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Theorem List for Metamath Proof Explorer - 36901-37000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdalawlem13 36901 Lemma for dalaw 36904. Special case to eliminate the requirement ((𝑃 𝑄) 𝑅) ∈ 𝑂 in dalawlem1 36889. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ ((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
 
Theoremdalawlem14 36902 Lemma for dalaw 36904. Combine dalawlem10 36898 and dalawlem13 36901. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ (((𝑃 𝑄) 𝑅) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑃 𝑄) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑄 𝑅) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑃))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
 
Theoremdalawlem15 36903 Lemma for dalaw 36904. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 36902, to obtain the elimination of the remaining conditions in dalawlem1 36889. (Contributed by NM, 6-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑂 = (LPlanes‘𝐾)       (((𝐾 ∈ HL ∧ ¬ (((𝑆 𝑇) 𝑈) ∈ 𝑂 ∧ (¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑆 𝑇) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑇 𝑈) ∧ ¬ ((𝑃 𝑆) (𝑄 𝑇)) (𝑈 𝑆))) ∧ ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆))))
 
Theoremdalaw 36904 Desargues's law, derived from Desargues's theorem dath 36754 and with no conditions on the atoms. If triples 𝑃, 𝑄, 𝑅 and 𝑆, 𝑇, 𝑈 are centrally perspective, i.e. ((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈), then they are axially perspective. Theorem 13.3 of [Crawley] p. 110. (Contributed by NM, 7-Oct-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) → (((𝑃 𝑆) (𝑄 𝑇)) (𝑅 𝑈) → ((𝑃 𝑄) (𝑆 𝑇)) (((𝑄 𝑅) (𝑇 𝑈)) ((𝑅 𝑃) (𝑈 𝑆)))))
 
SyntaxcpclN 36905 Extend class notation with projective subspace closure.
class PCl
 
Definitiondf-pclN 36906* 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 36953.) (Contributed by NM, 7-Sep-2013.)
PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥𝑦}))
 
TheorempclfvalN 36907* The projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (𝐾𝑉𝑈 = (𝑥 ∈ 𝒫 𝐴 {𝑦𝑆𝑥𝑦}))
 
TheorempclvalN 36908* Value of the projective subspace closure function. (Contributed by NM, 7-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) = {𝑦𝑆𝑋𝑦})
 
TheorempclclN 36909 Closure of the projective subspace closure function. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → (𝑈𝑋) ∈ 𝑆)
 
TheoremelpclN 36910* Membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)    &   𝑄 ∈ V       ((𝐾𝑉𝑋𝐴) → (𝑄 ∈ (𝑈𝑋) ↔ ∀𝑦𝑆 (𝑋𝑦𝑄𝑦)))
 
TheoremelpcliN 36911 Implication of membership in the projective subspace closure function. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (((𝐾𝑉𝑋𝑌𝑌𝑆) ∧ 𝑄 ∈ (𝑈𝑋)) → 𝑄𝑌)
 
TheorempclssN 36912 Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝑌𝑌𝐴) → (𝑈𝑋) ⊆ (𝑈𝑌))
 
TheorempclssidN 36913 A set of atoms is included in its projective subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴) → 𝑋 ⊆ (𝑈𝑋))
 
TheorempclidN 36914 The projective subspace closure of a projective subspace is itself. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝑆) → (𝑈𝑋) = 𝑋)
 
TheorempclbtwnN 36915 A projective subspace sandwiched between a set of atoms and the set's projective subspace closure equals the closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &   𝑈 = (PCl‘𝐾)       (((𝐾𝑉𝑋𝑆) ∧ (𝑌𝑋𝑋 ⊆ (𝑈𝑌))) → 𝑋 = (𝑈𝑌))
 
TheorempclunN 36916 The projective subspace closure of the union of two sets of atoms equals the closure of their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾𝑉𝑋𝐴𝑌𝐴) → (𝑈‘(𝑋𝑌)) = (𝑈‘(𝑋 + 𝑌)))
 
Theorempclun2N 36917 The projective subspace closure of the union of two subspaces equals their projective sum. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝑆𝑌𝑆) → (𝑈‘(𝑋𝑌)) = (𝑋 + 𝑌))
 
TheorempclfinN 36918* 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 36968. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐴) → (𝑈𝑋) = 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈𝑦))
 
TheorempclcmpatN 36919* The set of projective subspaces is compactly atomistic: if an atom is in the projective subspace closure of a set of atoms, it also belongs to the projective subspace closure of a finite subset of that set. Analogous to Lemma 3.3.10 of [PtakPulmannova] p. 74. (Contributed by NM, 10-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ AtLat ∧ 𝑋𝐴𝑃 ∈ (𝑈𝑋)) → ∃𝑦 ∈ Fin (𝑦𝑋𝑃 ∈ (𝑈𝑦)))
 
SyntaxcpolN 36920 Extend class notation with polarity of projective subspace $m$.
class 𝑃
 
Definitiondf-polarityN 36921* Define polarity of projective subspace, which is a kind of complement of the subspace. Item 2 in [Holland95] p. 222 bottom. For more generality, we define it for all subsets of atoms, not just projective subspaces. The intersection with Atoms‘𝑙 ensures it is defined when 𝑚 = ∅. (Contributed by NM, 23-Oct-2011.)
𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙) ↦ ((Atoms‘𝑙) ∩ 𝑝𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝)))))
 
TheorempolfvalN 36922* The projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       (𝐾𝐵𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 𝑝𝑚 (𝑀‘( 𝑝)))))
 
TheorempolvalN 36923* Value of the projective subspace polarity function. (Contributed by NM, 23-Oct-2011.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑃𝑋) = (𝐴 𝑝𝑋 (𝑀‘( 𝑝))))
 
Theorempolval2N 36924 Alternate expression for value of the projective subspace polarity function. Equation for polarity in [Holland95] p. 223. (Contributed by NM, 22-Jan-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑃𝑋) = (𝑀‘( ‘(𝑈𝑋))))
 
TheorempolsubN 36925 The polarity of a set of atoms is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ∈ 𝑆)
 
TheorempolssatN 36926 The polarity of a set of atoms is a set of atoms. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( 𝑋) ⊆ 𝐴)
 
Theorempol0N 36927 The polarity of the empty projective subspace is the whole space. (Contributed by NM, 29-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       (𝐾𝐵 → ( ‘∅) = 𝐴)
 
Theorempol1N 36928 The polarity of the whole projective subspace is the empty space. Remark in [Holland95] p. 223. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       (𝐾 ∈ HL → ( 𝐴) = ∅)
 
Theorem2pol0N 36929 The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
= (⊥𝑃𝐾)       (𝐾 ∈ HL → ( ‘( ‘∅)) = ∅)
 
TheorempolpmapN 36930 The polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑃‘(𝑀𝑋)) = (𝑀‘( 𝑋)))
 
Theorem2polpmapN 36931 Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → ( ‘( ‘(𝑀𝑋))) = (𝑀𝑋))
 
Theorem2polvalN 36932 Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → ( ‘( 𝑋)) = (𝑀‘(𝑈𝑋)))
 
Theorem2polssN 36933 A set of atoms is a subset of its double polarity. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → 𝑋 ⊆ ( ‘( 𝑋)))
 
Theorem3polN 36934 Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → ( ‘( ‘( 𝑆))) = ( 𝑆))
 
Theorempolcon3N 36935 Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))
 
Theorem2polcon4bN 36936 Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (( ‘( 𝑋)) ⊆ ( ‘( 𝑌)) ↔ ( 𝑌) ⊆ ( 𝑋)))
 
Theorempolcon2N 36937 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑌𝐴𝑋 ⊆ ( 𝑌)) → 𝑌 ⊆ ( 𝑋))
 
Theorempolcon2bN 36938 Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴𝑌𝐴) → (𝑋 ⊆ ( 𝑌) ↔ 𝑌 ⊆ ( 𝑋)))
 
Theorempclss2polN 36939 The projective subspace closure is a subset of closed subspace closure. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑈𝑋) ⊆ ( ‘( 𝑋)))
 
Theorempcl0N 36940 The projective subspace closure of the empty subspace. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
𝑈 = (PCl‘𝐾)       (𝐾 ∈ HL → (𝑈‘∅) = ∅)
 
Theorempcl0bN 36941 The projective subspace closure of the empty subspace. (Contributed by NM, 13-Sep-2013.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑈 = (PCl‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → ((𝑈𝑃) = ∅ ↔ 𝑃 = ∅))
 
TheorempmaplubN 36942 The LUB of a projective map is the projective map's argument. (Contributed by NM, 13-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑈 = (lub‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵) → (𝑈‘(𝑀𝑋)) = 𝑋)
 
TheoremsspmaplubN 36943 A set of atoms is a subset of the projective map of its LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → 𝑆 ⊆ (𝑀‘(𝑈𝑆)))
 
Theorem2pmaplubN 36944 Double projective map of an LUB. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝑈 = (lub‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈𝑆)))) = (𝑀‘(𝑈𝑆)))
 
TheorempaddunN 36945 The closure of the projective sum of two sets of atoms is the same as the closure of their union. (Closure is actually double polarity, which can be trivially inferred from this theorem using fveq2d 6668.) (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = ( ‘(𝑆𝑇)))
 
Theorempoldmj1N 36946 De Morgan's law for polarity of projective sum. (oldmj1 36239 analog.) (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴) → ( ‘(𝑆 + 𝑇)) = (( 𝑆) ∩ ( 𝑇)))
 
Theorempmapj2N 36947 The projective map of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &   𝑀 = (pmap‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝑀‘(𝑋 𝑌)) = ( ‘( ‘((𝑀𝑋) + (𝑀𝑌)))))
 
TheorempmapocjN 36948 The projective map of the orthocomplement of the join of two lattice elements. (Contributed by NM, 14-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐹 = (pmap‘𝐾)    &    + = (+𝑃𝐾)    &   𝑁 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐵𝑌𝐵) → (𝐹‘( ‘(𝑋 𝑌))) = (𝑁‘((𝐹𝑋) + (𝐹𝑌))))
 
TheorempolatN 36949 The polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
= (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ OL ∧ 𝑄𝐴) → (𝑃‘{𝑄}) = (𝑀‘( 𝑄)))
 
Theorem2polatN 36950 Double polarity of the singleton of an atom (i.e. a point). (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑄𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄})
 
TheorempnonsingN 36951 The intersection of a set of atoms and its polarity is empty. Definition of nonsingular in [Holland95] p. 214. (Contributed by NM, 29-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (⊥𝑃𝐾)       ((𝐾 ∈ HL ∧ 𝑋𝐴) → (𝑋 ∩ (𝑃𝑋)) = ∅)
 
SyntaxcpscN 36952 Extend class notation with set of all closed projective subspaces for a Hilbert lattice.
class PSubCl
 
Definitiondf-psubclN 36953* Define set of all closed projective subspaces, which are those sets of atoms that equal their double polarity. Based on definition in [Holland95] p. 223. (Contributed by NM, 23-Jan-2012.)
PSubCl = (𝑘 ∈ V ↦ {𝑠 ∣ (𝑠 ⊆ (Atoms‘𝑘) ∧ ((⊥𝑃𝑘)‘((⊥𝑃𝑘)‘𝑠)) = 𝑠)})
 
TheorempsubclsetN 36954* The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾𝐵𝐶 = {𝑠 ∣ (𝑠𝐴 ∧ ( ‘( 𝑠)) = 𝑠)})
 
TheoremispsubclN 36955 The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    = (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       (𝐾𝐷 → (𝑋𝐶 ↔ (𝑋𝐴 ∧ ( ‘( 𝑋)) = 𝑋)))
 
TheorempsubcliN 36956 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 36957 Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)
= (⊥𝑃𝐾)    &   𝐶 = (PSubCl‘𝐾)       ((𝐾𝐷𝑋𝐶) → ( ‘( 𝑋)) = 𝑋)
 
TheorempsubclsubN 36958 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 36959 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 36960 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 36961 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 36962 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 36963 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 36964 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 36965* 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 36966 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 36967 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 36968 The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 36918 and also pclcmpatN 36919. (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 36969 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 36970 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 36971 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 36972 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 36973 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 36974 Lemma for osumclN 36985. (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 36975 Lemma for osumclN 36985. (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 36976 Lemma for osumclN 36985. (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 36977 Lemma for osumclN 36985. (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 36978 Lemma for osumclN 36985. (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 36979 Lemma for osumclN 36985. Use atom exchange hlatexch1 36413 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 36980* Lemma for osumclN 36985. (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 36981 Lemma for osumclN 36985. (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 36982 Lemma for osumclN 36985. (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 36983 Lemma for osumclN 36985. Contradict osumcllem9N 36982. (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 36984 Lemma for osumclN 36985. (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 36985 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 36986 For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 36870 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 36987 Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 36971. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 36985. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)       (((𝐾 ∈ HL ∧ 𝑋𝐴) ∧ ( ‘( 𝑋)) = 𝑋) → (𝑋 + ( 𝑋)) = 𝐴)
 
Theorempexmidlem1N 36988 Lemma for pexmidN 36987. 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 36989 Lemma for pexmidN 36987. (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 36990 Lemma for pexmidN 36987. Use atom exchange hlatexch1 36413 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 36991* Lemma for pexmidN 36987. (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 36992 Lemma for pexmidN 36987. (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 36993 Lemma for pexmidN 36987. (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 36994 Lemma for pexmidN 36987. Contradict pexmidlem6N 36993. (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 36995 Lemma for pexmidN 36987. The contradiction of pexmidlem6N 36993 and pexmidlem7N 36994 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 36996 Excluded middle law for closed projective subspaces, which is equivalent to (and derived from) the orthomodular law poml4N 36971. 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 36997 Lemma for pl42N 37001. (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 36998 Lemma for pl42N 37001. (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 36999 Lemma for pl42N 37001. (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 37000 Lemma for pl42N 37001. (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))))))
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