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Theorem List for Metamath Proof Explorer - 34601-34700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemefr32snb 34601* Show closure of 𝑅 / 𝑠𝑁. (Contributed by NM, 28-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)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅 / 𝑠𝑁𝐵)
 
Theoremcdlemefr29bpre0N 34602* 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 34603* Show closure of the unique element in cdleme29c 34572. 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 34604* 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 34605* 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 34606* 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 34607* Value of (𝐹𝑅) when ¬ 𝑅 (𝑃 𝑄). TODO This may be useful for shortening others that now use riotasv 33153 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 34608 FIX COMMENT. TODO: see if this is the optimal utility theorem using lhpmat 34224. (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 34609* 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 34562 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 34610* 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 34611* 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 34612* 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 34613* 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 34614* 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 34615* Show closure of the unique element in cdleme29c 34572. (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 34616* 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 34617* 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 34618* 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 34224 here and elsewhere, and presence/absence of 𝑠 (𝑃 𝑄) term. Also, why can proof be shortened with cdleme27cl 34562? What is difference from cdlemefs27cl 34609? (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 34619* 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 34620* Value of (𝐹𝑅) when 𝑅 (𝑃 𝑄). TODO This may be useful for shortening others that now use riotasv 33153 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 34621* 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 34622* 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 34625 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 34623* 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 34624* Explicit expansion of cdlemefr45 34623. TODO: use to shorten cdlemefr45 34623 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 34625* 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 34626* Explicit expansion of cdlemefs45 34625. TODO: use to shorten cdlemefs45 34625 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 34627* Explicit expansion of cdlemefs45 34625. TODO: use to shorten cdlemefs45 34625 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 34628* 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 34629* 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 34630* 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 34631* 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 34632* 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 34633* 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 34634* 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 34635* 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 34636* 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 34637* 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 34638* 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 34639* 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 34640* 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 34641* 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 34642* 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 34643* 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 34644 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 34645 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 34646 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 34647 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 34648 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 34649 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 34650 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 34651 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 34652 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 34653 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 34654* 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 34630. 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 34655* 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 34656 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 34657 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 34658 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 34659 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 34660 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 34657 and cdleme37m 34658? (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 34661 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 34660 in convention of cdleme32sn1awN 34628. 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 34662 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 34660 in convention of cdleme32sn1awN 34628. 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 34663* 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 34628. (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 34664* 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 34665* 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 34666* Part of proof of Lemma E in [Crawley] p. 113. Apply cdleme40v 34665 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 34667 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 34668 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 34669 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 34670* 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 34671* 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 34672* Part of proof of Lemma E in [Crawley] p. 113. Show that f(r) is for combined cases; compare cdleme32snaw 34631. 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 34673* 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 34674* 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 34675* 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 34676* 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 34677* 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 34678* 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 34679* 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 34680* 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 34679 and ps-1 33671 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 34681* 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 34682* Part of proof of Lemma E in [Crawley] p. 113. Remove 𝑃𝑄 condition. TODO: FIX COMMENT. TODO: Use instead of cdleme42ke 34681 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 34683* 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 34684* 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 34683? Combine with cdleme42mN 34683? (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 34685 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 34686 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 34687 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 34688 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 34689 Conversion for 𝐺 to reuse 𝐹 theorems. TODO FIX COMMENT. TODO What other conversion theorems would be reused? e.g. cdlemeg46nlpq 34713? Find other hlatjcom 33562 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 34690 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 34691* 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 34692* 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 34693* 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 34694* 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 34695* Part of proof of Lemma D in [Crawley] p. 113. TODO: Can this replace uses of cdleme32a 34637? 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 34696* Remove 𝑃𝑄 condition in cdleme48fv 34695. TODO: Can this replace uses of cdleme32a 34637? 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 34695? (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 34697* 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 34698* TODO: fix comment. TODO: Remove unnecessary 𝑃𝑄 from cdleme48bw 34698 cdlemeg46c 34709 cdlemeg46fvaw 34712 cdlemeg46rgv 34724 cdlemeg46gfv 34726? cdleme48d 34731? 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 34699* 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 34700* 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))
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