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

Theoremcdleme41sn3a 36501* Show that 𝑅 / 𝑠𝑁 is under 𝑃 𝑄 when 𝑅 (𝑃 𝑄). (Contributed by NM, 19-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑌 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))    &   𝑍 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 (𝑃 𝑄))

Theoremcdleme32sn2awN 36502* Show that 𝑅 / 𝑠𝑁 is an atom not under 𝑊 when ¬ 𝑅 (𝑃 𝑄). (Contributed by NM, 6-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))

Theoremcdleme32snaw 36503* Show that 𝑅 / 𝑠𝑁 is an atom not under 𝑊. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))

Theoremcdleme32snb 36504* Show closure of 𝑅 / 𝑠𝑁. (Contributed by NM, 1-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝑅 / 𝑠𝑁𝐵)

Theoremcdleme32fva 36505* Part of proof of Lemma D in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. (Contributed by NM, 2-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)

Theoremcdleme32fva1 36506* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → (𝐹𝑅) = 𝑅 / 𝑠𝑁)

Theoremcdleme32fvaw 36507* Show that (𝐹𝑅) is an atom not under 𝑊 when 𝑅 is an atom not under 𝑊. (Contributed by NM, 18-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝐹𝑅) ∈ 𝐴 ∧ ¬ (𝐹𝑅) 𝑊))

Theoremcdleme32fvcl 36508* Part of proof of Lemma D in [Crawley] p. 113. Closure of the function 𝐹. (Contributed by NM, 10-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) → (𝐹𝑋) ∈ 𝐵)

Theoremcdleme32a 36509* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑁 (𝑋 𝑊)))

Theoremcdleme32b 36510* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑌) = (𝑁 (𝑌 𝑊)))

Theoremcdleme32c 36511* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 19-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑋 𝑊)) = 𝑋𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))

Theoremcdleme32d 36512* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))

Theoremcdleme32e 36513* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ (𝑠 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → (𝐹𝑋) (𝐹𝑌))

Theoremcdleme32f 36514* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑋𝐵𝑌𝐵) ∧ ¬ (𝑃𝑄 ∧ ¬ 𝑋 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑌 𝑊)) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))

Theoremcdleme32le 36515* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 20-Feb-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐶 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝐹𝑋) (𝐹𝑌))

Theoremcdleme35a 36516 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐹 𝑈) = (𝑅 𝑈))

Theoremcdleme35fnpq 36517 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝐹 (𝑃 𝑄))

Theoremcdleme35b 36518 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 ((𝑃 𝑅) 𝑊)) (𝑄 (𝑅 𝑈)))

Theoremcdleme35c 36519 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑄 𝐹) = (𝑄 ((𝑃 𝑅) 𝑊)))

Theoremcdleme35d 36520 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑄 𝐹) 𝑊) = ((𝑃 𝑅) 𝑊))

Theoremcdleme35e 36521 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝑃 ((𝑄 𝐹) 𝑊)) = (𝑃 𝑅))

Theoremcdleme35f 36522 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝑅 𝑈) (𝑃 𝑅)) = 𝑅)

Theoremcdleme35g 36523 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ((𝐹 𝑈) (𝑃 ((𝑄 𝐹) 𝑊))) = 𝑅)

Theoremcdleme35h 36524 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one outside of 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &   𝐺 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺)) → 𝑅 = 𝑆)

Theoremcdleme35h2 36525 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one outside of 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &   𝐺 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝐹𝐺)

Theoremcdleme35sn2aw 36526* Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one outside of 𝑃 𝑄 line case; compare cdleme32sn2awN 36502. TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)

Theoremcdleme35sn3a 36527* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → ¬ 𝑅 / 𝑠𝑁 (𝑃 𝑄))

Theoremcdleme36a 36528 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄))) → ¬ 𝑅 (𝑡 𝐸))

Theoremcdleme36m 36529 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 11-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝑉 = ((𝑡 𝐸) 𝑊)    &   𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))    &   𝐶 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐶) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)))) → 𝑅 = 𝑆)

Theoremcdleme37m 36530 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))    &   𝑉 = ((𝑡 𝐸) 𝑊)    &   𝑋 = ((𝑢 𝐷) 𝑊)    &   𝐶 = ((𝑆 𝑉) (𝐸 ((𝑡 𝑆) 𝑊)))    &   𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄)) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐶 = 𝐺)

Theoremcdleme38m 36531 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 13-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))    &   𝑉 = ((𝑡 𝐸) 𝑊)    &   𝑋 = ((𝑢 𝐷) 𝑊)    &   𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))    &   𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝐹 = 𝐺) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝑅 = 𝑆)

Theoremcdleme38n 36532 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. TODO shorter if proved directly from cdleme36m 36529 and cdleme37m 36530? (Contributed by NM, 14-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐷 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))    &   𝑉 = ((𝑡 𝐸) 𝑊)    &   𝑋 = ((𝑢 𝐷) 𝑊)    &   𝐹 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊)))    &   𝐺 = ((𝑆 𝑋) (𝐷 ((𝑢 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐹𝐺)

Theoremcdleme39a 36533 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. 𝐸, 𝑌, 𝐺, 𝑍 serve as f(t), f(u), ft(𝑅), ft(𝑆). Put hypotheses of cdleme38n 36532 in convention of cdleme32sn1awN 36500. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))    &   𝑉 = ((𝑡 𝐸) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑃 𝑄) ∧ (𝑡𝐴 ∧ ¬ 𝑡 𝑊))) → 𝐺 = ((𝑅 𝑉) (𝐸 ((𝑡 𝑅) 𝑊))))

Theoremcdleme39n 36534 Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. 𝐸, 𝑌, 𝐺, 𝑍 serve as f(t), f(u), ft(𝑅), ft(𝑆). Put hypotheses of cdleme38n 36532 in convention of cdleme32sn1awN 36500. TODO see if this hypothesis conversion would be better if done earlier. (Contributed by NM, 15-Mar-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))    &   𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝑌 ((𝑆 𝑢) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ ((𝑡𝐴 ∧ ¬ 𝑡 𝑊) ∧ ¬ 𝑡 (𝑃 𝑄)) ∧ ((𝑢𝐴 ∧ ¬ 𝑢 𝑊) ∧ ¬ 𝑢 (𝑃 𝑄)))) → 𝐺𝑍)

Theoremcdleme40m 36535* Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT Use proof idea from cdleme32sn1awN 36500. (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))    &   𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))    &   𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))    &   𝐹 = ((𝑃 𝑄) (𝑇 ((𝑆 𝑣) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ((𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆) ∧ (𝑣𝐴 ∧ ¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)))) → 𝑅 / 𝑠𝑁𝐹)

Theoremcdleme40n 36536* Part of proof of Lemma E in [Crawley] p. 113. Show that f(x) is one-to-one on 𝑃 𝑄 line. TODO: FIX COMMENT. TODO get rid of '.<' class? (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑌 = ((𝑃 𝑄) (𝐸 ((𝑅 𝑡) 𝑊)))    &   𝐶 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑌))    &   𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))    &   𝐹 = ((𝑃 𝑄) (𝑇 ((𝑆 𝑣) 𝑊)))    &   𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))    &   𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))    &   𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, < )    &   𝑍 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝐹))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑢𝑉)

Theoremcdleme40v 36537* Part of proof of Lemma E in [Crawley] p. 113. Change bound variables in 𝑆 / 𝑢𝑉 (but we use 𝑅 / 𝑢𝑉 for convenience since we have its hypotheses available). (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))    &   𝑇 = ((𝑣 𝑈) (𝑄 ((𝑃 𝑣) 𝑊)))    &   𝑋 = ((𝑃 𝑄) (𝑇 ((𝑢 𝑣) 𝑊)))    &   𝑂 = (𝑧𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑃 𝑄)) → 𝑧 = 𝑋))    &   𝑉 = if(𝑢 (𝑃 𝑄), 𝑂, 𝑌)       (𝑅𝐴𝑅 / 𝑠𝑁 = 𝑅 / 𝑢𝑉)

Theoremcdleme40w 36538* Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 36537 bound variable change to 𝑆 / 𝑢𝑉. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝑌 = ((𝑢 𝑈) (𝑄 ((𝑃 𝑢) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)

Theoremcdleme42a 36539 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 3-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑅 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝑅 𝑆) = (𝑅 𝑉))

Theoremcdleme42c 36540 Part of proof of Lemma E in [Crawley] p. 113. Match ¬ 𝑥 𝑊. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑅 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → ¬ (𝑅 𝑉) 𝑊)

Theoremcdleme42d 36541 Part of proof of Lemma E in [Crawley] p. 113. Match (𝑠 (𝑥 𝑊)) = 𝑥. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑅 𝑆) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) → (𝑅 ((𝑅 𝑉) 𝑊)) = (𝑅 𝑉))

Theoremcdleme41sn3aw 36542* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is different on and off the 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)

Theoremcdleme41sn4aw 36543* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for on and off 𝑃 𝑄 line. TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (¬ 𝑅 (𝑃 𝑄) ∧ 𝑆 (𝑃 𝑄) ∧ 𝑅𝑆)) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)

Theoremcdleme41snaw 36544* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for combined cases; compare cdleme32snaw 36503. TODO: FIX COMMENT. (Contributed by NM, 18-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → 𝑅 / 𝑠𝑁𝑆 / 𝑠𝑁)

Theoremcdleme41fva11 36545* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is one-to-one for r in W (r an atom not under w). TODO: FIX COMMENT. (Contributed by NM, 19-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → (𝐹𝑅) ≠ (𝐹𝑆))

Theoremcdleme42b 36546* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 6-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ (𝑃𝑄 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑅 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = (𝑅 / 𝑠𝑁 (𝑋 𝑊)))

Theoremcdleme42e 36547* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑃𝑄) → (𝐹‘(𝑅 𝑉)) = (𝑅 / 𝑠𝑁 ((𝑅 𝑉) 𝑊)))

Theoremcdleme42f 36548* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑃𝑄) → (𝐹‘(𝑅 𝑉)) = ((𝐹𝑅) 𝑉))

Theoremcdleme42g 36549* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑃𝑄) → (𝐹‘(𝑅 𝑆)) = ((𝐹𝑅) 𝑉))

Theoremcdleme42h 36550* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑃𝑄) → (𝐹𝑆) ((𝐹𝑅) 𝑉))

Theoremcdleme42i 36551* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 8-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑃𝑄) → ((𝐹𝑅) (𝐹𝑆)) ((𝐹𝑅) 𝑉))

Theoremcdleme42k 36552* Part of proof of Lemma E in [Crawley] p. 113. Since F ' S =/= F'R when S =/= R (i.e. 1-1); then ( ( F ' R ) .\/ ( F ' S ) ) is 2-dim therefore = ( ( F ' R ) .\/ V ) by cdleme42i 36551 and ps-1 35545 TODO: FIX COMMENT. (Contributed by NM, 20-Mar-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ 𝑅𝑆) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))

Theoremcdleme42ke 36553* Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑅𝑆 condition. TODO: FIX COMMENT. (Contributed by NM, 2-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))

Theoremcdleme42keg 36554* Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑃𝑄 condition. TODO: FIX COMMENT. TODO: Use instead of cdleme42ke 36553 and even combine with it? (Contributed by NM, 22-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝑉 = ((𝑅 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → ((𝐹𝑅) (𝐹𝑆)) = ((𝐹𝑅) 𝑉))

Theoremcdleme42mN 36555* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT . f preserves join: f(r s) = f(r) s, p. 115 10th line from bottom. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝐹‘(𝑅 𝑆)) = ((𝐹𝑅) (𝐹𝑆)))

Theoremcdleme42mgN 36556* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT . f preserves join: f(r s) = f(r) s, p. 115 10th line from bottom. TODO: Use instead of cdleme42mN 36555? Combine with cdleme42mN 36555? (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑠 𝑈) (𝑄 ((𝑃 𝑠) 𝑊)))    &   𝐸 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐺 = ((𝑃 𝑄) (𝐸 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐺))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐷)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊))) → (𝐹‘(𝑅 𝑆)) = ((𝐹𝑅) (𝐹𝑆)))

Theoremcdleme43aN 36557 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 penultimate line: g(f(r)) = (p v q) ^ (g(s) v v1). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑋 = ((𝑄 𝑃) 𝑊)    &   𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))    &   𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))    &   𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))    &   𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))    &   𝑉 = ((𝑍 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝐷) 𝑊)       ((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) → 𝐺 = ((𝑃 𝑄) (𝐷 𝑉)))

Theoremcdleme43bN 36558 Lemma for Lemma E in [Crawley] p. 113. g(s) is an atom not under w. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑋 = ((𝑄 𝑃) 𝑊)    &   𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))    &   𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))    &   𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))    &   𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))    &   𝑉 = ((𝑍 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝐷) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐷𝐴 ∧ ¬ 𝐷 𝑊))

Theoremcdleme43cN 36559 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 115 last line: r v g(s) = r v v2. (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑋 = ((𝑄 𝑃) 𝑊)    &   𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))    &   𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))    &   𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))    &   𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))    &   𝑉 = ((𝑍 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝐷) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝑅 𝐷) = (𝑅 𝑌))

Theoremcdleme43dN 36560 Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT p. 116 2nd line: f(r) v s = f(r) v f(g(s)). (Contributed by NM, 20-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑋 = ((𝑄 𝑃) 𝑊)    &   𝐶 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝐶 ((𝑅 𝑆) 𝑊)))    &   𝐷 = ((𝑆 𝑋) (𝑃 ((𝑄 𝑆) 𝑊)))    &   𝐺 = ((𝑄 𝑃) (𝐷 ((𝑍 𝑆) 𝑊)))    &   𝐸 = ((𝐷 𝑈) (𝑄 ((𝑃 𝐷) 𝑊)))    &   𝑉 = ((𝑍 𝑆) 𝑊)    &   𝑌 = ((𝑅 𝐷) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑍 𝑆) = (𝑍 𝐸))

Theoremcdleme46f2g2 36561 Conversion for 𝐺 to reuse 𝐹 theorems. TODO FIX COMMENT. TODO What other conversion theorems would be reused? e.g. cdlemeg46nlpq 36585? Find other hlatjcom 35436 uses giving 𝑄 𝑃. (Contributed by NM, 1-Apr-2013.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝑃 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑄 𝑃)))

Theoremcdleme46f2g1 36562 Conversion for 𝐺 to reuse 𝐹 theorems. TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013.)
= (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑄𝑃 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑄 𝑃) ∧ ¬ 𝑆 (𝑄 𝑃))))

Theoremcdleme17d2 36563* Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. 𝐹, 𝐺 represent f(s), fs(p) respectively. We show, in their notation, fs(p)=q. TODO: FIX COMMENT. (Contributed by NM, 5-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐹𝑃) = 𝑄)

Theoremcdleme17d3 36564* TODO: FIX COMMENT. (Contributed by NM, 5-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → (𝐹𝑃) = 𝑄)

Theoremcdleme17d4 36565* TODO: FIX COMMENT. (Contributed by NM, 11-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃 = 𝑄) → (𝐹𝑃) = 𝑄)

Theoremcdleme17d 36566* Part of proof of Lemma E in [Crawley] p. 114, first part of 4th paragraph. We show, in their notation, fs(p)=q. TODO FIX COMMENT. (Contributed by NM, 11-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐹𝑃) = 𝑄)

Theoremcdleme48fv 36567* Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 36509? TODO: Can this be used to help prove the 𝑅 or 𝑆 case where 𝑋 is an atom? (Contributed by NM, 8-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))

Theoremcdleme48fvg 36568* Remove 𝑃𝑄 condition in cdleme48fv 36567. TODO: Can this replace uses of cdleme32a 36509? TODO: Can this be used to help prove the 𝑅 or 𝑆 case where 𝑋 is an atom? TODO: Can this be proved more directly by eliminating 𝑃𝑄 in earlier theorems? Should this replace uses of cdleme48fv 36567? (Contributed by NM, 23-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑆) (𝑋 𝑊)))

Theoremcdleme46fvaw 36569* Show that (𝐹𝑅) is an atom not under 𝑊 when 𝑅 is an atom not under 𝑊. (Contributed by NM, 18-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ((𝐹𝑅) ∈ 𝐴 ∧ ¬ (𝐹𝑅) 𝑊))

Theoremcdleme48bw 36570* TODO: fix comment. TODO: Remove unnecessary 𝑃𝑄 from cdleme48bw 36570 cdlemeg46c 36581 cdlemeg46fvaw 36584 cdlemeg46rgv 36596 cdlemeg46gfv 36598? cdleme48d 36603? and possibly others they affect. (Contributed by NM, 9-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → ¬ (𝐹𝑋) 𝑊)

Theoremcdleme48b 36571* TODO: fix comment. (Contributed by NM, 8-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → ((𝐹𝑋) 𝑊) = (𝑋 𝑊))

Theoremcdleme46frvlpq 36572* Show that (𝐹𝑆) is not under 𝑃 𝑄 when 𝑆 isn't. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ (𝐹𝑆) (𝑃 𝑄))

Theoremcdleme46fsvlpq 36573* Show that (𝐹𝑅) is under 𝑃 𝑄 when 𝑅 is. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝐹𝑅) (𝑃 𝑄))

Theoremcdlemeg46fvcl 36574* TODO: fix comment. (Contributed by NM, 9-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑋𝐵) → (𝐺𝑋) ∈ 𝐵)

Theoremcdleme4gfv 36575* Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 36509? TODO: Can this be used to help prove the 𝑅 or 𝑆 case where 𝑋 is an atom? TODO: Would an antecedent transformer like cdleme46f2g2 36561 help? (Contributed by NM, 8-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑆 (𝑋 𝑊)) = 𝑋)) → (𝐺𝑋) = ((𝐺𝑆) (𝑋 𝑊)))

Theoremcdlemeg47b 36576* TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐺𝑆) = 𝑆 / 𝑣𝑁)

Theoremcdlemeg47rv 36577* Value of gs(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 3-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐺𝑅) = 𝑅 / 𝑢𝑆 / 𝑣𝑂)

Theoremcdlemeg47rv2 36578* Value of gs(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO: FIX COMMENT. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐺𝑅) = ((𝑄 𝑃) ((𝐺𝑆) ((𝑅 𝑆) 𝑊))))

Theoremcdlemeg49le 36579* Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 9-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝐺𝑋) (𝐺𝑌))

Theoremcdlemeg46bOLDN 36580* TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐺𝑆) = 𝑆 / 𝑣𝑁)

Theoremcdlemeg46c 36581* TODO FIX COMMENT. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → (𝐹‘(𝐺𝑆)) = 𝑆 / 𝑣𝑁 / 𝑡𝐷)

Theoremcdlemeg46rvOLDN 36582* Value of gs(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO FIX COMMENT. (Contributed by NM, 3-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐺𝑅) = 𝑅 / 𝑢𝑆 / 𝑣𝑂)

Theoremcdlemeg46rv2OLDN 36583* Value of gs(r) when r is an atom under pq and s is any atom not under pq, using very compact hypotheses. TODO FIX COMMENT. (Contributed by NM, 3-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐺𝑅) = ((𝑄 𝑃) ((𝐺𝑆) ((𝑅 𝑆) 𝑊))))

Theoremcdlemeg46fvaw 36584* Show that (𝐹𝑅) is an atom not under 𝑊 when 𝑅 is an atom not under 𝑊. (Contributed by NM, 1-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑃𝑄) → ((𝐺𝑅) ∈ 𝐴 ∧ ¬ (𝐺𝑅) 𝑊))

Theoremcdlemeg46nlpq 36585* Show that (𝐺𝑆) is not under 𝑃 𝑄 when 𝑆 isn't. (Contributed by NM, 3-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ ¬ 𝑆 (𝑃 𝑄)) → ¬ (𝐺𝑆) (𝑃 𝑄))

Theoremcdlemeg46ngfr 36586* TODO FIX COMMENT g(f(s))=s p. 115 4th line from bottom. (Contributed by NM, 4-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐺‘(𝐹𝑅)) = 𝑅)

Theoremcdlemeg46nfgr 36587* TODO FIX COMMENT f(g(s))=s p. 115 antepenultimate line. (Contributed by NM, 4-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐹‘(𝐺𝑅)) = 𝑅)

Theoremcdlemeg46sfg 36588* TODO FIX COMMENT f(r) s = f(r) g(s) p. 116 2nd line TODO: eliminate eqcomd? (Contributed by NM, 4-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝐹𝑅) 𝑆) = ((𝐹𝑅) (𝐹‘(𝐺𝑆))))

Theoremcdlemeg46fjgN 36589* NOT NEEDED? TODO FIX COMMENT. TODO eliminate eqcomd 2831? p. 116 2nd line. (Contributed by NM, 2-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝐹𝑅) (𝐹‘(𝐺𝑆))) = (𝐹‘(𝑅 (𝐺𝑆))))

Theoremcdlemeg46rjgN 36590* NOT NEEDED? TODO FIX COMMENT. r g(s) = r v2 p. 115 last line. (Contributed by NM, 2-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝑅 (𝐺𝑆)) = (𝑅 𝑌))

Theoremcdlemeg46fjv 36591* TODO FIX COMMENT f(r) f(g(s)) = f(r) v2 p. 116 2nd line. (Contributed by NM, 2-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝐹𝑅) (𝐹‘(𝐺𝑆))) = ((𝐹𝑅) 𝑌))

Theoremcdlemeg46fsfv 36592* TODO FIX COMMENT f(r) s = f(r) v2 p. 116 2nd line. (Contributed by NM, 2-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → ((𝐹𝑅) 𝑆) = ((𝐹𝑅) 𝑌))

Theoremcdlemeg46frv 36593* TODO FIX COMMENT. (f(r) v2) w = v2 p. 116 3rd line. (Contributed by NM, 2-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (((𝐹𝑅) 𝑌) 𝑊) = 𝑌)

Theoremcdlemeg46v1v2 36594* TODO FIX COMMENT v1 = v2 p. 116 3rd line. (Contributed by NM, 2-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)    &   𝑋 = (((𝐹𝑅) 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑋 = 𝑌)

Theoremcdlemeg46vrg 36595* TODO FIX COMMENT v1 r g(s) p. 116 3rd line. (Contributed by NM, 3-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑄 𝑃) 𝑊)    &   𝑁 = ((𝑣 𝑉) (𝑃 ((𝑄 𝑣) 𝑊)))    &   𝑂 = ((𝑄 𝑃) (𝑁 ((𝑢 𝑣) 𝑊)))    &   𝐺 = (𝑎𝐵 ↦ if((𝑄𝑃 ∧ ¬ 𝑎 𝑊), (𝑐𝐵𝑢𝐴 ((¬ 𝑢 𝑊 ∧ (𝑢 (𝑎 𝑊)) = 𝑎) → 𝑐 = (if(𝑢 (𝑄 𝑃), (𝑏𝐵𝑣𝐴 ((¬ 𝑣 𝑊 ∧ ¬ 𝑣 (𝑄 𝑃)) → 𝑏 = 𝑂)), 𝑢 / 𝑣𝑁) (𝑎 𝑊)))), 𝑎))    &   𝑌 = ((𝑅 (𝐺𝑆)) 𝑊)    &   𝑋 = (((𝐹𝑅) 𝑆) 𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑋 (𝑅 (𝐺𝑆)))

Theoremcdlemeg46rgv 36596* TODO FIX COMMENT r g(s) v1 p. 116 3rd line. (Contributed by NM, 3-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘