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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | dalawlem13 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)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem14 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)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem15 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)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalaw 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 ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))))) | ||
Syntax | cpclN 36905 | Extend class notation with projective subspace closure. |
class PCl | ||
Definition | df-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) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | pclfvalN 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) ⇒ ⊢ (𝐾 ∈ 𝑉 → 𝑈 = (𝑥 ∈ 𝒫 𝐴 ↦ ∩ {𝑦 ∈ 𝑆 ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | pclvalN 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∩ {𝑦 ∈ 𝑆 ∣ 𝑋 ⊆ 𝑦}) | ||
Theorem | pclclN 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ∈ 𝑆) | ||
Theorem | elpclN 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 ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → (𝑄 ∈ (𝑈‘𝑋) ↔ ∀𝑦 ∈ 𝑆 (𝑋 ⊆ 𝑦 → 𝑄 ∈ 𝑦))) | ||
Theorem | elpcliN 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ∈ 𝑆) ∧ 𝑄 ∈ (𝑈‘𝑋)) → 𝑄 ∈ 𝑌) | ||
Theorem | pclssN 36912 | Ordering is preserved by subspace closure. (Contributed by NM, 8-Sep-2013.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = (PCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ (𝑈‘𝑌)) | ||
Theorem | pclssidN 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ (𝑈‘𝑋)) | ||
Theorem | pclidN 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) → (𝑈‘𝑋) = 𝑋) | ||
Theorem | pclbtwnN 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑋 ∈ 𝑆) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑋 ⊆ (𝑈‘𝑌))) → 𝑋 = (𝑈‘𝑌)) | ||
Theorem | pclunN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ 𝑈 = (PCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑈‘(𝑋 + 𝑌))) | ||
Theorem | pclun2N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ 𝑈 = (PCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑈‘(𝑋 ∪ 𝑌)) = (𝑋 + 𝑌)) | ||
Theorem | pclfinN 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 ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) = ∪ 𝑦 ∈ (Fin ∩ 𝒫 𝑋)(𝑈‘𝑦)) | ||
Theorem | pclcmpatN 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 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑃 ∈ (𝑈‘𝑋)) → ∃𝑦 ∈ Fin (𝑦 ⊆ 𝑋 ∧ 𝑃 ∈ (𝑈‘𝑦))) | ||
Syntax | cpolN 36920 | Extend class notation with polarity of projective subspace $m$. |
class ⊥𝑃 | ||
Definition | df-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) ∩ ∩ 𝑝 ∈ 𝑚 ((pmap‘𝑙)‘((oc‘𝑙)‘𝑝))))) | ||
Theorem | polfvalN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐵 → 𝑃 = (𝑚 ∈ 𝒫 𝐴 ↦ (𝐴 ∩ ∩ 𝑝 ∈ 𝑚 (𝑀‘( ⊥ ‘𝑝))))) | ||
Theorem | polvalN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝐴 ∩ ∩ 𝑝 ∈ 𝑋 (𝑀‘( ⊥ ‘𝑝)))) | ||
Theorem | polval2N 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑃‘𝑋) = (𝑀‘( ⊥ ‘(𝑈‘𝑋)))) | ||
Theorem | polsubN 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝑆) | ||
Theorem | polssatN 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ⊆ 𝐴) | ||
Theorem | pol0N 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐵 → ( ⊥ ‘∅) = 𝐴) | ||
Theorem | pol1N 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ HL → ( ⊥ ‘𝐴) = ∅) | ||
Theorem | 2pol0N 36929 | The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅) | ||
Theorem | polpmapN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → (𝑃‘(𝑀‘𝑋)) = (𝑀‘( ⊥ ‘𝑋))) | ||
Theorem | 2polpmapN 36931 | Double polarity of a projective map. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ( ⊥ ‘( ⊥ ‘(𝑀‘𝑋))) = (𝑀‘𝑋)) | ||
Theorem | 2polvalN 36932 | Value of double polarity. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
⊢ 𝑈 = (lub‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑋)) = (𝑀‘(𝑈‘𝑋))) | ||
Theorem | 2polssN 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) | ||
Theorem | 3polN 36934 | Triple polarity cancels to a single polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘𝑆))) = ( ⊥ ‘𝑆)) | ||
Theorem | polcon3N 36935 | Contraposition law for polarity. Remark in [Holland95] p. 223. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) | ||
Theorem | 2polcon4bN 36936 | Contraposition law for polarity. (Contributed by NM, 6-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌)) ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) | ||
Theorem | polcon2N 36937 | Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → 𝑌 ⊆ ( ⊥ ‘𝑋)) | ||
Theorem | polcon2bN 36938 | Contraposition law for polarity. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋))) | ||
Theorem | pclss2polN 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑈 = (PCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑈‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) | ||
Theorem | pcl0N 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‘∅) = ∅) | ||
Theorem | pcl0bN 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 ∧ 𝑃 ⊆ 𝐴) → ((𝑈‘𝑃) = ∅ ↔ 𝑃 = ∅)) | ||
Theorem | pmaplubN 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 ∧ 𝑋 ∈ 𝐵) → (𝑈‘(𝑀‘𝑋)) = 𝑋) | ||
Theorem | sspmaplubN 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 ∧ 𝑆 ⊆ 𝐴) → 𝑆 ⊆ (𝑀‘(𝑈‘𝑆))) | ||
Theorem | 2pmaplubN 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 ∧ 𝑆 ⊆ 𝐴) → (𝑀‘(𝑈‘(𝑀‘(𝑈‘𝑆)))) = (𝑀‘(𝑈‘𝑆))) | ||
Theorem | paddunN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = ( ⊥ ‘(𝑆 ∪ 𝑇))) | ||
Theorem | poldmj1N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐴 ∧ 𝑇 ⊆ 𝐴) → ( ⊥ ‘(𝑆 + 𝑇)) = (( ⊥ ‘𝑆) ∩ ( ⊥ ‘𝑇))) | ||
Theorem | pmapj2N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑀‘(𝑋 ∨ 𝑌)) = ( ⊥ ‘( ⊥ ‘((𝑀‘𝑋) + (𝑀‘𝑌))))) | ||
Theorem | pmapocjN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ 𝑁 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐹‘( ⊥ ‘(𝑋 ∨ 𝑌))) = (𝑁‘((𝐹‘𝑋) + (𝐹‘𝑌)))) | ||
Theorem | polatN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ OL ∧ 𝑄 ∈ 𝐴) → (𝑃‘{𝑄}) = (𝑀‘( ⊥ ‘𝑄))) | ||
Theorem | 2polatN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴) → (𝑃‘(𝑃‘{𝑄})) = {𝑄}) | ||
Theorem | pnonsingN 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 = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → (𝑋 ∩ (𝑃‘𝑋)) = ∅) | ||
Syntax | cpscN 36952 | Extend class notation with set of all closed projective subspaces for a Hilbert lattice. |
class PSubCl | ||
Definition | df-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) ∧ ((⊥𝑃‘𝑘)‘((⊥𝑃‘𝑘)‘𝑠)) = 𝑠)}) | ||
Theorem | psubclsetN 36954* | The set of closed projective subspaces in a Hilbert lattice. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐵 → 𝐶 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑠)) = 𝑠)}) | ||
Theorem | ispsubclN 36955 | The predicate "is a closed projective subspace". (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝐶 ↔ (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))) | ||
Theorem | psubcliN 36956 | Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) | ||
Theorem | psubcli2N 36957 | Property of a closed projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.) |
⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | ||
Theorem | psubclsubN 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 ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆) | ||
Theorem | psubclssatN 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) ⇒ ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴) | ||
Theorem | pmapidclN 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 ∧ 𝑋 ∈ 𝐶) → (𝑀‘(𝑈‘𝑋)) = 𝑋) | ||
Theorem | 0psubclN 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) | ||
Theorem | 1psubclN 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 → 𝐴 ∈ 𝐶) | ||
Theorem | atpsubclN 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 ∧ 𝑄 ∈ 𝐴) → {𝑄} ∈ 𝐶) | ||
Theorem | pmapsubclN 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 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝐶) | ||
Theorem | ispsubcl2N 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 → (𝑋 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑋 = (𝑀‘𝑦))) | ||
Theorem | psubclinN 36966 | The intersection of two closed subspaces is closed. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) → (𝑋 ∩ 𝑌) ∈ 𝐶) | ||
Theorem | paddatclN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑄 ∈ 𝐴) → (𝑋 + {𝑄}) ∈ 𝐶) | ||
Theorem | pclfinclN 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 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑋 ∈ Fin) → (𝑈‘𝑋) ∈ 𝑆) | ||
Theorem | linepsubclN 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 ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝐶) | ||
Theorem | polsubclN 36970 | A polarity is a closed projective subspace. (Contributed by NM, 24-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) → ( ⊥ ‘𝑋) ∈ 𝐶) | ||
Theorem | poml4N 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) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))) | ||
Theorem | poml5N 36972 | Orthomodular law for projective lattices. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) ∩ ( ⊥ ‘𝑌)) = ( ⊥ ‘( ⊥ ‘𝑋))) | ||
Theorem | poml6N 36973 | Orthomodular law for projective lattices. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ 𝑌) → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋) | ||
Theorem | osumcllem1N 36974 | Lemma for osumclN 36985. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → (𝑈 ∩ 𝑀) = 𝑀) | ||
Theorem | osumcllem2N 36975 | Lemma for osumclN 36985. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝑈) → 𝑋 ⊆ (𝑈 ∩ 𝑀)) | ||
Theorem | osumcllem3N 36976 | Lemma for osumclN 36985. (Contributed by NM, 23-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑌 ∈ 𝐶 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (( ⊥ ‘𝑋) ∩ 𝑈) = 𝑌) | ||
Theorem | osumcllem4N 36977 | Lemma for osumclN 36985. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑌 ⊆ 𝐴 ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌)) → 𝑞 ≠ 𝑟) | ||
Theorem | osumcllem5N 36978 | Lemma for osumclN 36985. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + 𝑌)) | ||
Theorem | osumcllem6N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ 𝑌 ∧ 𝑞 ≤ (𝑟 ∨ 𝑝))) → 𝑝 ∈ (𝑋 + 𝑌)) | ||
Theorem | osumcllem7N 36980* | Lemma for osumclN 36985. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴) ∧ 𝑞 ∈ (𝑌 ∩ 𝑀)) → 𝑝 ∈ (𝑋 + 𝑌)) | ||
Theorem | osumcllem8N 36981 | Lemma for osumclN 36985. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝐴) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → (𝑌 ∩ 𝑀) = ∅) | ||
Theorem | osumcllem9N 36982 | Lemma for osumclN 36985. (Contributed by NM, 24-Mar-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅ ∧ 𝑝 ∈ 𝑈) ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 = 𝑋) | ||
Theorem | osumcllem10N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) & ⊢ 𝑈 = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌))) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ 𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ∈ (𝑋 + 𝑌)) → 𝑀 ≠ 𝑋) | ||
Theorem | osumcllem11N 36984 | Lemma for osumclN 36985. (Contributed by NM, 25-Mar-2012.) (New usage is discouraged.) |
⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ (𝑋 ⊆ ( ⊥ ‘𝑌) ∧ 𝑋 ≠ ∅)) → (𝑋 + 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 + 𝑌)))) | ||
Theorem | osumclN 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.) |
⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝐶 = (PSubCl‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝑋 ⊆ ( ⊥ ‘𝑌)) → (𝑋 + 𝑌) ∈ 𝐶) | ||
Theorem | pmapojoinN 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) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ ( ⊥ ‘𝑌)) → (𝑀‘(𝑋 ∨ 𝑌)) = ((𝑀‘𝑋) + (𝑀‘𝑌))) | ||
Theorem | pexmidN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) | ||
Theorem | pexmidlem1N 36988 | Lemma for pexmidN 36987. Holland's proof implicitly requires 𝑞 ≠ 𝑟, which we prove here. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋))) → 𝑞 ≠ 𝑟) | ||
Theorem | pexmidlem2N 36989 | Lemma for pexmidN 36987. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋) ∧ 𝑝 ≤ (𝑟 ∨ 𝑞))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) | ||
Theorem | pexmidlem3N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑟 ∈ 𝑋 ∧ 𝑞 ∈ ( ⊥ ‘𝑋)) ∧ 𝑞 ≤ (𝑟 ∨ 𝑝)) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) | ||
Theorem | pexmidlem4N 36991* | Lemma for pexmidN 36987. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑞 ∈ (( ⊥ ‘𝑋) ∩ 𝑀))) → 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋))) | ||
Theorem | pexmidlem5N 36992 | Lemma for pexmidN 36987. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → (( ⊥ ‘𝑋) ∩ 𝑀) = ∅) | ||
Theorem | pexmidlem6N 36993 | Lemma for pexmidN 36987. (Contributed by NM, 3-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 = 𝑋) | ||
Theorem | pexmidlem7N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) & ⊢ 𝑀 = (𝑋 + {𝑝}) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑝 ∈ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅ ∧ ¬ 𝑝 ∈ (𝑋 + ( ⊥ ‘𝑋)))) → 𝑀 ≠ 𝑋) | ||
Theorem | pexmidlem8N 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ∧ 𝑋 ≠ ∅)) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) | ||
Theorem | pexmidALTN 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) & ⊢ + = (+𝑃‘𝐾) & ⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴) ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) → (𝑋 + ( ⊥ ‘𝑋)) = 𝐴) | ||
Theorem | pl42lem1N 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) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝑋 ≤ ( ⊥ ‘𝑌) ∧ 𝑍 ≤ ( ⊥ ‘𝑊)) → (𝐹‘((((𝑋 ∨ 𝑌) ∧ 𝑍) ∨ 𝑊) ∧ 𝑉)) = (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉)))) | ||
Theorem | pl42lem2N 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) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (((𝐹‘𝑋) + (𝐹‘𝑌)) + (((𝐹‘𝑋) + (𝐹‘𝑊)) ∩ ((𝐹‘𝑌) + (𝐹‘𝑉)))) ⊆ (𝐹‘((𝑋 ∨ 𝑌) ∨ ((𝑋 ∨ 𝑊) ∧ (𝑌 ∨ 𝑉))))) | ||
Theorem | pl42lem3N 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) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → (((((𝐹‘𝑋) + (𝐹‘𝑌)) ∩ (𝐹‘𝑍)) + (𝐹‘𝑊)) ∩ (𝐹‘𝑉)) ⊆ ((((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑊)) ∩ (((𝐹‘𝑋) + (𝐹‘𝑌)) + (𝐹‘𝑉)))) | ||
Theorem | pl42lem4N 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) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵 ∧ 𝑉 ∈ 𝐵)) → ((𝑋 ≤ ( ⊥ ‘𝑌) ∧ 𝑍 ≤ ( ⊥ ‘𝑊)) → (𝐹‘((((𝑋 ∨ 𝑌) ∧ 𝑍) ∨ 𝑊) ∧ 𝑉)) ⊆ (𝐹‘((𝑋 ∨ 𝑌) ∨ ((𝑋 ∨ 𝑊) ∧ (𝑌 ∨ 𝑉)))))) |
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