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Theorem List for Metamath Proof Explorer - 39001-39100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemg13 39001 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΉ) = (π‘…β€˜πΊ) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg14f 39002 TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg14g 39003 TODO: FIX COMMENT. (Contributed by NM, 22-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (πΊβ€˜π‘ƒ) = 𝑃)) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg15a 39004 Eliminate the (πΉβ€˜π‘ƒ) β‰  𝑃 condition from cdlemg13 39001. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) = (π‘…β€˜πΊ) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg15 39005 Eliminate the ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) condition from cdlemg13 39001. TODO: FIX COMMENT. (Contributed by NM, 25-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘…β€˜πΉ) = (π‘…β€˜πΊ)) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg16 39006 Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 38990 "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's law dalaw 38235, in order to make this inference. This final step eliminates the (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ) condition from cdlemg12 38999. TODO: FIX COMMENT. TODO: should we also eliminate 𝑃 β‰  𝑄 here (or earlier)? Do it if we don't need to add it in for something else later. (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg16ALTN 39007 This version of cdlemg16 39006 uses cdlemg15a 39004 instead of cdlemg15 39005, in case cdlemg15 39005 ends up not being needed. TODO: FIX COMMENT. (Contributed by NM, 6-May-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑃 β‰  𝑄) ∧ (((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg16z 39008 Eliminate ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) condition from cdlemg16 39006. TODO: would it help to also eliminate 𝑃 β‰  𝑄 here or later? (Contributed by NM, 25-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg16zz 39009 Eliminate 𝑃 β‰  𝑄 from cdlemg16z 39008. TODO: Use this only if needed. (Contributed by NM, 26-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg17a 39010 TODO: FIX COMMENT. (Contributed by NM, 8-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐺 ∈ 𝑇 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄))) β†’ (πΊβ€˜π‘ƒ) ≀ (𝑃 ∨ 𝑄))
 
Theoremcdlemg17b 39011* 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 38639. (Contributed by NM, 8-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜π‘ƒ) = 𝑄)
 
Theoremcdlemg17dN 39012* TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑃 β‰  𝑄) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ∧ (πΊβ€˜π‘ƒ) β‰  𝑃)) β†’ (π‘…β€˜πΊ) = ((𝑃 ∨ 𝑄) ∧ π‘Š))
 
Theoremcdlemg17dALTN 39013 Same as cdlemg17dN 39012 with fewer antecedents but longer proof TODO: fix comment. (Contributed by NM, 9-May-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ (πΊβ€˜π‘ƒ) β‰  𝑃)) β†’ (π‘…β€˜πΊ) = ((𝑃 ∨ 𝑄) ∧ π‘Š))
 
Theoremcdlemg17e 39014* TODO: fix comment. (Contributed by NM, 8-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΊ)))
 
Theoremcdlemg17f 39015* TODO: fix comment. (Contributed by NM, 8-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)) = ((πΉβ€˜π‘ƒ) ∨ (πΊβ€˜(πΉβ€˜π‘ƒ))))
 
Theoremcdlemg17g 39016* TODO: fix comment. (Contributed by NM, 9-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜(πΉβ€˜π‘ƒ)) ≀ ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)))
 
Theoremcdlemg17h 39017* TODO: fix comment. (Contributed by NM, 10-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ 𝑆 ≀ ((πΉβ€˜π‘ƒ) ∨ (πΉβ€˜π‘„)))) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑆 = (πΉβ€˜π‘ƒ) ∨ 𝑆 = (πΉβ€˜π‘„)))
 
Theoremcdlemg17i 39018* TODO: fix comment. (Contributed by NM, 10-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜(πΉβ€˜π‘ƒ)) = (πΉβ€˜π‘„))
 
Theoremcdlemg17ir 39019* TODO: fix comment. (Contributed by NM, 13-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = (πΉβ€˜π‘„))
 
Theoremcdlemg17j 39020* TODO: fix comment. (Contributed by NM, 11-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜(πΉβ€˜π‘ƒ)) = (πΉβ€˜(πΊβ€˜π‘ƒ)))
 
Theoremcdlemg17pq 39021* Utility theorem for swapping 𝑃 and 𝑄. TODO: fix comment. (Contributed by NM, 11-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 β‰  𝑃) ∧ ((πΊβ€˜π‘„) β‰  𝑄 ∧ (π‘…β€˜πΊ) ≀ (𝑄 ∨ 𝑃) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑄 ∨ π‘Ÿ) = (𝑃 ∨ π‘Ÿ)))))
 
Theoremcdlemg17bq 39022* cdlemg17b 39011 with 𝑃 and 𝑄 swapped. Antecedent 𝐹 ∈ (π‘‡β€˜π‘Š) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 39011 also? (Contributed by NM, 13-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜π‘„) = 𝑃)
 
Theoremcdlemg17iqN 39023* cdlemg17i 39018 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑃 β‰  𝑄) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)) ∧ (πΊβ€˜π‘ƒ) β‰  𝑃)) β†’ (πΊβ€˜(πΉβ€˜π‘„)) = (πΉβ€˜π‘ƒ))
 
Theoremcdlemg17irq 39024* cdlemg17ir 39019 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΉβ€˜(πΊβ€˜π‘„)) = (πΉβ€˜π‘ƒ))
 
Theoremcdlemg17jq 39025* cdlemg17j 39020 with 𝑃 and 𝑄 swapped. (Contributed by NM, 13-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜(πΉβ€˜π‘„)) = (πΉβ€˜(πΊβ€˜π‘„)))
 
Theoremcdlemg17 39026* 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.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (πΊβ€˜((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))))) = ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„)))))
 
Theoremcdlemg18a 39027 Show two lines are different. TODO: fix comment. (Contributed by NM, 14-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ ((πΉβ€˜π‘„) ∨ (πΉβ€˜π‘ƒ)) β‰  (𝑃 ∨ 𝑄))) β†’ (𝑃 ∨ (πΉβ€˜π‘„)) β‰  (𝑄 ∨ (πΉβ€˜π‘ƒ)))
 
Theoremcdlemg18b 39028 Lemma for cdlemg18c 39029. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ (πΉβ€˜π‘ƒ) β‰  𝑄 ∧ ((πΉβ€˜π‘„) ∨ (πΉβ€˜π‘ƒ)) β‰  (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑃 ≀ (π‘ˆ ∨ (πΉβ€˜π‘„)))
 
Theoremcdlemg18c 39029 Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ (πΉβ€˜π‘ƒ) β‰  𝑄 ∧ ((πΉβ€˜π‘„) ∨ (πΉβ€˜π‘ƒ)) β‰  (𝑃 ∨ 𝑄))) β†’ ((𝑃 ∨ (πΉβ€˜π‘„)) ∧ (𝑄 ∨ (πΉβ€˜π‘ƒ))) ∈ 𝐴)
 
Theoremcdlemg18d 39030* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„)))) ∈ 𝐴)
 
Theoremcdlemg18 39031* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„)))) ≀ π‘Š)
 
Theoremcdlemg19a 39032* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ (𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„)))) = ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š))
 
Theoremcdlemg19 39033* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg20 39034* Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 23-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg21 39035* Version of cdlemg19 with (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) instead of (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑄) as a condition. (Contributed by NM, 23-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄 ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) ∧ ((π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg22 39036* cdlemg21 39035 with (πΉβ€˜π‘ƒ) β‰  𝑃 condition removed. (Contributed by NM, 23-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ ((π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄) ∧ ((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg24 39037* Combine cdlemg16z 39008 and cdlemg22 39036. TODO: Fix comment. (Contributed by NM, 24-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (((πΉβ€˜(πΊβ€˜π‘ƒ)) ∨ (πΉβ€˜(πΊβ€˜π‘„))) β‰  (𝑃 ∨ 𝑄) ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg37 39038* Use cdlemg8 38980 to eliminate the β‰  (𝑃 ∨ 𝑄) condition of cdlemg24 39037. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄 ∧ Β¬ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg25zz 39039 cdlemg16zz 39009 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑧) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑃 ∨ 𝑧))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑧 ∨ (πΉβ€˜(πΊβ€˜π‘§))) ∧ π‘Š))
 
Theoremcdlemg26zz 39040 cdlemg16zz 39009 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑄 ∨ 𝑧) ∧ Β¬ (π‘…β€˜πΊ) ≀ (𝑄 ∨ 𝑧))) β†’ ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š) = ((𝑧 ∨ (πΉβ€˜(πΊβ€˜π‘§))) ∧ π‘Š))
 
Theoremcdlemg27a 39041 For use with case when (𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)) or (𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)) is zero, letting us establish Β¬ 𝑧 ≀ π‘Š ∧ 𝑧 ≀ (𝑃 ∨ 𝑣) via 4atex 38425. TODO: Fix comment. (Contributed by NM, 28-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š)) ∧ (𝑧 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑧 ≀ (𝑃 ∨ 𝑣) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑧))
 
Theoremcdlemg28a 39042 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.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š)) ∧ ((𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ)) ∧ 𝑧 ≀ (𝑃 ∨ 𝑣) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑧 ∨ (πΉβ€˜(πΊβ€˜π‘§))) ∧ π‘Š))
 
Theoremcdlemg31b0N 39043 TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.β€˜πΎ)))
 
Theoremcdlemg31b0a 39044 TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 β‰  (π‘…β€˜πΉ))) β†’ (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.β€˜πΎ)))
 
Theoremcdlemg27b 39045 TODO: Fix comment. (Contributed by NM, 28-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝑧 β‰  𝑁)) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑧 ≀ (𝑃 ∨ 𝑣) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ Β¬ (π‘…β€˜πΉ) ≀ (𝑄 ∨ 𝑧))
 
Theoremcdlemg31a 39046 TODO: fix comment. (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇)) β†’ 𝑁 ≀ (𝑃 ∨ 𝑣))
 
Theoremcdlemg31b 39047 TODO: fix comment. (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇)) β†’ 𝑁 ≀ (𝑄 ∨ (π‘…β€˜πΉ)))
 
Theoremcdlemg31c 39048 Show that when 𝑁 is an atom, it is not under π‘Š. TODO: Is there a shorter direct proof? TODO: should we eliminate (πΉβ€˜π‘ƒ) β‰  𝑃 here? (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃 ∧ 𝑁 ∈ 𝐴)) β†’ Β¬ 𝑁 ≀ π‘Š)
 
Theoremcdlemg31d 39049 Eliminate (πΉβ€˜π‘ƒ) β‰  𝑃 from cdlemg31c 39048. TODO: Prove directly. TODO: do we need to eliminate (πΉβ€˜π‘ƒ) β‰  𝑃? It might be better to do this all at once at the end. See also cdlemg29 39054 versus cdlemg28 39053. (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑁 ∈ 𝐴)) β†’ Β¬ 𝑁 ≀ π‘Š)
 
Theoremcdlemg33b0 39050* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ 𝑁 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33c0 39051* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ 𝑧 ≀ (𝑃 ∨ 𝑣)))
 
Theoremcdlemg28b 39052* Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that Β¬ 𝑧 ≀ π‘Š is redundant here (but simplifies cdlemg28 39053.) (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣)) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ)) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃))) β†’ ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š) = ((𝑧 ∨ (πΉβ€˜(πΊβ€˜π‘§))) ∧ π‘Š))
 
Theoremcdlemg28 39053* 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 39054 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣)) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ)) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg29 39054* Eliminate (πΉβ€˜π‘ƒ) β‰  𝑃 and (πΊβ€˜π‘ƒ) β‰  𝑃 from cdlemg28 39053. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑧 ∈ 𝐴 ∧ Β¬ 𝑧 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂) ∧ 𝑧 ≀ (𝑃 ∨ 𝑣) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg33a 39055* TODO: Fix comment. (Contributed by NM, 29-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 β‰  𝑄 ∧ 𝑁 β‰  𝑂) ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33b 39056* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33c 39057* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.β€˜πΎ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33d 39058* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑁 = (0.β€˜πΎ) ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΊ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33e 39059* TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝑁 = (0.β€˜πΎ) ∧ 𝑂 = (0.β€˜πΎ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 β‰  𝑄 ∧ 𝑣 β‰  (π‘…β€˜πΉ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg33 39060* Combine cdlemg33b 39056, cdlemg33c 39057, cdlemg33d 39058, cdlemg33e 39059. TODO: Fix comment. (Contributed by NM, 30-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ βˆƒπ‘§ ∈ 𝐴 (Β¬ 𝑧 ≀ π‘Š ∧ (𝑧 β‰  𝑁 ∧ 𝑧 β‰  𝑂 ∧ 𝑧 ≀ (𝑃 ∨ 𝑣))))
 
Theoremcdlemg34 39061* Use cdlemg33 to eliminate 𝑧 from cdlemg29 39054. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΉ)))    &   π‘‚ = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (π‘…β€˜πΊ)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 β‰  𝑄) ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg35 39062* TODO: Fix comment. TODO: should we have a more general version of hlsupr 37735 to avoid the β‰  conditions? (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ βˆƒπ‘£ ∈ 𝐴 (𝑣 ≀ π‘Š ∧ (𝑣 β‰  (π‘…β€˜πΉ) ∧ 𝑣 β‰  (π‘…β€˜πΊ))))
 
Theoremcdlemg36 39063* Use cdlemg35 to eliminate 𝑣 from cdlemg34 39061. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ) ∧ βˆƒπ‘Ÿ ∈ 𝐴 (Β¬ π‘Ÿ ≀ π‘Š ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg38 39064 Use cdlemg37 39038 to eliminate βˆƒπ‘Ÿ ∈ 𝐴 from cdlemg36 39063. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄) ∧ (((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg39 39065 Eliminate β‰  conditions from cdlemg38 39064. TODO: Would this better be done at cdlemg35 39062? TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 β‰  𝑄)) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg40 39066 Eliminate 𝑃 β‰  𝑄 conditions from cdlemg39 39065. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ ((𝑃 ∨ (πΉβ€˜(πΊβ€˜π‘ƒ))) ∧ π‘Š) = ((𝑄 ∨ (πΉβ€˜(πΊβ€˜π‘„))) ∧ π‘Š))
 
Theoremcdlemg41 39067 Convert cdlemg40 39066 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) β†’ ((𝑃 ∨ ((𝐹 ∘ 𝐺)β€˜π‘ƒ)) ∧ π‘Š) = ((𝑄 ∨ ((𝐹 ∘ 𝐺)β€˜π‘„)) ∧ π‘Š))
 
Theoremltrnco 39068 The composition of two translations is a translation. Part of proof of Lemma G of [Crawley] p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (𝐹 ∘ 𝐺) ∈ 𝑇)
 
Theoremtrlcocnv 39069 Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘…β€˜(𝐹 ∘ ◑𝐺)) = (π‘…β€˜(𝐺 ∘ ◑𝐹)))
 
Theoremtrlcoabs 39070 Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (((𝐹 ∘ 𝐺)β€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) = ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)))
 
Theoremtrlcoabs2N 39071 Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))) = ((πΉβ€˜π‘ƒ) ∨ (πΊβ€˜π‘ƒ)))
 
Theoremtrlcoat 39072 The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)
𝐴 = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ)) β†’ (π‘…β€˜(𝐹 ∘ 𝐺)) ∈ 𝐴)
 
Theoremtrlcocnvat 39073 Commonly used special case of trlcoat 39072. (Contributed by NM, 1-Jul-2013.)
𝐴 = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ)) β†’ (π‘…β€˜(𝐹 ∘ ◑𝐺)) ∈ 𝐴)
 
Theoremtrlconid 39074 The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ)) β†’ (𝐹 ∘ 𝐺) β‰  ( I β†Ύ 𝐡))
 
Theoremtrlcolem 39075 Lemma for trlco 39076. (Contributed by NM, 1-Jun-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜(𝐹 ∘ 𝐺)) ≀ ((π‘…β€˜πΉ) ∨ (π‘…β€˜πΊ)))
 
Theoremtrlco 39076 The trace of a composition of translations is less than or equal to the join of their traces. Part of proof of Lemma G of [Crawley] p. 116, second paragraph on p. 117. (Contributed by NM, 2-Jun-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (π‘…β€˜(𝐹 ∘ 𝐺)) ≀ ((π‘…β€˜πΉ) ∨ (π‘…β€˜πΊ)))
 
Theoremtrlcone 39077 If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ) ∧ 𝐺 β‰  ( I β†Ύ 𝐡))) β†’ (π‘…β€˜πΉ) β‰  (π‘…β€˜(𝐹 ∘ 𝐺)))
 
Theoremcdlemg42 39078 Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ Β¬ (πΊβ€˜π‘ƒ) ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ)))
 
Theoremcdlemg43 39079 Part of proof of Lemma G of [Crawley] p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = (((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΊ))))
 
Theoremcdlemg44a 39080 Part of proof of Lemma G of [Crawley] p. 116, fourth line of third paragraph on p. 117: "so fg(p) = gf(p)." (Contributed by NM, 3-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΊβ€˜π‘ƒ) β‰  𝑃 ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = (πΊβ€˜(πΉβ€˜π‘ƒ)))
 
Theoremcdlemg44b 39081 Eliminate (πΉβ€˜π‘ƒ) β‰  𝑃, (πΊβ€˜π‘ƒ) β‰  𝑃 from cdlemg44a 39080. (Contributed by NM, 3-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ)) β†’ (πΉβ€˜(πΊβ€˜π‘ƒ)) = (πΊβ€˜(πΉβ€˜π‘ƒ)))
 
Theoremcdlemg44 39082 Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (Contributed by NM, 3-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ)) β†’ (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
 
Theoremcdlemg47a 39083 TODO: fix comment. TODO: Use this above in place of (πΉβ€˜π‘ƒ) = 𝑃 antecedents? (Contributed by NM, 5-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I β†Ύ 𝐡)) β†’ (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
 
Theoremcdlemg46 39084* Part of proof of Lemma G of [Crawley] p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ β„Ž ∈ 𝑇) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜β„Ž) β‰  (π‘…β€˜πΉ))) β†’ (π‘…β€˜(β„Ž ∘ 𝐹)) β‰  (π‘…β€˜πΉ))
 
Theoremcdlemg47 39085* Part of proof of Lemma G of [Crawley] p. 116, ninth line of third paragraph on p. 117: "we conclude that gf = fg." (Contributed by NM, 5-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (β„Ž ∈ 𝑇 ∧ (π‘…β€˜πΉ) = (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜β„Ž) β‰  (π‘…β€˜πΉ))) β†’ (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
 
Theoremcdlemg48 39086 Eliminate β„Ž from cdlemg47 39085. (Contributed by NM, 5-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) = (π‘…β€˜πΊ))) β†’ (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
 
Theoremltrncom 39087 Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
 
Theoremltrnco4 39088 Rearrange a composition of 4 translations, analogous to an4 655. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐸 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) β†’ ((𝐷 ∘ 𝐸) ∘ (𝐹 ∘ 𝐺)) = ((𝐷 ∘ 𝐹) ∘ (𝐸 ∘ 𝐺)))
 
Theoremtrljco 39089 Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)
∨ = (joinβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ ((π‘…β€˜πΉ) ∨ (π‘…β€˜(𝐹 ∘ 𝐺))) = ((π‘…β€˜πΉ) ∨ (π‘…β€˜πΊ)))
 
Theoremtrljco2 39090 Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
∨ = (joinβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) β†’ ((π‘…β€˜πΉ) ∨ (π‘…β€˜(𝐹 ∘ 𝐺))) = ((π‘…β€˜πΊ) ∨ (π‘…β€˜(𝐹 ∘ 𝐺))))
 
Syntaxctgrp 39091 Extend class notation with translation group.
class TGrp
 
Definitiondf-tgrp 39092* Define the class of all translation groups. π‘˜ is normally a member of HL. Each base set is the set of all lattice translations with respect to a hyperplane 𝑀, and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)
TGrp = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜π‘˜)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
 
Theoremtgrpfset 39093* The translation group maps for a lattice 𝐾. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (TGrpβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((LTrnβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€), 𝑔 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∘ 𝑔))⟩}))
 
Theoremtgrpset 39094* The translation group for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐺 = {⟨(Baseβ€˜ndx), π‘‡βŸ©, ⟨(+gβ€˜ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
 
Theoremtgrpbase 39095 The base set of the translation group is the set of all translations (for a fiducial co-atom π‘Š). (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    &   πΆ = (Baseβ€˜πΊ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝑇)
 
Theoremtgrpopr 39096* The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
 
Theoremtgrpov 39097 The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻 ∧ (𝑋 ∈ 𝑇 ∧ π‘Œ ∈ 𝑇)) β†’ (𝑋 + π‘Œ) = (𝑋 ∘ π‘Œ))
 
Theoremtgrpgrplem 39098 Lemma for tgrpgrp 39099. (Contributed by NM, 6-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜πΊ)    &   π΅ = (Baseβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐺 ∈ Grp)
 
Theoremtgrpgrp 39099 The translation group is a group. (Contributed by NM, 6-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐺 ∈ Grp)
 
Theoremtgrpabl 39100 The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   πΊ = ((TGrpβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐺 ∈ Abel)
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