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Theorem List for Metamath Proof Explorer - 37601-37700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremltrniotaidvalN 37601* Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑃)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝐹 = ( I ↾ 𝐵))
 
TheoremltrniotavalbN 37602* Value of the unique translation specified by a value. (Contributed by NM, 10-Mar-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) = 𝑄𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)))
 
Theoremcdlemeiota 37603* A translation is uniquely determined by one of its values. (Contributed by NM, 18-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐹𝑇) → 𝐹 = (𝑓𝑇 (𝑓𝑃) = (𝐹𝑃)))
 
Theoremcdlemg1ci2 37604* Any function of the form of the function constructed for cdleme 37578 is a translation. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)) → 𝐹𝑇)
 
Theoremcdlemg1cN 37605* Any translation belongs to the set of functions constructed for cdleme 37578. TODO: Fix comment. (Contributed by NM, 18-Apr-2013.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑇𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)))
 
Theoremcdlemg1cex 37606* Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 37581? (Contributed by NM, 17-Apr-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = (𝑓𝑇 (𝑓𝑝) = 𝑞))))
 
Theoremcdlemg2cN 37607* Any translation belongs to the set of functions constructed for cdleme 37578. TODO: Fix comment. (Contributed by NM, 22-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)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑄) → (𝐹𝑇𝐹 = 𝐺))
 
Theoremcdlemg2dN 37608* This theorem can be used to shorten 𝐺 = hypothesis. TODO: Fix comment. (Contributed by NM, 21-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)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑄 ((𝑃 𝑡) 𝑊)))    &   𝐸 = ((𝑃 𝑄) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑃𝑄 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑃 𝑄), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑃 𝑄)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑄)) → 𝐹 = 𝐺)
 
Theoremcdlemg2cex 37609* Any translation is one of our 𝐹 s. TODO: fix comment, move to its own block maybe? Would this help for cdlemf 37581? (Contributed by NM, 22-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑝 𝑞) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))    &   𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑝𝑞 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐹𝑇 ↔ ∃𝑝𝐴𝑞𝐴𝑝 𝑊 ∧ ¬ 𝑞 𝑊𝐹 = 𝐺)))
 
Theoremcdlemg2ce 37610* Utility theorem to eliminate p,q when converting theorems with explicit f. TODO: fix comment. (Contributed by NM, 22-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑝 𝑞) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))    &   𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑝𝑞 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   (𝐹 = 𝐺 → (𝜓𝜒))    &   ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑝𝐴 ∧ ¬ 𝑝 𝑊) ∧ (𝑞𝐴 ∧ ¬ 𝑞 𝑊)) ∧ 𝜑) → 𝜒)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝜑) → 𝜓)
 
Theoremcdlemg2jlemOLDN 37611* Part of proof of Lemma E in [Crawley] p. 113. TODO: FIX COMMENT. f preserves join: f(r s) = f(r) s, p. 115 10th line from bottom. TODO: Combine with cdlemg2jOLDN 37616? (Contributed by NM, 22-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)    &   𝑈 = ((𝑝 𝑞) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))    &   𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑝𝑞 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑃 𝑄)) = ((𝐹𝑃) (𝐹𝑄)))
 
Theoremcdlemg2fvlem 37612* Lemma for cdlemg2fv 37617. (Contributed by NM, 23-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑝 𝑞) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))    &   𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑝𝑞 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑃) (𝑋 𝑊)))
 
Theoremcdlemg2klem 37613* cdleme42keg 37504 with simpler hypotheses. TODO: FIX COMMENT. (Contributed by NM, 22-Apr-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((𝑝 𝑞) 𝑊)    &   𝐷 = ((𝑡 𝑈) (𝑞 ((𝑝 𝑡) 𝑊)))    &   𝐸 = ((𝑝 𝑞) (𝐷 ((𝑠 𝑡) 𝑊)))    &   𝐺 = (𝑥𝐵 ↦ if((𝑝𝑞 ∧ ¬ 𝑥 𝑊), (𝑧𝐵𝑠𝐴 ((¬ 𝑠 𝑊 ∧ (𝑠 (𝑥 𝑊)) = 𝑥) → 𝑧 = (if(𝑠 (𝑝 𝑞), (𝑦𝐵𝑡𝐴 ((¬ 𝑡 𝑊 ∧ ¬ 𝑡 (𝑝 𝑞)) → 𝑦 = 𝐸)), 𝑠 / 𝑡𝐷) (𝑥 𝑊)))), 𝑥))    &   𝑉 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑉))
 
Theoremcdlemg2idN 37614 Version of cdleme31id 37412 with simpler hypotheses. TODO: Fix comment. (Contributed by NM, 21-Apr-2013.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)       ((((𝐾 ∈ HL ∧ 𝑊𝐻𝐹𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = 𝑄𝑋𝐵) ∧ 𝑃 = 𝑄) → (𝐹𝑋) = 𝑋)
 
Theoremcdlemg3a 37615 Part of proof of Lemma G in [Crawley] p. 116, line 19. Show p q = p u. TODO: reformat cdleme0cp 37232 to match this, then replace with cdleme0cp 37232. (Contributed by NM, 19-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) → (𝑃 𝑄) = (𝑃 𝑈))
 
Theoremcdlemg2jOLDN 37616 TODO: Replace this with ltrnj 37150. (Contributed by NM, 22-Apr-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑃 𝑄)) = ((𝐹𝑃) (𝐹𝑄)))
 
Theoremcdlemg2fv 37617 Value of a translation in terms of an associated atom. cdleme48fvg 37518 with simpler hypotheses. TODO: Use ltrnj 37150 to vastly simplify. (Contributed by NM, 23-Apr-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) ∧ (𝐹𝑇 ∧ (𝑃 (𝑋 𝑊)) = 𝑋)) → (𝐹𝑋) = ((𝐹𝑃) (𝑋 𝑊)))
 
Theoremcdlemg2fv2 37618 Value of a translation in terms of an associated atom. TODO: FIX COMMENT. TODO: Is this useful elsewhere e.g. around cdlemeg46fjv 37541 that use more complex proofs? TODO: Use ltrnj 37150 to vastly simplify. (Contributed by NM, 23-Apr-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ 𝐹𝑇) → (𝐹‘(𝑅 𝑈)) = ((𝐹𝑅) 𝑈))
 
Theoremcdlemg2k 37619 cdleme42keg 37504 with simpler hypotheses. TODO: FIX COMMENT. TODO: derive from cdlemg3a 37615, cdlemg2fv2 37618, cdlemg2jOLDN 37616, ltrnel 37157? (Contributed by NM, 22-Apr-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑃) 𝑈))
 
Theoremcdlemg2kq 37620 cdlemg2k 37619 with 𝑃 and 𝑄 swapped. TODO: FIX COMMENT. (Contributed by NM, 15-May-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → ((𝐹𝑃) (𝐹𝑄)) = ((𝐹𝑄) 𝑈))
 
Theoremcdlemg2l 37621 TODO: FIX COMMENT. (Contributed by NM, 23-Apr-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑇𝐺𝑇)) → ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) = ((𝐹‘(𝐺𝑃)) 𝑈))
 
Theoremcdlemg2m 37622 TODO: FIX COMMENT. (Contributed by NM, 25-Apr-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &    = (meet‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝐹𝑇) → (((𝐹𝑃) (𝐹𝑄)) 𝑊) = 𝑈)
 
Theoremcdlemg5 37623* TODO: Is there a simpler more direct proof, that could be placed earlier e.g. near lhpexle 37023? 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 37624* 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 37023? Then replace cdlemb2 37059 with it. This is a more general version of cdlemb2 37059 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 37625 Properties of a translation of an element not under 𝑊. TODO: Fix comment. Can this be simplified? Perhaps derived from cdleme48bw 37520? 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 37626 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 37627 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 37628 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 37629 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 37630 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 37631 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 37632 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 37633 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 37634 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 37635 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 37636* TODO: FIX COMMENT. TODO: replace with cdlemg4 37635. (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 37637* TODO: FIX COMMENT. TODO: replace with cdlemg4 37635. (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 37638* 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 37639* 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 37640 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 37641 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 37642 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 37643 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 37644 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 37645 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 37646 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 37647 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 37648 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 37649 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 37650 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 37651 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 37652 The triples 𝑃, (𝐹‘(𝐺𝑃)), (𝐹𝑃)⟩ and 𝑄, (𝐹‘(𝐺𝑄)), (𝐹𝑄)⟩ are axially perspective by dalaw 36904. 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 37653 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 37654 TODO: FIX COMMENT. TODO: Can this be moved up as a stand-alone theorem in ltrn* area? TODO: Compare this proof to cdlemg2m 37622 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 37655 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 37656 TODO: FIX COMMENT. TODO: can proof using this be restructured to use cdlemg11a 37655? (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 37657 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 37658 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 37659 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 37660 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 37661 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 37662 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 37663 The triples 𝑃, (𝐹𝑃), (𝐹‘(𝐺𝑃))⟩ and 𝑄, (𝐹𝑄), (𝐹‘(𝐺𝑄))⟩ are axially perspective by dalaw 36904. 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 37664 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 37665 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 37666 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 37667 TODO: FIX COMMENT. TODO: Combine with cdlemg12f 37666. (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 37668 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 37669 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 37670 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 37671 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 37672 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 37673 Eliminate the (𝐹𝑃) ≠ 𝑃 condition from cdlemg13 37670. 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 37674 Eliminate the ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) condition from cdlemg13 37670. 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 37675 Part of proof of Lemma G of [Crawley] p. 116; 2nd line p. 117, which says that (our) cdlemg10 37659 "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 36904, in order to make this inference. This final step eliminates the (𝑅𝐹) ≠ (𝑅𝐺) condition from cdlemg12 37668. 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 37676 This version of cdlemg16 37675 uses cdlemg15a 37673 instead of cdlemg15 37674, in case cdlemg15 37674 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 37677 Eliminate ((𝐹‘(𝐺𝑃)) (𝐹‘(𝐺𝑄))) ≠ (𝑃 𝑄) condition from cdlemg16 37675. 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 37678 Eliminate 𝑃𝑄 from cdlemg16z 37677. 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 37679 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 37680* 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 37308. (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 37681* 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 37682 Same as cdlemg17dN 37681 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 37683* 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 37684* 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 37685* 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 37686* 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 37687* 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 37688* 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 37689* 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 37690* 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 37691* cdlemg17b 37680 with 𝑃 and 𝑄 swapped. Antecedent 𝐹 ∈ (𝑇𝑊) is redundant for easier use. TODO: should we have redundant antecedent for cdlemg17b 37680 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 37692* cdlemg17i 37687 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 37693* cdlemg17ir 37688 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 37694* cdlemg17j 37689 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 37695* 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 37696 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 37697 Lemma for cdlemg18c 37698. 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 37698 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 37699* 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 37700* 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)
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
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