑊 𝑋) = 1 )

Theoremlhpmcvr 36186 The meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 7-Dec-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (meet‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝑋 𝑊)𝐶𝑋)

Theoremlhpmcvr2 36187* Alternate way to express that the meet of a lattice hyperplane with an element not under it is covered by the element. (Contributed by NM, 9-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ (𝑝 (𝑋 𝑊)) = 𝑋))

Theoremlhpmcvr3 36188 Specialization of lhpmcvr2 36187. TODO: Use this to simplify many uses of (𝑃 (𝑋 𝑊)) = 𝑋 to become 𝑃 𝑋. (Contributed by NM, 6-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑋 ↔ (𝑃 (𝑋 𝑊)) = 𝑋))

Theoremlhpmcvr4N 36189 Specialization of lhpmcvr2 36187. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊𝑃 𝑋)) → ¬ 𝑃 𝑌)

Theoremlhpmcvr5N 36190* Specialization of lhpmcvr2 36187. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ ¬ 𝑝 𝑌 ∧ (𝑝 (𝑋 𝑊)) = 𝑋))

Theoremlhpmcvr6N 36191* Specialization of lhpmcvr2 36187. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ (𝑌𝐵 ∧ (𝑋 𝑌) 𝑊)) → ∃𝑝𝐴𝑝 𝑊 ∧ ¬ 𝑝 𝑌𝑝 𝑋))

Theoremlhpm0atN 36192 If the meet of a lattice hyperplane with a nonzero element is zero, the element is an atom. (Contributed by NM, 28-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋0 ∧ (𝑋 𝑊) = 0 )) → 𝑋𝐴)

Theoremlhpmat 36193 An element covered by the lattice unit, when conjoined with an atom not under it, equals the lattice zero. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 𝑊) = 0 )

Theoremlhpmatb 36194 An element covered by the lattice unit, when conjoined with an atom, equals zero iff the atom is not under it. (Contributed by NM, 15-Jun-2013.)
= (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴) → (¬ 𝑃 𝑊 ↔ (𝑃 𝑊) = 0 ))

Theoremlhp2at0 36195 Join and meet with different atoms under co-atom 𝑊. (Contributed by NM, 15-Jun-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ((𝑃 𝑈) 𝑉) = 0 )

Theoremlhp2atnle 36196 Inequality for 2 different atoms under co-atom 𝑊. (Contributed by NM, 17-Jun-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ (𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ¬ 𝑉 (𝑃 𝑈))

Theoremlhp2atne 36197 Inequality for joins with 2 different atoms under co-atom 𝑊. (Contributed by NM, 22-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝑈𝐴𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ 𝑈𝑉) → (𝑃 𝑈) ≠ (𝑄 𝑉))

Theoremlhp2at0nle 36198 Inequality for 2 different atoms (or an atom and zero) under co-atom 𝑊. (Contributed by NM, 28-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑈𝑉) ∧ ((𝑈𝐴𝑈 = 0 ) ∧ 𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ¬ 𝑉 (𝑃 𝑈))

Theoremlhp2at0ne 36199 Inequality for joins with 2 different atoms (or an atom and zero) under co-atom 𝑊. (Contributed by NM, 28-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (((𝑈𝐴𝑈 = 0 ) ∧ 𝑈 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) ∧ 𝑈𝑉) → (𝑃 𝑈) ≠ (𝑄 𝑉))

Theoremlhpelim 36200 Eliminate an atom not under a lattice hyperplane. TODO: Look at proofs using lhpmat 36193 to see if this can be used to shorten them. (Contributed by NM, 27-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) → ((𝑃 (𝑋 𝑊)) 𝑊) = (𝑋 𝑊))

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