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Theorem List for Metamath Proof Explorer - 34801-34900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemg5 34801* TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 34199? TODO: The hypothesis is unused. FIX COMMENT. (Contributed by NM, 26-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ∃𝑞𝐴 (𝑃𝑞 ∧ ¬ 𝑞 𝑊))
 
Theoremcdlemb3 34802* Given two atoms not under the fiducial co-atom 𝑊, there is a third. Lemma B in [Crawley] p. 112. TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 34199? Then replace cdlemb2 34235 with it. This is a more general version of cdlemb2 34235 without 𝑃𝑄 condition. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → ∃𝑟𝐴𝑟 𝑊 ∧ ¬ 𝑟 (𝑃 𝑄)))
 
Theoremcdlemg7fvbwN 34803 Properties of a translation of an element not under 𝑊. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 34698? Done with a *ltrn* theorem? (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ 𝐹𝑇) → ((𝐹𝑋) ∈ 𝐵 ∧ ¬ (𝐹𝑋) 𝑊))
 
Theoremcdlemg4a 34804 TODO: FIX COMMENT If fg(p) = p, then tr f = tr g. (Contributed by NM, 23-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝐹‘(𝐺𝑃)) = 𝑃) → (𝑅𝐹) = (𝑅𝐺))
 
Theoremcdlemg4b1 34805 TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺𝑇) → (𝑃 𝑉) = (𝑃 (𝐺𝑃)))
 
Theoremcdlemg4b2 34806 TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺𝑇) → ((𝐺𝑃) 𝑉) = (𝑃 (𝐺𝑃)))
 
Theoremcdlemg4b12 34807 TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺𝑇) → ((𝐺𝑃) 𝑉) = (𝑃 𝑉))
 
Theoremcdlemg4c 34808 TODO: FIX COMMENT. (Contributed by NM, 24-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐺𝑇) ∧ ¬ 𝑄 (𝑃 𝑉)) → ¬ (𝐺𝑄) (𝑃 𝑉))
 
Theoremcdlemg4d 34809 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ¬ (𝐺𝑄) ((𝐺𝑃) (𝐹‘(𝐺𝑃))))
 
Theoremcdlemg4e 34810 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)    &    = (meet‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = (((𝐺𝑄) (𝑅𝐹)) ((𝐹‘(𝐺𝑃)) (((𝐺𝑃) (𝐺𝑄)) 𝑊))))
 
Theoremcdlemg4f 34811 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)    &    = (meet‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = ((𝑄 𝑉) (𝑃 ((𝑃 𝑄) 𝑊))))
 
Theoremcdlemg4g 34812 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)    &    = (meet‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = ((𝑄 𝑉) (𝑃 𝑄)))
 
Theoremcdlemg4 34813 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = 𝑄)
 
Theoremcdlemg6a 34814* TODO: FIX COMMENT. TODO: replace with cdlemg4 34813. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑟 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑟)) = 𝑟)
 
Theoremcdlemg6b 34815* TODO: FIX COMMENT. TODO: replace with cdlemg4 34813. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ 𝑄 (𝑟 𝑉) ∧ (𝐹‘(𝐺𝑟)) = 𝑟)) → (𝐹‘(𝐺𝑄)) = 𝑄)
 
Theoremcdlemg6c 34816* TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 𝑉)) → (𝐹‘(𝐺𝑄)) = 𝑄))
 
Theoremcdlemg6d 34817* TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (((𝑟𝐴 ∧ ¬ 𝑟 𝑊) ∧ ¬ 𝑟 (𝑃 (𝐺𝑃))) → (𝐹‘(𝐺𝑄)) = 𝑄))
 
Theoremcdlemg6e 34818 TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    = (join‘𝐾)    &   𝑉 = (𝑅𝐺)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑄 (𝑃 𝑉) ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = 𝑄)
 
Theoremcdlemg6 34819 TODO: FIX COMMENT. (Contributed by NM, 27-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑄)) = 𝑄)
 
Theoremcdlemg7fvN 34820 Value of a translation composition in terms of an associated atom. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹‘(𝐺𝑋)) = ((𝐹‘(𝐺𝑃)) (𝑋 𝑊)))
 
Theoremcdlemg7aN 34821 TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑋)) = 𝑋)
 
Theoremcdlemg7N 34822 TODO: FIX COMMENT. (Contributed by NM, 28-Apr-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑋𝐵) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → (𝐹‘(𝐺𝑋)) = 𝑋)
 
Theoremcdlemg8a 34823 TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹‘(𝐺𝑃)) = 𝑃)) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg8b 34824 TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑃 (𝐹‘(𝐺𝑃))) = (𝑃 𝑄))
 
Theoremcdlemg8c 34825 TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → (𝑄 (𝐹‘(𝐺𝑄))) = (𝑃 𝑄))
 
Theoremcdlemg8d 34826 TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄) ∧ (𝐹‘(𝐺𝑃)) ≠ 𝑃)) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg8 34827 TODO: FIX COMMENT. (Contributed by NM, 29-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg9a 34828 TODO: FIX COMMENT. (Contributed by NM, 1-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 𝑈) ((𝐹‘(𝐺𝑃)) 𝑈)) ((𝐺𝑃) 𝑈))
 
Theoremcdlemg9b 34829 The triples 𝑃, (𝐹‘(𝐺𝑃)), (𝐹𝑃)⟩ and 𝑄, (𝐹‘(𝐺𝑄)), (𝐹𝑄)⟩ are centrally perspective. TODO: FIX COMMENT. (Contributed by NM, 1-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 𝑄) ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄)))) ((𝐺𝑃) (𝐺𝑄)))
 
Theoremcdlemg9 34830 The triples 𝑃, (𝐹‘(𝐺𝑃)), (𝐹𝑃)⟩ and 𝑄, (𝐹‘(𝐺𝑄)), (𝐹𝑄)⟩ are axially perspective by dalaw 34080. Part of Lemma G of [Crawley] p. 116, last 2 lines. TODO: FIX COMMENT. (Contributed by NM, 1-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) ((((𝐹‘(𝐺𝑃)) (𝐺𝑃)) ((𝐹‘(𝐺𝑄)) (𝐺𝑄))) (((𝐺𝑃) 𝑃) ((𝐺𝑄) 𝑄))))
 
Theoremcdlemg10b 34831 TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? (Contributed by NM, 4-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
 
Theoremcdlemg10bALTN 34832 TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 34800 and pick best, if moved to ltrn* area. (Contributed by NM, 4-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = ((𝑃 𝑄) 𝑊))
 
Theoremcdlemg11a 34833 TODO: FIX COMMENT. (Contributed by NM, 4-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → (𝐹‘(𝐺𝑃)) ≠ 𝑃)
 
Theoremcdlemg11aq 34834 TODO: FIX COMMENT. TODO: can proof using this be restructured to use cdlemg11a 34833? (Contributed by NM, 4-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → (𝐹‘(𝐺𝑄)) ≠ 𝑄)
 
Theoremcdlemg10c 34835 TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in trl* area? (Contributed by NM, 4-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑅𝐹) ((𝐺𝑃) (𝐺𝑄)) ↔ (𝑅𝐹) (𝑃 𝑄)))
 
Theoremcdlemg10a 34836 TODO: FIX COMMENT. (Contributed by NM, 3-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) ((𝑅𝐹) (𝑅𝐺)))
 
Theoremcdlemg10 34837 TODO: FIX COMMENT. (Contributed by NM, 4-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) 𝑊)
 
Theoremcdlemg11b 34838 TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (𝑃 𝑄) ≠ ((𝐺𝑃) (𝐺𝑄)))
 
Theoremcdlemg12a 34839 TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ (𝑃 𝑈) ≠ ((𝐺𝑃) 𝑈))) → ((𝑃 𝑈) ((𝐺𝑃) 𝑈)) ((𝐹‘(𝐺𝑃)) 𝑈))
 
Theoremcdlemg12b 34840 The triples 𝑃, (𝐹𝑃), (𝐹‘(𝐺𝑃))⟩ and 𝑄, (𝐹𝑄), (𝐹‘(𝐺𝑄))⟩ are centrally perspective. TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 𝑄) ((𝐺𝑃) (𝐺𝑄))) ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))))
 
Theoremcdlemg12c 34841 The triples 𝑃, (𝐹𝑃), (𝐹‘(𝐺𝑃))⟩ and 𝑄, (𝐹𝑄), (𝐹‘(𝐺𝑄))⟩ are axially perspective by dalaw 34080. TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐺𝑃)) (𝑄 (𝐺𝑄))) ((((𝐺𝑃) (𝐹‘(𝐺𝑃))) ((𝐺𝑄) (𝐹‘(𝐺𝑄)))) (((𝐹‘(𝐺𝑃)) 𝑃) ((𝐹‘(𝐺𝑄)) 𝑄))))
 
Theoremcdlemg12d 34842 TODO: FIX COMMENT. (Contributed by NM, 5-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → (𝑅𝐺) ((𝑅𝐹) (((𝐹‘(𝐺𝑃)) 𝑃) ((𝐹‘(𝐺𝑄)) 𝑄))))
 
Theoremcdlemg12e 34843 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    0 = (0.‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄) ∧ (𝑅𝐹) ≠ (𝑅𝐺))) → (((𝐹‘(𝐺𝑃)) 𝑃) ((𝐹‘(𝐺𝑄)) 𝑄)) ≠ 0 )
 
Theoremcdlemg12f 34844 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄)) ∧ (𝑅𝐹) ≠ (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊))
 
Theoremcdlemg12g 34845 TODO: FIX COMMENT. TODO: Combine with cdlemg12f 34844. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄)) ∧ (𝑅𝐹) ≠ (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) = ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊))
 
Theoremcdlemg12 34846 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄)) ∧ (𝑅𝐹) ≠ (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg13a 34847 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝑅𝐹) = (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → (𝑃 (𝐹‘(𝐺𝑃))) = ((𝐺𝑃) (𝐹‘(𝐺𝑃))))
 
Theoremcdlemg13 34848 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝑅𝐹) = (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg14f 34849 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐹𝑃) = 𝑃)) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg14g 34850 TODO: FIX COMMENT. (Contributed by NM, 22-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇 ∧ (𝐺𝑃) = 𝑃)) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg15a 34851 Eliminate the (𝐹𝑃) ≠ 𝑃 condition from cdlemg13 34848. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇) ∧ ((𝑅𝐹) = (𝑅𝐺) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg15 34852 Eliminate the ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) condition from cdlemg13 34848. TODO: FIX COMMENT. (Contributed by NM, 25-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑅𝐹) = (𝑅𝐺)) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg16 34853 Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 34837 "implies (2)" (of p. 116). No details are provided by the authors, so there may be a shorter proof; but ours requires the 14 lemmas, one using Desargues' law dalaw 34080, in order to make this inference. This final step eliminates the (𝑅𝐹) ≠ (𝑅𝐺) condition from cdlemg12 34846. TODO: FIX COMMENT. TODO: should we also eliminate 𝑃𝑄 here (or earlier)? Do it if we don't need to add it in for something else later. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg16ALTN 34854 This version of cdlemg16 34853 uses cdlemg15a 34851 instead of cdlemg15 34852, in case cdlemg15 34852 ends up not being needed. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻 ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ (((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg16z 34855 Eliminate ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) condition from cdlemg16 34853. TODO: would it help to also eliminate 𝑃𝑄 here or later? (Contributed by NM, 25-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg16zz 34856 Eliminate 𝑃𝑄 from cdlemg16z 34855. TODO: Use this only if needed. (Contributed by NM, 26-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ (𝑅𝐹) (𝑃 𝑄) ∧ ¬ (𝑅𝐺) (𝑃 𝑄))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg17a 34857 TODO: FIX COMMENT. (Contributed by NM, 8-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇 ∧ (𝑅𝐺) (𝑃 𝑄))) → (𝐺𝑃) (𝑃 𝑄))
 
Theoremcdlemg17b 34858* Part of proof of Lemma G in [Crawley] p. 117, 4th line. Whenever (in their terminology) p q/0 (i.e. the sublattice from 0 to p q) contains precisely three atoms and g is not the identity, g(p) = q. See also comments under cdleme0nex 34485. (Contributed by NM, 8-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑃) = 𝑄)
 
Theoremcdlemg17dN 34859* TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻𝐺𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ∧ (𝐺𝑃) ≠ 𝑃)) → (𝑅𝐺) = ((𝑃 𝑄) 𝑊))
 
Theoremcdlemg17dALTN 34860 Same as cdlemg17dN 34859 with fewer antecedents but longer proof TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻𝐺𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴𝑃𝑄) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ (𝐺𝑃) ≠ 𝑃)) → (𝑅𝐺) = ((𝑃 𝑄) 𝑊))
 
Theoremcdlemg17e 34861* TODO: fix comment. (Contributed by NM, 8-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) (𝑅𝐺)))
 
Theoremcdlemg17f 34862* TODO: fix comment. (Contributed by NM, 8-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) (𝐺‘(𝐹𝑃))))
 
Theoremcdlemg17g 34863* TODO: fix comment. (Contributed by NM, 9-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺‘(𝐹𝑃)) ((𝐹𝑃) (𝐹𝑄)))
 
Theoremcdlemg17h 34864* TODO: fix comment. (Contributed by NM, 10-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝑄𝑆 ((𝐹𝑃) (𝐹𝑄)))) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑆 = (𝐹𝑃) ∨ 𝑆 = (𝐹𝑄)))
 
Theoremcdlemg17i 34865* TODO: fix comment. (Contributed by NM, 10-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺‘(𝐹𝑃)) = (𝐹𝑄))
 
Theoremcdlemg17ir 34866* TODO: fix comment. (Contributed by NM, 13-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹‘(𝐺𝑃)) = (𝐹𝑄))
 
Theoremcdlemg17j 34867* TODO: fix comment. (Contributed by NM, 11-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺‘(𝐹𝑃)) = (𝐹‘(𝐺𝑃)))
 
Theoremcdlemg17pq 34868* Utility theorem for swapping 𝑃 and 𝑄. TODO: fix comment. (Contributed by NM, 11-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑄𝑃) ∧ ((𝐺𝑄) ≠ 𝑄 ∧ (𝑅𝐺) (𝑄 𝑃) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑄 𝑟) = (𝑃 𝑟)))))
 
Theoremcdlemg17bq 34869* cdlemg17b 34858 with 𝑃 and 𝑄 swapped. Antecedent 𝐹 ∈ (𝑇𝑊) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 34858 also? (Contributed by NM, 13-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺𝑄) = 𝑃)
 
Theoremcdlemg17iqN 34870* cdlemg17i 34865 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻 ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)) ∧ (𝐺𝑃) ≠ 𝑃)) → (𝐺‘(𝐹𝑄)) = (𝐹𝑃))
 
Theoremcdlemg17irq 34871* cdlemg17ir 34866 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐹‘(𝐺𝑄)) = (𝐹𝑃))
 
Theoremcdlemg17jq 34872* cdlemg17j 34867 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺‘(𝐹𝑄)) = (𝐹‘(𝐺𝑄)))
 
Theoremcdlemg17 34873* Part of Lemma G of [Crawley] p. 117, lines 7 and 8. We show an argument whose value at 𝐺 equals itself. TODO: fix comment. (Contributed by NM, 12-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝐺𝑃) ≠ 𝑃 ∧ (𝑅𝐺) (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝐺‘((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄))))) = ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))))
 
Theoremcdlemg18a 34874 Show two lines are different. TODO: fix comment. (Contributed by NM, 14-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝐹𝑇) ∧ (𝑃𝑄 ∧ ((𝐹𝑄) (𝐹𝑃)) ≠ (𝑃 𝑄))) → (𝑃 (𝐹𝑄)) ≠ (𝑄 (𝐹𝑃)))
 
Theoremcdlemg18b 34875 Lemma for cdlemg18c 34876. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝑃𝑄 ∧ (𝐹𝑃) ≠ 𝑄 ∧ ((𝐹𝑄) (𝐹𝑃)) ≠ (𝑃 𝑄))) → ¬ 𝑃 (𝑈 (𝐹𝑄)))
 
Theoremcdlemg18c 34876 Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝑃𝑄 ∧ (𝐹𝑃) ≠ 𝑄 ∧ ((𝐹𝑄) (𝐹𝑃)) ≠ (𝑃 𝑄))) → ((𝑃 (𝐹𝑄)) (𝑄 (𝐹𝑃))) ∈ 𝐴)
 
Theoremcdlemg18d 34877* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑇𝐺𝑇) ∧ 𝑃𝑄 ∧ (𝐺𝑃) ≠ 𝑃) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) ∈ 𝐴)
 
Theoremcdlemg18 34878* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑇𝐺𝑇) ∧ 𝑃𝑄 ∧ (𝐺𝑃) ≠ 𝑃) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) 𝑊)
 
Theoremcdlemg19a 34879* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑇𝐺𝑇) ∧ 𝑃𝑄 ∧ (𝐺𝑃) ≠ 𝑃) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) (𝑄 (𝐹‘(𝐺𝑄)))) = ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊))
 
Theoremcdlemg19 34880* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑇𝐺𝑇) ∧ 𝑃𝑄 ∧ (𝐺𝑃) ≠ 𝑃) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg20 34881* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 23-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝑅𝐺) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg21 34882* Version of cdlemg19 with (𝑅𝐹) (𝑃 𝑄) instead of (𝑅𝐺) (𝑃 𝑄) as a condition. (Contributed by NM, 23-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑇𝐺𝑇) ∧ 𝑃𝑄 ∧ (𝐹𝑃) ≠ 𝑃) ∧ ((𝑅𝐹) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg22 34883* cdlemg21 34882 with (𝐹𝑃) ≠ 𝑃 condition removed. (Contributed by NM, 23-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ ((𝑅𝐹) (𝑃 𝑄) ∧ ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg24 34884* Combine cdlemg16z 34855 and cdlemg22 34883. TODO: Fix comment. (Contributed by NM, 24-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇𝑃𝑄) ∧ (((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg37 34885* Use cdlemg8 34827 to eliminate the ≠ (𝑃 𝑄) condition of cdlemg24 34884. (Contributed by NM, 31-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇𝑃𝑄 ∧ ¬ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
 
Theoremcdlemg25zz 34886 cdlemg16zz 34856 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ (𝑅𝐹) (𝑃 𝑧) ∧ ¬ (𝑅𝐺) (𝑃 𝑧))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑧 (𝐹‘(𝐺𝑧))) 𝑊))
 
Theoremcdlemg26zz 34887 cdlemg16zz 34856 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ 𝐹𝑇) ∧ (𝐺𝑇 ∧ ¬ (𝑅𝐹) (𝑄 𝑧) ∧ ¬ (𝑅𝐺) (𝑄 𝑧))) → ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊) = ((𝑧 (𝐹‘(𝐺𝑧))) 𝑊))
 
Theoremcdlemg27a 34888 For use with case when (𝑃 𝑣) (𝑄 (𝑅𝐹)) or (𝑃 𝑣) (𝑄 (𝑅𝐹)) is zero, letting us establish ¬ 𝑧 𝑊𝑧 (𝑃 𝑣) via 4atex 34270. TODO: Fix comment. (Contributed by NM, 28-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑣𝐴𝑣 𝑊)) ∧ (𝑧𝐴𝐹𝑇) ∧ (𝑣 ≠ (𝑅𝐹) ∧ 𝑧 (𝑃 𝑣) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ (𝑅𝐹) (𝑃 𝑧))
 
Theoremcdlemg28a 34889 Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑣𝐴𝑣 𝑊)) ∧ ((𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ 𝐹𝑇𝐺𝑇) ∧ ((𝑣 ≠ (𝑅𝐹) ∧ 𝑣 ≠ (𝑅𝐺)) ∧ 𝑧 (𝑃 𝑣) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝐺𝑃) ≠ 𝑃))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑧 (𝐹‘(𝐺𝑧))) 𝑊))
 
Theoremcdlemg31b0N 34890 TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       (((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ 𝑣 ≠ (𝑅𝐹) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑁𝐴𝑁 = (0.‘𝐾)))
 
Theoremcdlemg31b0a 34891 TODO: Fix comment. (Contributed by NM, 30-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑣𝐴𝑣 𝑊)) ∧ (𝐹𝑇𝑣 ≠ (𝑅𝐹))) → (𝑁𝐴𝑁 = (0.‘𝐾)))
 
Theoremcdlemg27b 34892 TODO: Fix comment. (Contributed by NM, 28-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑧𝐴 ∧ (𝑣𝐴𝑣 𝑊) ∧ (𝐹𝑇𝑧𝑁)) ∧ (𝑣 ≠ (𝑅𝐹) ∧ 𝑧 (𝑃 𝑣) ∧ (𝐹𝑃) ≠ 𝑃)) → ¬ (𝑅𝐹) (𝑄 𝑧))
 
Theoremcdlemg31a 34893 TODO: fix comment. (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑣𝐴𝐹𝑇)) → 𝑁 (𝑃 𝑣))
 
Theoremcdlemg31b 34894 TODO: fix comment. (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑣𝐴𝐹𝑇)) → 𝑁 (𝑄 (𝑅𝐹)))
 
Theoremcdlemg31c 34895 Show that when 𝑁 is an atom, it is not under 𝑊. TODO: Is there a shorter direct proof? TODO: should we eliminate (𝐹𝑃) ≠ 𝑃 here? (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ 𝐹𝑇) ∧ (𝑣 ≠ (𝑅𝐹) ∧ (𝐹𝑃) ≠ 𝑃𝑁𝐴)) → ¬ 𝑁 𝑊)
 
Theoremcdlemg31d 34896 Eliminate (𝐹𝑃) ≠ 𝑃 from cdlemg31c 34895. TODO: Prove directly. TODO: do we need to eliminate (𝐹𝑃) ≠ 𝑃? It might be better to do this all at once at the end. See also cdlemg29 34901 vs. cdlemg28 34900. (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑣𝐴𝑣 𝑊)) ∧ (𝐹𝑇𝑣 ≠ (𝑅𝐹) ∧ 𝑁𝐴)) → ¬ 𝑁 𝑊)
 
Theoremcdlemg33b0 34897* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ 𝑁𝐴𝐹𝑇) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊 ∧ (𝑧𝑁𝑧 (𝑃 𝑣))))
 
Theoremcdlemg33c0 34898* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ 𝐹𝑇) ∧ (𝑃𝑄𝑣 ≠ (𝑅𝐹) ∧ ∃𝑟𝐴𝑟 𝑊 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → ∃𝑧𝐴𝑧 𝑊𝑧 (𝑃 𝑣)))
 
Theoremcdlemg28b 34899* Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that ¬ 𝑧 𝑊 is redundant here (but simplifies cdlemg28 34900.) (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))    &   𝑂 = ((𝑃 𝑣) (𝑄 (𝑅𝐺)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣)) ∧ (𝑣 ≠ (𝑅𝐹) ∧ 𝑣 ≠ (𝑅𝐺)) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝐺𝑃) ≠ 𝑃))) → ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊) = ((𝑧 (𝐹‘(𝐺𝑧))) 𝑊))
 
Theoremcdlemg28 34900* Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 34901 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑁 = ((𝑃 𝑣) (𝑄 (𝑅𝐹)))    &   𝑂 = ((𝑃 𝑣) (𝑄 (𝑅𝐺)))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝑣𝐴𝑣 𝑊) ∧ (𝑧𝐴 ∧ ¬ 𝑧 𝑊) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑧𝑁𝑧𝑂𝑧 (𝑃 𝑣)) ∧ (𝑣 ≠ (𝑅𝐹) ∧ 𝑣 ≠ (𝑅𝐺)) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝐺𝑃) ≠ 𝑃))) → ((𝑃 (𝐹‘(𝐺𝑃))) 𝑊) = ((𝑄 (𝐹‘(𝐺𝑄))) 𝑊))
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