P. 366 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	paddcom 36501	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 36502	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 36503	A projective subspace sum is a superset of its first summand. (ssun1 4075 analog.) (Contributed by NM, 3-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
Theorem	sspadd2 36504	A projective subspace sum is a superset of its second summand. (ssun2 4076 analog.) (Contributed by NM, 3-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))
Theorem	paddss1 36505	Subset law for projective subspace sum. (unss1 4082 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 36506	Subset law for projective subspace sum. (unss2 4084 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 36507	Subset law for projective subspace sum. (unss12 4085 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 36508	Lemma for paddass 36526. (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 36509	Lemma for paddass 36526. (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 36510*	Lemma for paddass 36526. Restate projective space axiom ps-2 36166. (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 36511*	Lemma for paddass 36526. Combine paddasslem1 36508, paddasslem2 36509, and paddasslem3 36510. (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 36512	Lemma for paddass 36526. 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 36513	Lemma for paddass 36526. (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 36514	Lemma for paddass 36526. Combine paddasslem5 36512 and paddasslem6 36513. (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 36515	Lemma for paddass 36526. (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 36516	Lemma for paddass 36526. Combine paddasslem7 36514 and paddasslem8 36515. (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 36517	Lemma for paddass 36526. Use paddasslem4 36511 to eliminate 𝑠 from paddasslem9 36516. (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 36518	Lemma for paddass 36526. 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 36519	Lemma for paddass 36526. 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 36520	Lemma for paddass 36526. 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 36521	Lemma for paddass 36526. Remove 𝑝 ≠ 𝑧, 𝑥 ≠ 𝑦, and ¬ 𝑟 ≤ (𝑥 ∨ 𝑦) from antecedent of paddasslem10 36517, using paddasslem11 36518, paddasslem12 36519, and paddasslem13 36520. (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 36522	Lemma for paddass 36526. Use elpaddn0 36488 to eliminate 𝑦 and 𝑧 from paddasslem14 36521. (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 36523	Lemma for paddass 36526. Use elpaddn0 36488 to eliminate 𝑥 and 𝑟 from paddasslem15 36522. (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 36524	Lemma for paddass 36526. 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 36525	Lemma for paddass 36526. Combine paddasslem16 36523 and paddasslem17 36524. (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 36526	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 36527	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 36528	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 36529	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 36530	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 36531	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 36532	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 36533	Subset law for projective subspace sum. (unss 4087 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 36534*	Lemma for pmod1i 36536. (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 36535	Lemma for pmod1i 36536. (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 36536	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 36537	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 36538	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 36539	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 36540	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 36541	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 36542	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 36543	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 36544	A version of the modular law pmod1i 36536 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 36545	Version of modular law pmod1i 36536 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 36546	Version of modular law pmod1i 36536 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 36547	Version of modular law pmod1i 36536 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 36548	Version of modular law pmod1i 36536 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 36549	Version of modular law pmod2iN 36537 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 36550	Version of modular law pmod2iN 36537 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 36551	Version of modular law pmod1i 36536 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 36552	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 36553	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 36554	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 36555	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 36556	Lemma for llnexchb2 36557. (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 36557	Line exchange property (compare cvlatexchb2 36023 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 36558	Line exchange property (compare cvlatexch2 36025 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 36559	Lemma for dalaw 36574. Special case of dath2 36425, where 𝐶 is replaced by ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)). The remaining lemmas will eliminate the conditions on the atoms imposed by dath2 36425. (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 36560	Lemma for dalaw 36574. 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 36561	Lemma for dalaw 36574. First piece of dalawlem5 36563. (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 36562	Lemma for dalaw 36574. Second piece of dalawlem5 36563. (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 36563	Lemma for dalaw 36574. Special case to eliminate the requirement ¬ (𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑃 ∨ 𝑄) in dalawlem1 36559. (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 36564	Lemma for dalaw 36574. First piece of dalawlem8 36566. (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 36565	Lemma for dalaw 36574. Second piece of dalawlem8 36566. (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 36566	Lemma for dalaw 36574. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑄 ∨ 𝑅) in dalawlem1 36559. (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 36567	Lemma for dalaw 36574. Special case to eliminate the requirement ¬ ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇)) ≤ (𝑅 ∨ 𝑃) in dalawlem1 36559. (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 36568	Lemma for dalaw 36574. Combine dalawlem5 36563, dalawlem8 36566, 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 36569	Lemma for dalaw 36574. First part of dalawlem13 36571. (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 36570	Lemma for dalaw 36574. Second part of dalawlem13 36571. (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 36571	Lemma for dalaw 36574. Special case to eliminate the requirement ((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 in dalawlem1 36559. (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 36572	Lemma for dalaw 36574. Combine dalawlem10 36568 and dalawlem13 36571. (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 36573	Lemma for dalaw 36574. Swap variable triples 𝑃𝑄𝑅 and 𝑆𝑇𝑈 in dalawlem14 36572, to obtain the elimination of the remaining conditions in dalawlem1 36559. (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 36574	Desargues's law, derived from Desargues's theorem dath 36424 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 36575	Extend class notation with projective subspace closure.
class PCl
Definition	df-pclN 36576*	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 36623.) (Contributed by NM, 7-Sep-2013.)
⊢ PCl = (𝑘 ∈ V ↦ (𝑥 ∈ 𝒫 (Atoms‘𝑘) ↦ ∩ {𝑦 ∈ (PSubSp‘𝑘) ∣ 𝑥 ⊆ 𝑦}))
Theorem	pclfvalN 36577*	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 36578*	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 36579	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 36580*	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 36581	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 36582	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 36583	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 36584	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 36585	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 36586	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 36587	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 36588*	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 36638. (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 36589*	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 36590	Extend class notation with polarity of projective subspace $m$.
class ⊥𝑃
Definition	df-polarityN 36591*	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 36592*	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 36593*	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 36594	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 36595	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 36596	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 36597	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 36598	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 36599	The closed subspace closure of the empty set. (Contributed by NM, 12-Sep-2013.) (New usage is discouraged.)
⊢ ⊥ = (⊥𝑃‘𝐾)    ⇒   ⊢ (𝐾 ∈ HL → ( ⊥ ‘( ⊥ ‘∅)) = ∅)
Theorem	polpmapN 36600	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)))