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Theorem List for Metamath Proof Explorer - 37701-37800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcdlemefr29bpre0N 37701* TODO fix comment. (Contributed by NM, 28-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)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (∀𝑠𝐴 (((¬ 𝑠 𝑊 ∧ ¬ 𝑠 (𝑃 𝑄)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))

Theoremcdlemefr29clN 37702* Show closure of the unique element in cdleme29c 37671. TODO fix comment. TODO Not needed? (Contributed by NM, 29-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)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑂𝐵)

Theoremcdleme43frv1snN 37703* Value of 𝑅 / 𝑠𝑁 when ¬ 𝑅 (𝑃 𝑄). (Contributed by NM, 30-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)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁 = 𝑋)

Theoremcdlemefr32fvaN 37704* Part of proof of Lemma E in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. TODO: FIX COMMENT. (Contributed by NM, 29-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)))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (𝑁 (𝑥 𝑊))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)

Theoremcdlemefr32fva1 37705* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-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)))    &   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)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐹𝑅) = 𝑅 / 𝑠𝑁)

Theoremcdlemefr31fv1 37706* Value of (𝐹𝑅) when ¬ 𝑅 (𝑃 𝑄). TODO This may be useful for shortening others that now use riotasv 36254 3d . TODO: FIX COMMENT. (Contributed by NM, 30-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)))    &   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))    &   𝑋 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐹𝑅) = 𝑋)

Theoremcdlemefs29pre00N 37707 FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 37325. (Contributed by NM, 27-Mar-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑅 (𝑃 𝑄)) ∧ 𝑠𝐴) → (((¬ 𝑠 𝑊𝑠 (𝑃 𝑄)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) ↔ (¬ 𝑠 𝑊 ∧ (𝑠 (𝑅 𝑊)) = 𝑅)))

Theoremcdlemefs27cl 37708* Part of proof of Lemma E in [Crawley] p. 113. Closure of 𝑁. TODO FIX COMMENT This is the start of a re-proof of cdleme27cl 37661 etc. with the 𝑠 (𝑃 𝑄) condition (so as to not have the 𝐶 hypothesis). (Contributed by NM, 24-Mar-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)))    &   𝐼 = (𝑢𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑢 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑠𝐴 ∧ ¬ 𝑠 𝑊) ∧ 𝑠 (𝑃 𝑄) ∧ 𝑃𝑄)) → 𝑁𝐵)

Theoremcdlemefs32sn1aw 37709* Show that 𝑅 / 𝑠𝑁 is an atom not under 𝑊 when 𝑅 (𝑃 𝑄). (Contributed by NM, 24-Mar-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)))    &   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 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))

Theoremcdlemefs32snb 37710* Show closure of 𝑅 / 𝑠𝑁. (Contributed by NM, 24-Mar-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)))    &   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 / 𝑠𝑁𝐵)

Theoremcdlemefs29bpre0N 37711* TODO: FIX COMMENT. (Contributed by NM, 26-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 = ((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)) → (∀𝑠𝐴 (((¬ 𝑠 𝑊𝑠 (𝑃 𝑄)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))) ↔ 𝑧 = 𝑅 / 𝑠𝑁))

Theoremcdlemefs29bpre1N 37712* TODO: FIX COMMENT. (Contributed by NM, 27-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 = ((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)) → ∃𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝑠 (𝑃 𝑄)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))

Theoremcdlemefs29cpre1N 37713* TODO: FIX COMMENT. (Contributed by NM, 26-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 = ((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)) → ∃!𝑧𝐵𝑠𝐴 (((¬ 𝑠 𝑊𝑠 (𝑃 𝑄)) ∧ (𝑠 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))

Theoremcdlemefs29clN 37714* Show closure of the unique element in cdleme29c 37671. (Contributed by NM, 27-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 = ((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 (𝑅 𝑊)) = 𝑅) → 𝑧 = (𝑁 (𝑅 𝑊))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑂𝐵)

Theoremcdleme43fsv1snlem 37715* Value of 𝑅 / 𝑠𝑁 when 𝑅 (𝑃 𝑄). (Contributed by NM, 30-Mar-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)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑌 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝑌 ((𝑅 𝑆) 𝑊)))    &   𝑉 = ((𝑃 𝑄) (𝐷 ((𝑅 𝑡) 𝑊)))    &   𝑋 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝑉))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 / 𝑠𝑁 = 𝑍)

Theoremcdleme43fsv1sn 37716* Value of 𝑅 / 𝑠𝑁 when 𝑅 (𝑃 𝑄). (Contributed by NM, 30-Mar-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)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))    &   𝑁 = if(𝑠 (𝑃 𝑄), 𝐼, 𝐶)    &   𝑌 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝑌 ((𝑅 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → 𝑅 / 𝑠𝑁 = 𝑍)

Theoremcdlemefs32fvaN 37717* Part of proof of Lemma E in [Crawley] p. 113. Value of 𝐹 at an atom not under 𝑊. TODO: FIX COMMENT. TODO: consolidate uses of lhpmat 37325 here and elsewhere, and presence/absence of 𝑠 (𝑃 𝑄) term. Also, why can proof be shortened with cdleme27cl 37661? What is difference from cdlemefs27cl 37708? (Contributed by NM, 29-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 = ((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))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑥𝑂 = 𝑅 / 𝑠𝑁)

Theoremcdlemefs32fva1 37718* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. (Contributed by NM, 29-Mar-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)))    &   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)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝑅 (𝑃 𝑄)) → (𝐹𝑅) = 𝑅 / 𝑠𝑁)

Theoremcdlemefs31fv1 37719* Value of (𝐹𝑅) when 𝑅 (𝑃 𝑄). TODO This may be useful for shortening others that now use riotasv 36254 3d . TODO: FIX COMMENT. ***END OF VALUE AT ATOM STUFF TO REPLACE ONES BELOW***
```       "cdleme3xsn1aw" decreased using "cdlemefs32sn1aw"
"cdleme32sn1aw" decreased from 3302 to 36 using "cdlemefs32sn1aw".
"cdleme32sn2aw" decreased from 1687 to 26 using "cdlemefr32sn2aw".
"cdleme32snaw" decreased from 376 to 375 using "cdlemefs32sn1aw".
"cdleme32snaw" decreased from 375 to 368 using "cdlemefr32sn2aw".
"cdleme35sn3a" decreased from 547 to 523 using "cdleme43frv1sn".
```
(Contributed by NM, 27-Mar-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)))    &   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))    &   𝑌 = ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊)))    &   𝑍 = ((𝑃 𝑄) (𝑌 ((𝑅 𝑆) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = 𝑍)

Theoremcdlemefr44 37720* Value of f(r) when r is an atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefr45 instead? TODO: FIX COMMENT. (Contributed by NM, 31-Mar-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)))    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → (𝐹𝑅) = 𝑅 / 𝑡𝐷)

Theoremcdlemefs44 37721* Value of fs(r) when r is an atom under pq and s is any atom not under pq, using more compact hypotheses. TODO: eliminate and use cdlemefs45 37724 instead TODO: FIX COMMENT. (Contributed by NM, 31-Mar-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)))    &   𝑂 = (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), 𝐼, 𝑠 / 𝑡𝐷) (𝑥 𝑊))))    &   𝐹 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), 𝑂, 𝑥))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐼 = (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = 𝑅 / 𝑠𝑆 / 𝑡𝐸)

Theoremcdlemefr45 37722* Value of f(r) when r is an 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)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 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) = 𝑅 / 𝑡𝐷)

Theoremcdlemefr45e 37723* Explicit expansion of cdlemefr45 37722. TODO: use to shorten cdlemefr45 37722 uses? TODO: FIX COMMENT. (Contributed by NM, 10-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)))    &   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) = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊))))

Theoremcdlemefs45 37724* Value of fs(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)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 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))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = 𝑅 / 𝑠𝑆 / 𝑡𝐸)

Theoremcdlemefs45ee 37725* Explicit expansion of cdlemefs45 37724. TODO: use to shorten cdlemefs45 37724 uses? Should ((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) be assigned to a hypothesis letter? TODO: FIX COMMENT. (Contributed by NM, 10-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)))    &   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))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = ((𝑃 𝑄) (((𝑆 𝑈) (𝑄 ((𝑃 𝑆) 𝑊))) ((𝑅 𝑆) 𝑊))))

Theoremcdlemefs45eN 37726* Explicit expansion of cdlemefs45 37724. TODO: use to shorten cdlemefs45 37724 uses? TODO: FIX COMMENT. (Contributed by NM, 10-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)))    &   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))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊)) ∧ (𝑅 (𝑃 𝑄) ∧ ¬ 𝑆 (𝑃 𝑄))) → (𝐹𝑅) = ((𝑃 𝑄) ((𝐹𝑆) ((𝑅 𝑆) 𝑊))))

Theoremcdleme32sn1awN 37727* 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)    &   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 / 𝑠𝑁𝐴 ∧ ¬ 𝑅 / 𝑠𝑁 𝑊))

Theoremcdleme41sn3a 37728* 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 37729* 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 37730* 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 37731* 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 37732* 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 37733* 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 37734* 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 37735* 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 37736* 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 37737* 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 37738* 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 37739* 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 37740* 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 37741* 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 37742* 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 37743 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 37744 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 37745 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 37746 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 37747 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 37748 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 37749 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 37750 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 37751 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 37752 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 37753* 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 37729. 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 37754* 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 37755 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 37756 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 37757 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 37758 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 37759 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 37756 and cdleme37m 37757? (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 37760 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 37759 in convention of cdleme32sn1awN 37727. 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 37761 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 37759 in convention of cdleme32sn1awN 37727. 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 37762* 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 37727. (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 37763* 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 37764* 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 37765* Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 37764 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 37766 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 37767 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 37768 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 37769* 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 37770* 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 37771* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for combined cases; compare cdleme32snaw 37730. 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 37772* 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 37773* 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 37774* 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 37775* 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 37776* 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 37777* 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 37778* 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 37779* 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 37778 and ps-1 36772 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 37780* 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 37781* Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑃𝑄 condition. TODO: FIX COMMENT. TODO: Use instead of cdleme42ke 37780 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 37782* 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 37783* 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 37782? Combine with cdleme42mN 37782? (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 37784 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 37785 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 37786 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 37787 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 37788 Conversion for 𝐺 to reuse 𝐹 theorems. TODO FIX COMMENT. TODO What other conversion theorems would be reused? e.g. cdlemeg46nlpq 37812? Find other hlatjcom 36663 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 37789 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 37790* 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 37791* 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 37792* 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 37793* 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 37794* Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 37736? 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 37795* Remove 𝑃𝑄 condition in cdleme48fv 37794. TODO: Can this replace uses of cdleme32a 37736? 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 37794? (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 37796* 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 37797* TODO: fix comment. TODO: Remove unnecessary 𝑃𝑄 from cdleme48bw 37797 cdlemeg46c 37808 cdlemeg46fvaw 37811 cdlemeg46rgv 37823 cdlemeg46gfv 37825? cdleme48d 37830? 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 37798* 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 37799* 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 37800* 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))

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