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Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdalem48 36301 Lemma for dath 36317. Analogue of dalem45 36298 for 𝑃𝑄. (Contributed by NM, 16-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝜓) → ¬ 𝑐 (𝑃 𝑄))

Theoremdalem49 36302 Lemma for dath 36317. Analogue of dalem45 36298 for 𝑄𝑅. (Contributed by NM, 16-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝜓) → ¬ 𝑐 (𝑄 𝑅))

Theoremdalem50 36303 Lemma for dath 36317. Analogue of dalem45 36298 for 𝑅𝑃. (Contributed by NM, 16-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝜓) → ¬ 𝑐 (𝑅 𝑃))

Theoremdalem51 36304 Lemma for dath 36317. 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 𝑌𝑍, to eventually prove the case where 𝑌 = 𝑍. (Contributed by NM, 16-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝑌 = 𝑍𝜓) → ((((𝐾 ∈ HL ∧ 𝑐𝐴) ∧ (𝐺𝐴𝐻𝐴𝐼𝐴) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) ∧ (((𝐺 𝐻) 𝐼) ∈ 𝑂𝑌𝑂) ∧ ((¬ 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))

Theoremdalem52 36305 Lemma for dath 36317. Lines 𝐺𝐻 and 𝑃𝑄 intersect at an atom. (Contributed by NM, 8-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) ∈ 𝐴)

Theoremdalem53 36306 Lemma for dath 36317. The auxliary axis of perspectivity 𝐵 is a line (analogous to the actual axis of perspectivity 𝑋 in dalem15 36259. (Contributed by NM, 8-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → 𝐵𝑁)

Theoremdalem54 36307 Lemma for dath 36317. Line 𝐺𝐻 intersects the auxiliary axis of perspectivity 𝐵. (Contributed by NM, 8-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) 𝐵) ∈ 𝐴)

Theoremdalem55 36308 Lemma for dath 36317. Lines 𝐺𝐻 and 𝑃𝑄 intersect at the auxiliary line 𝐵 (later shown to be an axis of perspectivity; see dalem60 36313). (Contributed by NM, 8-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑃 𝑄)) = ((𝐺 𝐻) 𝐵))

Theoremdalem56 36309 Lemma for dath 36317. Analogue of dalem55 36308 for line 𝑆𝑇. (Contributed by NM, 8-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → ((𝐺 𝐻) (𝑆 𝑇)) = ((𝐺 𝐻) 𝐵))

Theoremdalem57 36310 Lemma for dath 36317. Axis of perspectivity point 𝐷 is on the auxiliary line 𝐵. (Contributed by NM, 9-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → 𝐷 𝐵)

Theoremdalem58 36311 Lemma for dath 36317. Analogue of dalem57 36310 for 𝐸. (Contributed by NM, 10-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → 𝐸 𝐵)

Theoremdalem59 36312 Lemma for dath 36317. Analogue of dalem57 36310 for 𝐹. (Contributed by NM, 10-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → 𝐹 𝐵)

Theoremdalem60 36313 Lemma for dath 36317. 𝐵 is an axis of perspectivity (almost). (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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)       ((𝜑𝑌 = 𝑍𝜓) → (𝐷 𝐸) = 𝐵)

Theoremdalem61 36314 Lemma for dath 36317. 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)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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 ↔ ((𝑐𝐴𝑑𝐴) ∧ ¬ 𝑐 𝑌 ∧ (𝑑𝑐 ∧ ¬ 𝑑 𝑌𝐶 (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))       ((𝜑𝑌 = 𝑍𝜓) → 𝐹 (𝐷 𝐸))

Theoremdalem62 36315 Lemma for dath 36317. Eliminate the condition 𝜓 containing dummy variables 𝑐 and 𝑑. (Contributed by NM, 11-Aug-2012.)
(𝜑 ↔ (((𝐾 ∈ HL ∧ 𝐶 ∈ (Base‘𝐾)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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))       ((𝜑𝑌 = 𝑍) → 𝐹 (𝐷 𝐸))

Theoremdalem63 36316 Lemma for dath 36317. 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)) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (𝑌𝑂𝑍𝑂) ∧ ((¬ 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))       (𝜑𝐹 (𝐷 𝐸))

Theoremdath 36317 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 36241.

For a visual demonstration, see the "Desargues's theorem" applet at http://www.dynamicgeometry.com/JavaSketchpad/Gallery.html. The points I, J, and K there define the axis of perspectivity.

See theorem dalaw 36467 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 ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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))

Theoremdath2 36318 Version of Desargues's theorem dath 36317 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 ∧ 𝐶𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑆𝐴𝑇𝐴𝑈𝐴)) ∧ (((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))

Theoremlineset 36319* The set of lines in a Hilbert lattice. (Contributed by NM, 19-Sep-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)       (𝐾𝐵𝑁 = {𝑠 ∣ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑠 = {𝑝𝐴𝑝 (𝑞 𝑟)})})

Theoremisline 36320* The predicate "is a line". (Contributed by NM, 19-Sep-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)       (𝐾𝐷 → (𝑋𝑁 ↔ ∃𝑞𝐴𝑟𝐴 (𝑞𝑟𝑋 = {𝑝𝐴𝑝 (𝑞 𝑟)})))

Theoremislinei 36321* Condition implying "is a line". (Contributed by NM, 3-Feb-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑁 = (Lines‘𝐾)       (((𝐾𝐷𝑄𝐴𝑅𝐴) ∧ (𝑄𝑅𝑋 = {𝑝𝐴𝑝 (𝑄 𝑅)})) → 𝑋𝑁)

TheorempointsetN 36322* The set of points in a Hilbert lattice. (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (Points‘𝐾)       (𝐾𝐵𝑃 = {𝑝 ∣ ∃𝑎𝐴 𝑝 = {𝑎}})

TheoremispointN 36323* The predicate "is a point". (Contributed by NM, 2-Oct-2011.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (Points‘𝐾)       (𝐾𝐷 → (𝑋𝑃 ↔ ∃𝑎𝐴 𝑋 = {𝑎}))

TheorematpointN 36324 The singleton of an atom is a point. (Contributed by NM, 14-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑃 = (Points‘𝐾)       ((𝐾𝐷𝑋𝐴) → {𝑋} ∈ 𝑃)

Theorempsubspset 36325* 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)       (𝐾𝐵𝑆 = {𝑠 ∣ (𝑠𝐴 ∧ ∀𝑝𝑠𝑞𝑠𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑠))})

Theoremispsubsp 36326* The predicate "is a projective subspace". (Contributed by NM, 2-Oct-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝑋𝑞𝑋𝑟𝐴 (𝑟 (𝑝 𝑞) → 𝑟𝑋))))

Theoremispsubsp2 36327* The predicate "is a projective subspace". (Contributed by NM, 13-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (𝐾𝐷 → (𝑋𝑆 ↔ (𝑋𝐴 ∧ ∀𝑝𝐴 (∃𝑞𝑋𝑟𝑋 𝑝 (𝑞 𝑟) → 𝑝𝑋))))

Theorempsubspi 36328* Property of a projective subspace. (Contributed by NM, 13-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ ∃𝑞𝑋𝑟𝑋 𝑃 (𝑞 𝑟)) → 𝑃𝑋)

Theorempsubspi2N 36329 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)       (((𝐾𝐷𝑋𝑆𝑃𝐴) ∧ (𝑄𝑋𝑅𝑋𝑃 (𝑄 𝑅))) → 𝑃𝑋)

Theorem0psubN 36330 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)       (𝐾𝑉 → ∅ ∈ 𝑆)

TheoremsnatpsubN 36331 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 ∧ 𝑃𝐴) → {𝑃} ∈ 𝑆)

TheorempointpsubN 36332 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 ∧ 𝑋𝑃) → 𝑋𝑆)

TheoremlinepsubN 36333 A line is a projective subspace. (Contributed by NM, 16-Oct-2011.) (New usage is discouraged.)
𝑁 = (Lines‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝑁) → 𝑋𝑆)

TheorematpsubN 36334 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)       (𝐾𝑉𝐴𝑆)

Theorempsubssat 36335 A projective subspace consists of atoms. (Contributed by NM, 4-Nov-2011.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)       ((𝐾𝐵𝑋𝑆) → 𝑋𝐴)

TheorempsubatN 36336 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)       ((𝐾𝐵𝑋𝑆𝑌𝑋) → 𝑌𝐴)

Theorempmapfval 36337* The projective map of a Hilbert lattice. (Contributed by NM, 2-Oct-2011.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       (𝐾𝐶𝑀 = (𝑥𝐵 ↦ {𝑎𝐴𝑎 𝑥}))

Theorempmapval 36338* 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)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) = {𝑎𝐴𝑎 𝑋})

Theoremelpmap 36339 Member of a projective map. (Contributed by NM, 27-Jan-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑃 ∈ (𝑀𝑋) ↔ (𝑃𝐴𝑃 𝑋)))

Theorempmapssat 36340 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)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐴)

TheorempmapssbaN 36341 A weakening of pmapssat 36340 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐵)

Theorempmaple 36342 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 ∧ 𝑋𝐵𝑌𝐵) → (𝑋 𝑌 ↔ (𝑀𝑋) ⊆ (𝑀𝑌)))

Theorempmap11 36343 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 ∧ 𝑋𝐵𝑌𝐵) → ((𝑀𝑋) = (𝑀𝑌) ↔ 𝑋 = 𝑌))

Theorempmapat 36344 The projective map of an atom. (Contributed by NM, 25-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑀𝑃) = {𝑃})

Theoremelpmapat 36345 Member of the projective map of an atom. (Contributed by NM, 27-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &   𝑀 = (pmap‘𝐾)       ((𝐾 ∈ HL ∧ 𝑃𝐴) → (𝑋 ∈ (𝑀𝑃) ↔ 𝑋 = 𝑃))

Theorempmap0 36346 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 → (𝑀0 ) = ∅)

Theorempmapeq0 36347 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 ∧ 𝑋𝐵) → ((𝑀𝑋) = ∅ ↔ 𝑋 = 0 ))

Theorempmap1N 36348 Value of the projective map of a Hilbert lattice at lattice unit. 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 → (𝑀1 ) = 𝐴)

Theorempmapsub 36349 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 ∧ 𝑋𝐵) → (𝑀𝑋) ∈ 𝑆)

Theorempmapglbx 36350* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb 36351, 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 ∧ ∀𝑖𝐼 𝑆𝐵𝐼 ≠ ∅) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = 𝑖𝐼 (𝑀𝑆))

Theorempmapglb 36351* 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 ∧ 𝑆𝐵𝑆 ≠ ∅) → (𝑀‘(𝐺𝑆)) = 𝑥𝑆 (𝑀𝑥))

Theorempmapglb2N 36352* 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 ∧ 𝑆𝐵) → (𝑀‘(𝐺𝑆)) = (𝐴 𝑥𝑆 (𝑀𝑥)))

Theorempmapglb2xN 36353* The projective map of the GLB of a set of lattice elements. Index-set version of pmapglb2N 36352, 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 ∧ ∀𝑖𝐼 𝑆𝐵) → (𝑀‘(𝐺‘{𝑦 ∣ ∃𝑖𝐼 𝑦 = 𝑆})) = (𝐴 𝑖𝐼 (𝑀𝑆)))

Theorempmapmeet 36354 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 ∧ 𝑋𝐵𝑌𝐵) → (𝑃‘(𝑋 𝑌)) = ((𝑃𝑋) ∩ (𝑃𝑌)))

Theoremisline2 36355* 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 → (𝑋𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑀‘(𝑝 𝑞)))))

Theoremlinepmap 36356 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 ∧ 𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → (𝑀‘(𝑃 𝑄)) ∈ 𝑁)

Theoremisline3 36357* 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 ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴𝑞𝐴 (𝑝𝑞𝑋 = (𝑝 𝑞))))

Theoremisline4N 36358* 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 ∧ 𝑋𝐵) → ((𝑀𝑋) ∈ 𝑁 ↔ ∃𝑝𝐴 𝑝𝐶𝑋))

Theoremlneq2at 36359 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 ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) ∧ (𝑃𝐴𝑄𝐴𝑃𝑄) ∧ (𝑃 𝑋𝑄 𝑋)) → 𝑋 = (𝑃 𝑄))

TheoremlnatexN 36360* 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 ∧ 𝑋𝐵 ∧ (𝑀𝑋) ∈ 𝑁) → ∃𝑞𝐴 (𝑞𝑃𝑞 𝑋))

TheoremlnjatN 36361* 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 ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → ∃𝑞𝐴 (𝑞𝑃𝑋 = (𝑃 𝑞)))

TheoremlncvrelatN 36362 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 ∧ 𝑋𝐵𝑃𝐵) ∧ ((𝑀𝑋) ∈ 𝑁𝑃𝐶𝑋)) → 𝑃𝐴)

Theoremlncvrat 36363 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 ∧ 𝑋𝐵𝑃𝐴) ∧ ((𝑀𝑋) ∈ 𝑁𝑃 𝑋)) → 𝑃𝐶𝑋)

Theoremlncmp 36364 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 ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝑀𝑋) ∈ 𝑁 ∧ (𝑀𝑌) ∈ 𝑁)) → (𝑋 𝑌𝑋 = 𝑌))

Theorem2lnat 36365 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 ∧ 𝑋𝐵𝑌𝐵) ∧ ((𝐹𝑋) ∈ 𝑁 ∧ (𝐹𝑌) ∈ 𝑁) ∧ (𝑋𝑌 ∧ (𝑋 𝑌) ≠ 0 )) → (𝑋 𝑌) ∈ 𝐴)

Theorem2atm2atN 36366 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → ((𝑅 𝑃) (𝑅 𝑄)) ≠ 0 )

Theorem2llnma1b 36367 Generalization of 2llnma1 36368. (Contributed by NM, 26-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑋𝐵𝑃𝐴𝑄𝐴) ∧ ¬ 𝑄 (𝑃 𝑋)) → ((𝑃 𝑋) (𝑃 𝑄)) = 𝑃)

Theorem2llnma1 36368 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝑃) (𝑄 𝑅)) = 𝑄)

Theorem2llnma3r 36369 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃 𝑅) ≠ (𝑄 𝑅)) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)

Theorem2llnma2 36370 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑅 𝑃) (𝑅 𝑄)) = 𝑅)

Theorem2llnma2rN 36371 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 ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → ((𝑃 𝑅) (𝑄 𝑅)) = 𝑅)

20.24.13  Construction of a vector space from a Hilbert lattice

Theoremcdlema1N 36372 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 ∧ 𝑋𝐵𝑌𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋𝑄 𝑌) ∧ ((𝐹𝑌) ∈ 𝑁 ∧ (𝑋 𝑌) ∈ 𝐴 ∧ ¬ 𝑄 𝑋))) → (𝑋 𝑅) = (𝑋 𝑌))

Theoremcdlema2N 36373 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 ∧ 𝑋𝐵) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ((𝑅𝑃𝑅 (𝑃 𝑄)) ∧ (𝑃 𝑋 ∧ ¬ 𝑄 𝑋))) → (𝑅 𝑋) = 0 )

Theoremcdlemblem 36374 Lemma for cdlemb 36375. (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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) ∧ (𝑢𝐴 ∧ (𝑢𝑉𝑢 < 𝑋)) ∧ (𝑟𝐴 ∧ (𝑟𝑃𝑟𝑢𝑟 (𝑃 𝑢)))) → (¬ 𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))

Theoremcdlemb 36375* 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 ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑋𝐵𝑃𝑄) ∧ (𝑋𝐶 1 ∧ ¬ 𝑃 𝑋 ∧ ¬ 𝑄 𝑋)) → ∃𝑟𝐴𝑟 𝑋 ∧ ¬ 𝑟 (𝑃 𝑄)))

Syntaxcpadd 36376 Extend class notation with projective subspace sum.
class +𝑃

Definitiondf-padd 36377* 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) ↦ ((𝑚𝑛) ∪ {𝑝 ∈ (Atoms‘𝑙) ∣ ∃𝑞𝑚𝑟𝑛 𝑝(le‘𝑙)(𝑞(join‘𝑙)𝑟)})))

Theorempaddfval 36378* Projective subspace sum operation. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (𝐾𝐵+ = (𝑚 ∈ 𝒫 𝐴, 𝑛 ∈ 𝒫 𝐴 ↦ ((𝑚𝑛) ∪ {𝑝𝐴 ∣ ∃𝑞𝑚𝑟𝑛 𝑝 (𝑞 𝑟)})))

Theorempaddval 36379* Projective subspace sum operation value. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = ((𝑋𝑌) ∪ {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)}))

Theoremelpadd 36380* Member of a projective subspace sum. (Contributed by NM, 29-Dec-2011.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑆 ∈ (𝑋 + 𝑌) ↔ ((𝑆𝑋𝑆𝑌) ∨ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟)))))

Theoremelpaddn0 36381* Member of projective subspace sum of nonempty sets. (Contributed by NM, 3-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝐴 ∧ ∃𝑞𝑋𝑟𝑌 𝑆 (𝑞 𝑟))))

Theorempaddvaln0N 36382* 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)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = {𝑝𝐴 ∣ ∃𝑞𝑋𝑟𝑌 𝑝 (𝑞 𝑟)})

Theoremelpaddri 36383 Condition implying membership in a projective subspace sum. (Contributed by NM, 8-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) ∧ (𝑄𝑋𝑅𝑌) ∧ (𝑆𝐴𝑆 (𝑄 𝑅))) → 𝑆 ∈ (𝑋 + 𝑌))

TheoremelpaddatriN 36384 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)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑅𝑋𝑆𝐴𝑆 (𝑅 𝑄))) → 𝑆 ∈ (𝑋 + {𝑄}))

Theoremelpaddat 36385* Membership in a projective subspace sum with a point. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ 𝑋 ≠ ∅) → (𝑆 ∈ (𝑋 + {𝑄}) ↔ (𝑆𝐴 ∧ ∃𝑝𝑋 𝑆 (𝑝 𝑄))))

TheoremelpaddatiN 36386* 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)    &    + = (+𝑃𝐾)       (((𝐾 ∈ Lat ∧ 𝑋𝐴𝑄𝐴) ∧ (𝑋 ≠ ∅ ∧ 𝑅 ∈ (𝑋 + {𝑄}))) → ∃𝑝𝑋 𝑅 (𝑝 𝑄))

Theoremelpadd2at 36387 Membership in a projective subspace sum of two points. (Contributed by NM, 29-Jan-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑄𝐴𝑅𝐴) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ (𝑆𝐴𝑆 (𝑄 𝑅))))

Theoremelpadd2at2 36388 Membership in a projective subspace sum of two points. (Contributed by NM, 8-Mar-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ (𝑄𝐴𝑅𝐴𝑆𝐴)) → (𝑆 ∈ ({𝑄} + {𝑅}) ↔ 𝑆 (𝑄 𝑅)))

TheorempaddunssN 36389 Projective subspace sum includes the set union of its arguments. (Contributed by NM, 12-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋𝑌) ⊆ (𝑋 + 𝑌))

Theoremelpadd0 36390 Member of projective subspace sum with at least one empty set. (Contributed by NM, 29-Dec-2011.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑆 ∈ (𝑋 + 𝑌) ↔ (𝑆𝑋𝑆𝑌)))

Theorempaddval0 36391 Projective subspace sum with at least one empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       (((𝐾𝐵𝑋𝐴𝑌𝐴) ∧ ¬ (𝑋 ≠ ∅ ∧ 𝑌 ≠ ∅)) → (𝑋 + 𝑌) = (𝑋𝑌))

Theorempadd01 36392 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (𝑋 + ∅) = 𝑋)

Theorempadd02 36393 Projective subspace sum with an empty set. (Contributed by NM, 11-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴) → (∅ + 𝑋) = 𝑋)

Theorempaddcom 36394 Projective subspace sum commutes. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾 ∈ Lat ∧ 𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Theorempaddssat 36395 A projective subspace sum is a set of atoms. (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → (𝑋 + 𝑌) ⊆ 𝐴)

Theoremsspadd1 36396 A projective subspace sum is a superset of its first summand. (ssun1 4031 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑋 + 𝑌))

Theoremsspadd2 36397 A projective subspace sum is a superset of its second summand. (ssun2 4032 analog.) (Contributed by NM, 3-Jan-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑋𝐴𝑌𝐴) → 𝑋 ⊆ (𝑌 + 𝑋))

Theorempaddss1 36398 Subset law for projective subspace sum. (unss1 4037 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑍)))

Theorempaddss2 36399 Subset law for projective subspace sum. (unss2 4039 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑍𝐴) → (𝑋𝑌 → (𝑍 + 𝑋) ⊆ (𝑍 + 𝑌)))

Theorempaddss12 36400 Subset law for projective subspace sum. (unss12 4040 analog.) (Contributed by NM, 7-Mar-2012.)
𝐴 = (Atoms‘𝐾)    &    + = (+𝑃𝐾)       ((𝐾𝐵𝑌𝐴𝑊𝐴) → ((𝑋𝑌𝑍𝑊) → (𝑋 + 𝑍) ⊆ (𝑌 + 𝑊)))

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