P. 398 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39701-39800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dalem47 39701	Lemma for dath 39718. Analogue of dalem45 39699 for 𝐼𝐺. (Contributed by NM, 16-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ¬ 𝑐 ≤ (𝐼 ∨ 𝐺))
Theorem	dalem48 39702	Lemma for dath 39718. Analogue of dalem45 39699 for 𝑃𝑄. (Contributed by NM, 16-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑃 ∨ 𝑄))
Theorem	dalem49 39703	Lemma for dath 39718. Analogue of dalem45 39699 for 𝑄𝑅. (Contributed by NM, 16-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑄 ∨ 𝑅))
Theorem	dalem50 39704	Lemma for dath 39718. Analogue of dalem45 39699 for 𝑅𝑃. (Contributed by NM, 16-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → ¬ 𝑐 ≤ (𝑅 ∨ 𝑃))
Theorem	dalem51 39705	Lemma for dath 39718. Construct the condition 𝜑 with 𝑐, 𝐺𝐻𝐼, and 𝑌 in place of 𝐶, 𝑌, and 𝑍 respectively. This lets us reuse the special case of Desargues's theorem where 𝑌 ≠ 𝑍, to eventually prove the case where 𝑌 = 𝑍. (Contributed by NM, 16-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((((𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴) ∧ (𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) ∧ (((𝐺 ∨ 𝐻) ∨ 𝐼) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂) ∧ ((¬ 𝑐 ≤ (𝐺 ∨ 𝐻) ∧ ¬ 𝑐 ≤ (𝐻 ∨ 𝐼) ∧ ¬ 𝑐 ≤ (𝐼 ∨ 𝐺)) ∧ (¬ 𝑐 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝑐 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝑐 ≤ (𝑅 ∨ 𝑃)) ∧ (𝑐 ≤ (𝐺 ∨ 𝑃) ∧ 𝑐 ≤ (𝐻 ∨ 𝑄) ∧ 𝑐 ≤ (𝐼 ∨ 𝑅)))) ∧ ((𝐺 ∨ 𝐻) ∨ 𝐼) ≠ 𝑌))
Theorem	dalem52 39706	Lemma for dath 39718. Lines 𝐺𝐻 and 𝑃𝑄 intersect at an atom. (Contributed by NM, 8-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) ∈ 𝐴)
Theorem	dalem53 39707	Lemma for dath 39718. The auxiliary axis of perspectivity 𝐵 is a line (analogous to the actual axis of perspectivity 𝑋 in dalem15 39660. (Contributed by NM, 8-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑁 = (LLines‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐵 ∈ 𝑁)
Theorem	dalem54 39708	Lemma for dath 39718. Line 𝐺𝐻 intersects the auxiliary axis of perspectivity 𝐵. (Contributed by NM, 8-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ 𝐵) ∈ 𝐴)
Theorem	dalem55 39709	Lemma for dath 39718. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 39714). (Contributed by NM, 8-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑃 ∨ 𝑄)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
Theorem	dalem56 39710	Lemma for dath 39718. Analogue of dalem55 39709 for line 𝑆𝑇. (Contributed by NM, 8-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → ((𝐺 ∨ 𝐻) ∧ (𝑆 ∨ 𝑇)) = ((𝐺 ∨ 𝐻) ∧ 𝐵))
Theorem	dalem57 39711	Lemma for dath 39718. Axis of perspectivity point 𝐷 is on the auxiliary line 𝐵. (Contributed by NM, 9-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐷 ≤ 𝐵)
Theorem	dalem58 39712	Lemma for dath 39718. Analogue of dalem57 39711 for 𝐸. (Contributed by NM, 10-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐸 ≤ 𝐵)
Theorem	dalem59 39713	Lemma for dath 39718. Analogue of dalem57 39711 for 𝐹. (Contributed by NM, 10-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ 𝐵)
Theorem	dalem60 39714	Lemma for dath 39718. 𝐵 is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐺 = ((𝑐 ∨ 𝑃) ∧ (𝑑 ∨ 𝑆))    &   ⊢ 𝐻 = ((𝑐 ∨ 𝑄) ∧ (𝑑 ∨ 𝑇))    &   ⊢ 𝐼 = ((𝑐 ∨ 𝑅) ∧ (𝑑 ∨ 𝑈))    &   ⊢ 𝐵 = (((𝐺 ∨ 𝐻) ∨ 𝐼) ∧ 𝑌)    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → (𝐷 ∨ 𝐸) = 𝐵)
Theorem	dalem61 39715	Lemma for dath 39718. Show that atoms 𝐷, 𝐸, and 𝐹 lie on the same line (axis of perspectivity). Eliminate hypotheses containing dummy atoms 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ (𝜓 ↔ ((𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴) ∧ ¬ 𝑐 ≤ 𝑌 ∧ (𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ (𝑐 ∨ 𝑑))))    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓) → 𝐹 ≤ (𝐷 ∨ 𝐸))
Theorem	dalem62 39716	Lemma for dath 39718. Eliminate the condition 𝜓 containing dummy variables 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    ⇒   ⊢ ((𝜑 ∧ 𝑌 = 𝑍) → 𝐹 ≤ (𝐷 ∨ 𝐸))
Theorem	dalem63 39717	Lemma for dath 39718. Combine the cases where 𝑌 and 𝑍 are different planes with the case where 𝑌 and 𝑍 are the same plane. (Contributed by NM, 11-Aug-2012.)
⊢ (𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))))    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝑌 = ((𝑃 ∨ 𝑄) ∨ 𝑅)    &   ⊢ 𝑍 = ((𝑆 ∨ 𝑇) ∨ 𝑈)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    ⇒   ⊢ (𝜑 → 𝐹 ≤ (𝐷 ∨ 𝐸))
Theorem	dath 39718	Desargues's theorem of projective geometry (proved for a Hilbert lattice). Assume each triple of atoms (points) 𝑃𝑄𝑅 and 𝑆𝑇𝑈 forms a triangle (i.e. determines a plane). Assume that lines 𝑃𝑆, 𝑄𝑇, and 𝑅𝑈 meet at a "center of perspectivity" 𝐶. (We also assume that 𝐶 is not on any of the 6 lines forming the two triangles.) Then the atoms 𝐷 = (𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇), 𝐸 = (𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈), 𝐹 = (𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆) are colinear, forming an "axis of perspectivity". Our proof roughly follows Theorem 2.7.1, p. 78 in Beutelspacher and Rosenbaum, Projective Geometry: From Foundations to Applications, Cambridge University Press (1988). Unlike them, we do not assume that 𝐶 is an atom to make this theorem slightly more general for easier future use. However, we prove that 𝐶 must be an atom in dalemcea 39642. For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html 39642. The points I, J, and K there define the axis of perspectivity. See Theorems dalaw 39868 for Desargues's law, which eliminates all of the preconditions on the atoms except for central perspectivity. This is Metamath 100 proof #87. (Contributed by NM, 20-Aug-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → 𝐹 ≤ (𝐷 ∨ 𝐸))
Theorem	dath2 39719	Version of Desargues's theorem dath 39718 with a different variable ordering. (Contributed by NM, 7-Oct-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑂 = (LPlanes‘𝐾)    &   ⊢ 𝐷 = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   ⊢ 𝐸 = ((𝑄 ∨ 𝑅) ∧ (𝑇 ∨ 𝑈))    &   ⊢ 𝐹 = ((𝑅 ∨ 𝑃) ∧ (𝑈 ∨ 𝑆))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝐶 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴)) ∧ (((𝑃 ∨ 𝑄) ∨ 𝑅) ∈ 𝑂 ∧ ((𝑆 ∨ 𝑇) ∨ 𝑈) ∈ 𝑂) ∧ ((¬ 𝐶 ≤ (𝑃 ∨ 𝑄) ∧ ¬ 𝐶 ≤ (𝑄 ∨ 𝑅) ∧ ¬ 𝐶 ≤ (𝑅 ∨ 𝑃)) ∧ (¬ 𝐶 ≤ (𝑆 ∨ 𝑇) ∧ ¬ 𝐶 ≤ (𝑇 ∨ 𝑈) ∧ ¬ 𝐶 ≤ (𝑈 ∨ 𝑆)) ∧ (𝐶 ≤ (𝑃 ∨ 𝑆) ∧ 𝐶 ≤ (𝑄 ∨ 𝑇) ∧ 𝐶 ≤ (𝑅 ∨ 𝑈)))) → 𝐷 ≤ (𝐸 ∨ 𝐹))
Theorem	lineset 39720*	The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → 𝑁 = {𝑠 ∣ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑠 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})})
Theorem	isline 39721*	The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑁 ↔ ∃𝑞 ∈ 𝐴 ∃𝑟 ∈ 𝐴 (𝑞 ≠ 𝑟 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑞 ∨ 𝑟)})))
Theorem	islinei 39722*	Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ≠ 𝑅 ∧ 𝑋 = {𝑝 ∈ 𝐴 ∣ 𝑝 ≤ (𝑄 ∨ 𝑅)})) → 𝑋 ∈ 𝑁)
Theorem	pointsetN 39723*	The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (Points‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → 𝑃 = {𝑝 ∣ ∃𝑎 ∈ 𝐴 𝑝 = {𝑎}})
Theorem	ispointN 39724*	The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (Points‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑃 ↔ ∃𝑎 ∈ 𝐴 𝑋 = {𝑎}))
Theorem	atpointN 39725	The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (Points‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐴) → {𝑋} ∈ 𝑃)
Theorem	psubspset 39726*	The set of projective subspaces in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐵 → 𝑆 = {𝑠 ∣ (𝑠 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑠 ∀𝑞 ∈ 𝑠 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑠))})
Theorem	ispsubsp 39727*	The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝑋 ∀𝑞 ∈ 𝑋 ∀𝑟 ∈ 𝐴 (𝑟 ≤ (𝑝 ∨ 𝑞) → 𝑟 ∈ 𝑋))))
Theorem	ispsubsp2 39728*	The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐷 → (𝑋 ∈ 𝑆 ↔ (𝑋 ⊆ 𝐴 ∧ ∀𝑝 ∈ 𝐴 (∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑝 ≤ (𝑞 ∨ 𝑟) → 𝑝 ∈ 𝑋))))
Theorem	psubspi 39729*	Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ ∃𝑞 ∈ 𝑋 ∃𝑟 ∈ 𝑋 𝑃 ≤ (𝑞 ∨ 𝑟)) → 𝑃 ∈ 𝑋)
Theorem	psubspi2N 39730	Property of a projective subspace. (Contributed by NM, 13-Jan-2012.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ∧ 𝑃 ∈ 𝐴) ∧ (𝑄 ∈ 𝑋 ∧ 𝑅 ∈ 𝑋 ∧ 𝑃 ≤ (𝑄 ∨ 𝑅))) → 𝑃 ∈ 𝑋)
Theorem	0psubN 39731	The empty set is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → ∅ ∈ 𝑆)
Theorem	snatpsubN 39732	The singleton of an atom is a projective subspace. (Contributed by NM, 9-Sep-2013.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴) → {𝑃} ∈ 𝑆)
Theorem	pointpsubN 39733	A point (singleton of an atom) is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
⊢ 𝑃 = (Points‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ AtLat ∧ 𝑋 ∈ 𝑃) → 𝑋 ∈ 𝑆)
Theorem	linepsubN 39734	A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝑁) → 𝑋 ∈ 𝑆)
Theorem	atpsubN 39735	The set of all atoms is a projective subspace. Remark below Definition 15.1 of [MaedaMaeda] p. 61. (Contributed by NM, 13-Oct-2011.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆)
Theorem	psubssat 39736	A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆) → 𝑋 ⊆ 𝐴)
Theorem	psubatN 39737	A member of a projective subspace is an atom. (Contributed by NM, 4-Nov-2011.) (New usage is discouraged.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑋) → 𝑌 ∈ 𝐴)
Theorem	pmapfval 39738*	The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝐶 → 𝑀 = (𝑥 ∈ 𝐵 ↦ {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑥}))
Theorem	pmapval 39739*	Value of the projective map of a Hilbert lattice. Definition in Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 2-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) = {𝑎 ∈ 𝐴 ∣ 𝑎 ≤ 𝑋})
Theorem	elpmap 39740	Member of a projective map. (Contributed by NM, 27-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑃 ∈ (𝑀‘𝑋) ↔ (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑋)))
Theorem	pmapssat 39741	The projective map of a Hilbert lattice is a set of atoms. (Contributed by NM, 14-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐴)
Theorem	pmapssbaN 39742	A weakening of pmapssat 39741 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐶 ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ⊆ 𝐵)
Theorem	pmaple 39743	The projective map of a Hilbert lattice preserves ordering. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑀‘𝑋) ⊆ (𝑀‘𝑌)))
Theorem	pmap11 39744	The projective map of a Hilbert lattice is one-to-one. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑀‘𝑋) = (𝑀‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	pmapat 39745	The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑀‘𝑃) = {𝑃})
Theorem	elpmapat 39746	Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴) → (𝑋 ∈ (𝑀‘𝑃) ↔ 𝑋 = 𝑃))
Theorem	pmap0 39747	Value of the projective map of a Hilbert lattice at lattice zero. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (𝐾 ∈ AtLat → (𝑀‘ 0 ) = ∅)
Theorem	pmapeq0 39748	A projective map value is zero iff its argument is lattice zero. (Contributed by NM, 27-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) = ∅ ↔ 𝑋 = 0 ))
Theorem	pmap1N 39749	Value of the projective map of a Hilbert lattice at lattice unity. Part of Theorem 15.5.1 of [MaedaMaeda] p. 62. (Contributed by NM, 22-Oct-2011.) (New usage is discouraged.)
⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (𝐾 ∈ OP → (𝑀‘ 1 ) = 𝐴)
Theorem	pmapsub 39750	The projective map of a Hilbert lattice maps to projective subspaces. Part of Theorem 15.5 of [MaedaMaeda] p. 62. (Contributed by NM, 17-Oct-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑆 = (PSubSp‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵) → (𝑀‘𝑋) ∈ 𝑆)
Theorem	pmapglbx 39751*	The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 39752, where we read 𝑆 as 𝑆(𝑖). Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵 ∧ 𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆))
Theorem	pmapglb 39752*	The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. (Contributed by NM, 5-Dec-2011.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) → (𝑀‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥))
Theorem	pmapglb2N 39753*	The projective map of the GLB of a set of lattice elements 𝑆. Variant of Theorem 15.5.2 of [MaedaMaeda] p. 62. Allows 𝑆 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑆 ⊆ 𝐵) → (𝑀‘(𝐺‘𝑆)) = (𝐴 ∩ ∩ 𝑥 ∈ 𝑆 (𝑀‘𝑥)))
Theorem	pmapglb2xN 39754*	The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 39753, where we read 𝑆 as 𝑆(𝑖). Extension of Theorem 15.5.2 of [MaedaMaeda] p. 62 that allows 𝐼 = ∅. (Contributed by NM, 21-Jan-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ ∀𝑖 ∈ 𝐼 𝑆 ∈ 𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖 ∈ 𝐼 𝑦 = 𝑆})) = (𝐴 ∩ ∩ 𝑖 ∈ 𝐼 (𝑀‘𝑆)))
Theorem	pmapmeet 39755	The projective map of a meet. (Contributed by NM, 25-Jan-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑃 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑃‘(𝑋 ∧ 𝑌)) = ((𝑃‘𝑋) ∩ (𝑃‘𝑌)))
Theorem	isline2 39756*	Definition of line in terms of projective map. (Contributed by NM, 25-Jan-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (𝐾 ∈ Lat → (𝑋 ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑀‘(𝑝 ∨ 𝑞)))))
Theorem	linepmap 39757	A line described with a projective map. (Contributed by NM, 3-Feb-2012.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ Lat ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 ≠ 𝑄) → (𝑀‘(𝑃 ∨ 𝑄)) ∈ 𝑁)
Theorem	isline3 39758*	Definition of line in terms of original lattice elements. (Contributed by NM, 29-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 (𝑝 ≠ 𝑞 ∧ 𝑋 = (𝑝 ∨ 𝑞))))
Theorem	isline4N 39759*	Definition of line in terms of original lattice elements. (Contributed by NM, 16-Jun-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) → ((𝑀‘𝑋) ∈ 𝑁 ↔ ∃𝑝 ∈ 𝐴 𝑝𝐶𝑋))
Theorem	lneq2at 39760	A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑋)) → 𝑋 = (𝑃 ∨ 𝑄))
Theorem	lnatexN 39761*	There is an atom in a line different from any other. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ (𝑀‘𝑋) ∈ 𝑁) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑞 ≤ 𝑋))
Theorem	lnjatN 39762*	Given an atom in a line, there is another atom which when joined equals the line. (Contributed by NM, 30-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → ∃𝑞 ∈ 𝐴 (𝑞 ≠ 𝑃 ∧ 𝑋 = (𝑃 ∨ 𝑞)))
Theorem	lncvrelatN 39763	A lattice element covered by a line is an atom. (Contributed by NM, 28-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃𝐶𝑋)) → 𝑃 ∈ 𝐴)
Theorem	lncvrat 39764	A line covers the atoms it contains. (Contributed by NM, 30-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ 𝑃 ≤ 𝑋)) → 𝑃𝐶𝑋)
Theorem	lncmp 39765	If two lines are comparable, they are equal. (Contributed by NM, 30-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝑀 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑀‘𝑋) ∈ 𝑁 ∧ (𝑀‘𝑌) ∈ 𝑁)) → (𝑋 ≤ 𝑌 ↔ 𝑋 = 𝑌))
Theorem	2lnat 39766	Two intersecting lines intersect at an atom. (Contributed by NM, 30-Apr-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝐹‘𝑋) ∈ 𝑁 ∧ (𝐹‘𝑌) ∈ 𝑁) ∧ (𝑋 ≠ 𝑌 ∧ (𝑋 ∧ 𝑌) ≠ 0 )) → (𝑋 ∧ 𝑌) ∈ 𝐴)
Theorem	2atm2atN 39767	Two joins with a common atom have a nonzero meet. (Contributed by NM, 4-Jul-2012.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) → ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) ≠ 0 )
Theorem	2llnma1b 39768	Generalization of 2llnma1 39769. (Contributed by NM, 26-Apr-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ ¬ 𝑄 ≤ (𝑃 ∨ 𝑋)) → ((𝑃 ∨ 𝑋) ∧ (𝑃 ∨ 𝑄)) = 𝑃)
Theorem	2llnma1 39769	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 11-Oct-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄)) → ((𝑄 ∨ 𝑃) ∧ (𝑄 ∨ 𝑅)) = 𝑄)
Theorem	2llnma3r 39770	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 30-Apr-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ∨ 𝑅) ≠ (𝑄 ∨ 𝑅)) → ((𝑃 ∨ 𝑅) ∧ (𝑄 ∨ 𝑅)) = 𝑅)
Theorem	2llnma2 39771	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 28-May-2012.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑅 ∨ 𝑃) ∧ (𝑅 ∨ 𝑄)) = 𝑅)
Theorem	2llnma2rN 39772	Two different intersecting lines (expressed in terms of atoms) meet at their common point (atom). (Contributed by NM, 2-May-2013.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ≠ 𝑄 ∧ ¬ 𝑅 ≤ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ 𝑅) ∧ (𝑄 ∨ 𝑅)) = 𝑅)
21.28.13  Construction of a vector space from a Hilbert lattice
Theorem	cdlema1N 39773	A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 29-Apr-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑁 = (Lines‘𝐾)    &   ⊢ 𝐹 = (pmap‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ 𝑄 ≤ 𝑌) ∧ ((𝐹‘𝑌) ∈ 𝑁 ∧ (𝑋 ∧ 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑋 ∨ 𝑅) = (𝑋 ∨ 𝑌))
Theorem	cdlema2N 39774	A condition for required for proof of Lemma A in [Crawley] p. 112. (Contributed by NM, 9-May-2012.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ ((𝑅 ≠ 𝑃 ∧ 𝑅 ≤ (𝑃 ∨ 𝑄)) ∧ (𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋))) → (𝑅 ∧ 𝑋) = 0 )
Theorem	cdlemblem 39775	Lemma for cdlemb 39776. (Contributed by NM, 8-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ < = (lt‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑋)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) ∧ (𝑢 ∈ 𝐴 ∧ (𝑢 ≠ 𝑉 ∧ 𝑢 < 𝑋)) ∧ (𝑟 ∈ 𝐴 ∧ (𝑟 ≠ 𝑃 ∧ 𝑟 ≠ 𝑢 ∧ 𝑟 ≤ (𝑃 ∨ 𝑢)))) → (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
Theorem	cdlemb 39776*	Given two atoms not less than or equal to an element covered by 1, there is a third. Lemma B in [Crawley] p. 112. (Contributed by NM, 8-May-2012.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 1 = (1.‘𝐾)    &   ⊢ 𝐶 = ( ⋖ ‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑋 ∈ 𝐵 ∧ 𝑃 ≠ 𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 ≤ 𝑋 ∧ ¬ 𝑄 ≤ 𝑋)) → ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑋 ∧ ¬ 𝑟 ≤ (𝑃 ∨ 𝑄)))
Syntax	cpadd 39777	Extend class notation with projective subspace sum.
class +𝑃
Definition	df-padd 39778*	Define projective sum of two subspaces (or more generally two sets of atoms), which is the union of all lines generated by pairs of atoms from each subspace. Lemma 16.2 of [MaedaMaeda] p. 68. For convenience, our definition is generalized to apply to empty sets. (Contributed by NM, 29-Dec-2011.)
⊢ +𝑃 = (𝑙 ∈ V ↦ (𝑚 ∈ 𝒫 (Atoms‘𝑙), 𝑛 ∈ 𝒫 (Atoms‘𝑙) ↦ ((𝑚 ∪ 𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞 ∈ 𝑚 ∃𝑟 ∈ 𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))
Theorem	paddfval 39779*	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 39780*	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 39781*	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 39782*	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 39783*	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 39784	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 39785	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 39786*	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 39787*	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 39788	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 39789	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 39790	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 39791	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 39792	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 39793	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 39794	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 39795	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 39796	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 39797	A projective subspace sum is a superset of its first summand. (ssun1 4187 analog.) (Contributed by NM, 3-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))
Theorem	sspadd2 39798	A projective subspace sum is a superset of its second summand. (ssun2 4188 analog.) (Contributed by NM, 3-Jan-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑋 ⊆ 𝐴 ∧ 𝑌 ⊆ 𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))
Theorem	paddss1 39799	Subset law for projective subspace sum. (unss1 4194 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 39800	Subset law for projective subspace sum. (unss2 4196 analog.) (Contributed by NM, 7-Mar-2012.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ + = (+𝑃‘𝐾)    ⇒   ⊢ ((𝐾 ∈ 𝐵 ∧ 𝑌 ⊆ 𝐴 ∧ 𝑍 ⊆ 𝐴) → (𝑋 ⊆ 𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))