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Theorem | dalawlem1 37001 | Lemma for dalaw 37016. Special case of dath2 36867, where 𝐶 is replaced by ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 36867. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑂 = (LPlanes‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ 𝑂) ∧ ((¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃)) ∧ (¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑆 ∨ 𝑇) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑇 ∨ 𝑈) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑈 ∨ 𝑆)) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈))) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem2 37002 | Lemma for dalaw 37016. Utility lemma that breaks ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) into a join of two pieces. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴)) → ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇)) ≤ ((((𝑃 ∨ 𝑄) ∨ 𝑇) ∧ 𝑆) ∨ (((𝑃 ∨ 𝑄) ∨ 𝑆) ∧ 𝑇))) | ||
Theorem | dalawlem3 37003 | Lemma for dalaw 37016. First piece of dalawlem5 37005. (Contributed by NM, 4-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑄 ∨ 𝑇) ∨ 𝑃) ∧ 𝑆) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem4 37004 | Lemma for dalaw 37016. Second piece of dalawlem5 37005. (Contributed by NM, 4-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑆) ∨ 𝑄) ∧ 𝑇) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem5 37005 | Lemma for dalaw 37016. Special case to eliminate the requirement ¬ (𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) in dalawlem1 37001. (Contributed by NM, 4-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem6 37006 | Lemma for dalaw 37016. First piece of dalawlem8 37008. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) → (((𝑃 ∨ 𝑄) ∨ 𝑇) ∧ 𝑆) ≤ (((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈)) ∨ ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆)))) | ||
Theorem | dalawlem7 37007 | Lemma for dalaw 37016. Second piece of dalawlem8 37008. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem8 37008 | Lemma for dalaw 37016. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) in dalawlem1 37001. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem9 37009 | Lemma for dalaw 37016. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃) in dalawlem1 37001. (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem10 37010 | Lemma for dalaw 37016. Combine dalawlem5 37005, dalawlem8 37008, and dalawlem9 . (Contributed by NM, 6-Oct-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ ¬ (¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) ∧ ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃)) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem11 37011 | Lemma for dalaw 37016. First part of dalawlem13 37013. (Contributed by NM, 17-Sep-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑃 ≤ (𝑄 ∨ 𝑅) ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem12 37012 | Lemma for dalaw 37016. Second part of dalawlem13 37013. (Contributed by NM, 17-Sep-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑄 = 𝑅 ∧ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑈)) ∧ (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 | dalawlem13 37013 | Lemma for dalaw 37016. Special case to eliminate the requirement ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 in dalawlem1 37001. (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 37014 | Lemma for dalaw 37016. Combine dalawlem10 37010 and dalawlem13 37013. (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 37015 | Lemma for dalaw 37016. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 37014, to obtain the elimination of the remaining conditions in dalawlem1 37001. (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 37016 | Desargues's law, derived from Desargues's theorem dath 36866 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 37017 | Extend class notation with projective subspace closure. |
class PCl | ||
Definition | df-pclN 37018* | 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 37065.) (Contributed by NM, 7-Sep-2013.) |
⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | pclfvalN 37019* | 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 37020* | 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 37021 | 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 37022* | 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 37023 | 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 37024 | 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 37025 | 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 37026 | 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 37027 | 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 37028 | 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 37029 | 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 37030* | 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 37080. (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 37031* | 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 37032 | Extend class notation with polarity of projective subspace $m$. |
class ⊥𝑃 | ||
Definition | df-polarityN 37033* | 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 37034* | 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 37035* | 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 37036 | 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 37037 | 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 37038 | 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 37039 | 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 37040 | 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 37041 | The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.) |
⊢ ⊥ = (⊥𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅) | ||
Theorem | polpmapN 37042 | 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 37043 | 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 37044 | 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 37045 | 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 37046 | 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 37047 | 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 37048 | 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 37049 | 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 37050 | 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 37051 | 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 37052 | 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 37053 | 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 37054 | 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 37055 | 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 37056 | 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 37057 | 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 37058 | De Morgan's law for polarity of projective sum. (oldmj1 36351 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 37059 | 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 37060 | 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 37061 | 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 37062 | 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 37063 | 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 37064 | Extend class notation with set of all closed projective subspaces for a Hilbert lattice. |
class PSubCl | ||
Definition | df-psubclN 37065* | 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 37066* | 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 37067 | 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 37068 | 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 37069 | 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 37070 | 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 37071 | 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 37072 | 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 37073 | 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 37074 | 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 37075 | 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 37076 | 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 37077* | 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 37078 | 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 37079 | 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 37080 | The projective subspace closure of a finite set of atoms is a closed subspace. Compare the (non-closed) subspace version pclfinN 37030 and also pclcmpatN 37031. (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 37081 | 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 37082 | 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 37083 | 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 37084 | 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 37085 | 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 37086 | Lemma for osumclN 37097. (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 37087 | Lemma for osumclN 37097. (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 37088 | Lemma for osumclN 37097. (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 37089 | Lemma for osumclN 37097. (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 37090 | Lemma for osumclN 37097. (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 37091 | Lemma for osumclN 37097. Use atom exchange hlatexch1 36525 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 37092* | Lemma for osumclN 37097. (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 37093 | Lemma for osumclN 37097. (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 37094 | Lemma for osumclN 37097. (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 37095 | Lemma for osumclN 37097. Contradict osumcllem9N 37094. (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 37096 | Lemma for osumclN 37097. (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 37097 | 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 37098 | For orthogonal elements, projective map of join equals projective sum. Compare pmapjoin 36982 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 37099 | Excluded middle law for closed projective subspaces, which can be shown to be equivalent to (and derivable from) the orthomodular law poml4N 37083. Lemma 3.3(2) in [Holland95] p. 215, which we prove as a special case of osumclN 37097. (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 37100 | Lemma for pexmidN 37099. 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) |
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