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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | paddfval 34101* | Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝐵 → + = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝 ≤ (𝑞 ∨ 𝑟)}))) | ||
Theorem | paddval 34102* | Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = ((𝑋 ∪ 𝑌) ∪ {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)})) | ||
Theorem | elpadd 34103* | Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑆 ∈ (𝑋 + 𝑌) ↔ ((𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌) ∨ (𝑆 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑆 ≤ (𝑞 ∨ 𝑟))))) | ||
Theorem | elpaddn0 34104* | Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑆 ≤ (𝑞 ∨ 𝑟)))) | ||
Theorem | paddvaln0N 34105* | Projective subspace sum operation value for nonempty sets. (Contributed by NM, 27-Jan-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝 ∈ 𝐴 ∣ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑌 𝑝 ≤ (𝑞 ∨ 𝑟)}) | ||
Theorem | elpaddri 34106 | Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑌) ∧ (𝑆 ∈ 𝐴 ∧ 𝑆 ≤ (𝑄 ∨ 𝑅))) → 𝑆 ∈ (𝑋 + 𝑌)) | ||
Theorem | elpaddatriN 34107 | Condition implying membership in a projective subspace sum with a point. (Contributed by NM, 1-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝑋 ∧ 𝑆 ∈ 𝐴 ∧ 𝑆 ≤ (𝑅 ∨ 𝑄))) → 𝑆 ∈ (𝑋 + {𝑄})) | ||
Theorem | elpaddat 34108* | Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆 ∈ 𝐴 ∧ ∃𝑝 ∈ 𝑋 𝑆 ≤ (𝑝 ∨ 𝑄)))) | ||
Theorem | elpaddatiN 34109* | Consequence of membership in a projective subspace sum with a point. (Contributed by NM, 2-Feb-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝 ∈ 𝑋 𝑅 ≤ (𝑝 ∨ 𝑄)) | ||
Theorem | elpadd2at 34110 | Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ (𝑆 ∈ 𝐴 ∧ 𝑆 ≤ (𝑄 ∨ 𝑅)))) | ||
Theorem | elpadd2at2 34111 | Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴)) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ 𝑆 ≤ (𝑄 ∨ 𝑅))) | ||
Theorem | paddunssN 34112 | Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 ∪ 𝑌) ⊆ (𝑋 + 𝑌)) | ||
Theorem | elpadd0 34113 | Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆 ∈ 𝑋 ∨ 𝑆 ∈ 𝑌))) | ||
Theorem | paddval0 34114 | Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = (𝑋 ∪ 𝑌)) | ||
Theorem | padd01 34115 | Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (𝑋 + ∅) = 𝑋) | ||
Theorem | padd02 34116 | Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴) → (∅ + 𝑋) = 𝑋) | ||
Theorem | paddcom 34117 | Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) | ||
Theorem | paddssat 34118 | A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → (𝑋 + 𝑌) ⊆ 𝐴) | ||
Theorem | sspadd1 34119 | A projective subspace sum is a superset of its first summand. (ssun1 3738 analog.) (Contributed by NM, 3-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌)) | ||
Theorem | sspadd2 34120 | A projective subspace sum is a superset of its second summand. (ssun2 3739 analog.) (Contributed by NM, 3-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋)) | ||
Theorem | paddss1 34121 | Subset law for projective subspace sum. (unss1 3744 analog.) (Contributed by NM, 7-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 ⊆ 𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍))) | ||
Theorem | paddss2 34122 | Subset law for projective subspace sum. (unss2 3746 analog.) (Contributed by NM, 7-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 ⊆ 𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌))) | ||
Theorem | paddss12 34123 | Subset law for projective subspace sum. (unss12 3747 analog.) (Contributed by NM, 7-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴) → ((𝑋 ⊆ 𝑌 ∧ 𝑍 ⊆ 𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊))) | ||
Theorem | paddasslem1 34124 | Lemma for paddass 34142. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ 𝑥 ≠ 𝑦) ∧ ¬ 𝑟 ≤ (𝑥 ∨ 𝑦)) → ¬ 𝑥 ≤ (𝑟 ∨ 𝑦)) | ||
Theorem | paddasslem2 34125 | Lemma for paddass 34142. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → 𝑧 ≤ (𝑟 ∨ 𝑦)) | ||
Theorem | paddasslem3 34126* | Lemma for paddass 34142. Restate projective space axiom ps-2 33782. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑥 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((¬ 𝑥 ≤ (𝑟 ∨ 𝑦) ∧ 𝑝 ≠ 𝑧) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑧 ≤ (𝑟 ∨ 𝑦))) → ∃𝑠 ∈ 𝐴 (𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) | ||
Theorem | paddasslem4 34127* | Lemma for paddass 34142. Combine paddasslem1 34124, paddasslem2 34125, and paddasslem3 34126. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ (𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦 ∧ ¬ 𝑟 ≤ (𝑥 ∨ 𝑦))) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧))) → ∃𝑠 ∈ 𝐴 (𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧))) | ||
Theorem | paddasslem5 34128 | Lemma for paddass 34142. Show 𝑠 ≠ 𝑧 by contradiction. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑟 ∈ 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦))) → 𝑠 ≠ 𝑧) | ||
Theorem | paddasslem6 34129 | Lemma for paddass 34142. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑝 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ 𝑧 ∈ 𝐴) ∧ (𝑠 ≠ 𝑧 ∧ 𝑠 ≤ (𝑝 ∨ 𝑧))) → 𝑝 ≤ (𝑠 ∨ 𝑧)) | ||
Theorem | paddasslem7 34130 | Lemma for paddass 34142. Combine paddasslem5 34128 and paddasslem6 34129. (Contributed by NM, 9-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ ((¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦)) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧))) → 𝑝 ≤ (𝑠 ∨ 𝑧)) | ||
Theorem | paddasslem8 34131 | Lemma for paddass 34142. (Contributed by NM, 8-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑠 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑠 ∨ 𝑧))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem9 34132 | Lemma for paddass 34142. Combine paddasslem7 34130 and paddasslem8 34131. (Contributed by NM, 9-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)) ∧ (𝑠 ∈ 𝐴 ∧ 𝑠 ≤ (𝑥 ∨ 𝑦) ∧ 𝑠 ≤ (𝑝 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem10 34133 | Lemma for paddass 34142. Use paddasslem4 34127 to eliminate 𝑠 from paddasslem9 34132. (Contributed by NM, 9-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧 ∧ 𝑥 ≠ 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (¬ 𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem11 34134 | Lemma for paddass 34142. The case when 𝑝 = 𝑧. (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑝 = 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) ∧ 𝑧 ∈ 𝑍) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem12 34135 | Lemma for paddass 34142. The case when 𝑥 = 𝑦. (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑥 = 𝑦) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem13 34136 | Lemma for paddass 34142. The case when 𝑟 ≤ (𝑥 ∨ 𝑦). (Unlike the proof in Maeda and Maeda, we don't need 𝑥 ≠ 𝑦.) (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑝 ≠ 𝑧) ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌) ∧ (𝑟 ≤ (𝑥 ∨ 𝑦) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem14 34137 | Lemma for paddass 34142. Remove 𝑝 ≠ 𝑧, 𝑥 ≠ 𝑦, and ¬ 𝑟 ≤ (𝑥 ∨ 𝑦) from antecedent of paddasslem10 34133, using paddasslem11 34134, paddasslem12 34135, and paddasslem13 34136. (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑝 ∈ 𝐴 ∧ 𝑟 ∈ 𝐴)) ∧ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑌 ∧ 𝑧 ∈ 𝑍) ∧ (𝑝 ≤ (𝑥 ∨ 𝑟) ∧ 𝑟 ≤ (𝑦 ∨ 𝑧)))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem15 34138 | Lemma for paddass 34142. Use elpaddn0 34104 to eliminate 𝑦 and 𝑧 from paddasslem14 34137. (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅)) ∧ (𝑝 ∈ 𝐴 ∧ (𝑥 ∈ 𝑋 ∧ 𝑟 ∈ (𝑌 + 𝑍)) ∧ 𝑝 ≤ (𝑥 ∨ 𝑟))) → 𝑝 ∈ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem16 34139 | Lemma for paddass 34142. Use elpaddn0 34104 to eliminate 𝑥 and 𝑟 from paddasslem15 34138. (Contributed by NM, 11-Jan-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem17 34140 | Lemma for paddass 34142. The case when at least one sum argument is empty. (Contributed by NM, 12-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) ∧ ¬ ((𝑋 ≠ ∅ ∧ (𝑌 + 𝑍) ≠ ∅) ∧ (𝑌 ≠ ∅ ∧ 𝑍 ≠ ∅))) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddasslem18 34141 | Lemma for paddass 34142. Combine paddasslem16 34139 and paddasslem17 34140. (Contributed by NM, 12-Jan-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + 𝑍)) ⊆ ((𝑋 + 𝑌) + 𝑍)) | ||
Theorem | paddass 34142 | Projective subspace sum is associative. Equation 16.2.1 of [MaedaMaeda] p. 68. In our version, the subspaces do not have to be nonempty. (Contributed by NM, 29-Dec-2011.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍))) | ||
Theorem | padd12N 34143 | Commutative/associative law for projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍))) | ||
Theorem | padd4N 34144 | Rearrangement of 4 terms in a projective subspace sum. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ⊆ 𝐴 ∧ 𝑊 ⊆ 𝐴)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊))) | ||
Theorem | paddidm 34145 | Projective subspace sum is idempotent. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 13-Jan-2012.) |
⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → (𝑋 + 𝑋) = 𝑋) | ||
Theorem | paddclN 34146 | The projective sum of two subspaces is a subspace. Part of Lemma 16.2 of [MaedaMaeda] p. 68. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.) |
⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 + 𝑌) ∈ 𝑆) | ||
Theorem | paddssw1 34147 | Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) → (𝑋 + 𝑌) ⊆ (𝑍 + 𝑍))) | ||
Theorem | paddssw2 34148 | Subset law for projective subspace sum valid for all subsets of atoms. (Contributed by NM, 14-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → ((𝑋 + 𝑌) ⊆ 𝑍 → (𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍))) | ||
Theorem | paddss 34149 | Subset law for projective subspace sum. (unss 3749 analog.) (Contributed by NM, 7-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ 𝐵 ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → ((𝑋 ⊆ 𝑍 ∧ 𝑌 ⊆ 𝑍) ↔ (𝑋 + 𝑌) ⊆ 𝑍)) | ||
Theorem | pmodlem1 34150* | Lemma for pmod1i 34152. (Contributed by NM, 9-Mar-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) ∧ (𝑍 ∈ 𝑆 ∧ 𝑋 ⊆ 𝑍 ∧ 𝑝 ∈ 𝑍) ∧ (𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑌 ∧ 𝑝 ≤ (𝑞 ∨ 𝑟))) → 𝑝 ∈ (𝑋 + (𝑌 ∩ 𝑍))) | ||
Theorem | pmodlem2 34151 | Lemma for pmod1i 34152. (Contributed by NM, 9-Mar-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆) ∧ 𝑋 ⊆ 𝑍) → ((𝑋 + 𝑌) ∩ 𝑍) ⊆ (𝑋 + (𝑌 ∩ 𝑍))) | ||
Theorem | pmod1i 34152 | The modular law holds in a projective subspace. (Contributed by NM, 10-Mar-2012.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ∈ 𝑆)) → (𝑋 ⊆ 𝑍 → ((𝑋 + 𝑌) ∩ 𝑍) = (𝑋 + (𝑌 ∩ 𝑍)))) | ||
Theorem | pmod2iN 34153 | Dual of the modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑍 ⊆ 𝑋 → ((𝑋 ∩ 𝑌) + 𝑍) = (𝑋 ∩ (𝑌 + 𝑍)))) | ||
Theorem | pmodN 34154 | The modular law for projective subspaces. (Contributed by NM, 26-Mar-2012.) (New usage is discouraged.) |
⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑆 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴)) → (𝑋 ∩ (𝑌 + (𝑋 ∩ 𝑍))) = ((𝑋 ∩ 𝑌) + (𝑋 ∩ 𝑍))) | ||
Theorem | pmodl42N 34155 | Lemma derived from modular law. (Contributed by NM, 8-Apr-2012.) (New usage is discouraged.) |
⊢ 𝑆 = (PSubSp‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) ∧ (𝑍 ∈ 𝑆 ∧ 𝑊 ∈ 𝑆)) → (((𝑋 + 𝑌) + 𝑍) ∩ ((𝑋 + 𝑌) + 𝑊)) = ((𝑋 + 𝑌) + ((𝑋 + 𝑍) ∩ (𝑌 + 𝑊)))) | ||
Theorem | pmapjoin 34156 | The projective map of the join of two lattice elements. Part of Equation 15.5.3 of [MaedaMaeda] p. 63. (Contributed by NM, 27-Jan-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑀‘𝑋) + (𝑀‘𝑌)) ⊆ (𝑀‘(𝑋 ∨ 𝑌))) | ||
Theorem | pmapjat1 34157 | The projective map of the join of a lattice element and an atom. (Contributed by NM, 28-Jan-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑋 ∨ 𝑄)) = ((𝑀‘𝑋) + (𝑀‘𝑄))) | ||
Theorem | pmapjat2 34158 | The projective map of the join of an atom with a lattice element. (Contributed by NM, 12-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴) → (𝑀‘(𝑄 ∨ 𝑋)) = ((𝑀‘𝑄) + (𝑀‘𝑋))) | ||
Theorem | pmapjlln1 34159 | The projective map of the join of a lattice element and a lattice line (expressed as the join 𝑄 ∨ 𝑅 of two atoms). (Contributed by NM, 16-Sep-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑀 = (pmap‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → (𝑀‘(𝑋 ∨ (𝑄 ∨ 𝑅))) = ((𝑀‘𝑋) + (𝑀‘(𝑄 ∨ 𝑅)))) | ||
Theorem | hlmod1i 34160 | A version of the modular law pmod1i 34152 that holds in a Hilbert lattice. (Contributed by NM, 13-May-2012.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐹 = (pmap‘𝐾) & ⊢ + = (+𝑃‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑍 ∧ (𝐹‘(𝑋 ∨ 𝑌)) = ((𝐹‘𝑋) + (𝐹‘𝑌))) → ((𝑋 ∨ 𝑌) ∧ 𝑍) = (𝑋 ∨ (𝑌 ∧ 𝑍)))) | ||
Theorem | atmod1i1 34161 | Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 11-May-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑌) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) | ||
Theorem | atmod1i1m 34162 | Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when an element meets an atom. (Contributed by NM, 2-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ (𝑋 ∧ 𝑃) ≤ 𝑍) → ((𝑋 ∧ 𝑃) ∨ (𝑌 ∧ 𝑍)) = (((𝑋 ∧ 𝑃) ∨ 𝑌) ∧ 𝑍)) | ||
Theorem | atmod1i2 34163 | Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ (𝑃 ∧ 𝑌)) = ((𝑋 ∨ 𝑃) ∧ 𝑌)) | ||
Theorem | llnmod1i2 34164 | Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 ∨ 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ ((𝑃 ∨ 𝑄) ∧ 𝑌)) = ((𝑋 ∨ (𝑃 ∨ 𝑄)) ∧ 𝑌)) | ||
Theorem | atmod2i1 34165 | Version of modular law pmod2iN 34153 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → ((𝑋 ∧ 𝑌) ∨ 𝑃) = (𝑋 ∧ (𝑌 ∨ 𝑃))) | ||
Theorem | atmod2i2 34166 | Version of modular law pmod2iN 34153 that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 14-May-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ 𝑃) ∨ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑌))) | ||
Theorem | llnmod2i2 34167 | Version of modular law pmod1i 34152 that holds in a Hilbert lattice, when one element is a lattice line (expressed as the join 𝑃 ∨ 𝑄). (Contributed by NM, 16-Sep-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑌 ≤ 𝑋) → ((𝑋 ∧ (𝑃 ∨ 𝑄)) ∨ 𝑌) = (𝑋 ∧ ((𝑃 ∨ 𝑄) ∨ 𝑌))) | ||
Theorem | atmod3i1 34168 | Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑋) → (𝑃 ∨ (𝑋 ∧ 𝑌)) = (𝑋 ∧ (𝑃 ∨ 𝑌))) | ||
Theorem | atmod3i2 34169 | Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ∨ (𝑌 ∧ 𝑃)) = (𝑌 ∧ (𝑋 ∨ 𝑃))) | ||
Theorem | atmod4i1 34170 | Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 10-Jun-2012.) (Revised by Mario Carneiro, 10-May-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑃 ≤ 𝑌) → ((𝑋 ∧ 𝑌) ∨ 𝑃) = ((𝑋 ∨ 𝑃) ∧ 𝑌)) | ||
Theorem | atmod4i2 34171 | Version of modular law that holds in a Hilbert lattice, when one element is an atom. (Contributed by NM, 4-Jun-2012.) (Revised by Mario Carneiro, 10-Mar-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → ((𝑃 ∧ 𝑌) ∨ 𝑋) = ((𝑃 ∨ 𝑋) ∧ 𝑌)) | ||
Theorem | llnexchb2lem 34172 | Lemma for llnexchb2 34173. (Contributed by NM, 17-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑁 = (LLines‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑋) ∧ (𝑋 ∧ 𝑌) ∈ 𝐴) → ((𝑋 ∧ 𝑌) ≤ (𝑃 ∨ 𝑄) ↔ (𝑋 ∧ 𝑌) = (𝑋 ∧ (𝑃 ∨ 𝑄)))) | ||
Theorem | llnexchb2 34173 | Line exchange property (compare cvlatexchb2 33640 for atoms). (Contributed by NM, 17-Nov-2012.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑁 = (LLines‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) ≤ 𝑍 ↔ (𝑋 ∧ 𝑌) = (𝑋 ∧ 𝑍))) | ||
Theorem | llnexch2N 34174 | Line exchange property (compare cvlatexch2 33642 for atoms). (Contributed by NM, 18-Nov-2012.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑁 = (LLines‘𝐾) ⇒ ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝑁 ∧ 𝑌 ∈ 𝑁 ∧ 𝑍 ∈ 𝑁) ∧ ((𝑋 ∧ 𝑌) ∈ 𝐴 ∧ 𝑋 ≠ 𝑍)) → ((𝑋 ∧ 𝑌) ≤ 𝑍 → (𝑋 ∧ 𝑍) ≤ 𝑌)) | ||
Theorem | dalawlem1 34175 | Lemma for dalaw 34190. Special case of dath2 34041, where 𝐶 is replaced by ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 34041. (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 34176 | Lemma for dalaw 34190. 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 34177 | Lemma for dalaw 34190. First piece of dalawlem5 34179. (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 34178 | Lemma for dalaw 34190. Second piece of dalawlem5 34179. (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 34179 | Lemma for dalaw 34190. Special case to eliminate the requirement ¬ (𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) in dalawlem1 34175. (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 34180 | Lemma for dalaw 34190. First piece of dalawlem8 34182. (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 34181 | Lemma for dalaw 34190. Second piece of dalawlem8 34182. (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 34182 | Lemma for dalaw 34190. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) in dalawlem1 34175. (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 34183 | Lemma for dalaw 34190. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃) in dalawlem1 34175. (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 34184 | Lemma for dalaw 34190. Combine dalawlem5 34179, dalawlem8 34182, 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 34185 | Lemma for dalaw 34190. First part of dalawlem13 34187. (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 34186 | Lemma for dalaw 34190. Second part of dalawlem13 34187. (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 34187 | Lemma for dalaw 34190. Special case to eliminate the requirement ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 in dalawlem1 34175. (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 34188 | Lemma for dalaw 34190. Combine dalawlem10 34184 and dalawlem13 34187. (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 34189 | Lemma for dalaw 34190. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 34188, to obtain the elimination of the remaining conditions in dalawlem1 34175. (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 34190 | Desargues' law, derived from Desargues' theorem dath 34040 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 34191 | Extend class notation with projective subspace closure. |
class PCl | ||
Definition | df-pclN 34192* | 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 34239.) (Contributed by NM, 7-Sep-2013.) |
⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦})) | ||
Theorem | pclfvalN 34193* | 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 34194* | 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 34195 | 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 34196* | 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 34197 | 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 34198 | 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 34199 | 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 34200 | 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) |
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