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Theorem List for Metamath Proof Explorer - 35101-35200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcdlemk26b-3 35101* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵) ∧ (𝑅𝐹) = (𝑅𝑁))) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ∃𝑥𝑇 ((𝑥 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑥) ≠ (𝑅𝐹) ∧ (𝑅𝑥) ≠ (𝑅𝐺)) ∧ (𝑥𝑌𝐺) ∈ 𝑇))
 
Theoremcdlemk26-3 35102* Part of proof of Lemma K of [Crawley] p. 118. Eliminate the 𝑥 requirements from cdlemk25-3 35100. (Contributed by NM, 10-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐷𝑇𝑁𝑇) ∧ (𝐺𝑇𝐶𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐶 ≠ ( I ↾ 𝐵))) ∧ (((𝑅𝐺) ≠ (𝑅𝐶) ∧ (𝑅𝐶) ≠ (𝑅𝐹) ∧ (𝑅𝐷) ≠ (𝑅𝐹)) ∧ (𝑅𝐺) ≠ (𝑅𝐷))) → ((𝐷𝑌𝐺)‘𝑃) = ((𝐶𝑌𝐺)‘𝑃))
 
Theoremcdlemk27-3 35103* Part of proof of Lemma K of [Crawley] p. 118. Eliminate the 𝑃 from the conclusion of cdlemk25-3 35100. (Contributed by NM, 10-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐷𝑇𝑁𝑇) ∧ (𝐺𝑇𝐶𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐷 ≠ ( I ↾ 𝐵)) ∧ (𝐺 ≠ ( I ↾ 𝐵) ∧ 𝐶 ≠ ( I ↾ 𝐵))) ∧ (((𝑅𝐺) ≠ (𝑅𝐶) ∧ (𝑅𝐶) ≠ (𝑅𝐹) ∧ (𝑅𝐷) ≠ (𝑅𝐹)) ∧ (𝑅𝐺) ≠ (𝑅𝐷))) → (𝐷𝑌𝐺) = (𝐶𝑌𝐺))
 
Theoremcdlemk28-3 35104* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 14-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → ∃𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → 𝑧 = (𝑏𝑌𝐺)))
 
Theoremcdlemk33N 35105* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. TODO: not needed, is embodied in cdlemk34 35106. (Contributed by NM, 18-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → 𝑧 = (𝑏𝑌𝐺)))       (((𝐾 ∈ HL ∧ 𝑊𝐻 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊))) → 𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = ((𝑏𝑌𝐺)‘𝑃))))
 
Theoremcdlemk34 35106* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. Part of attempt to simplify hypotheses. (Contributed by NM, 18-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → 𝑧 = (𝑏𝑌𝐺)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → 𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = ((𝑃 (𝑅𝐺)) (((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹)))) (𝑅‘(𝐺𝑏)))))))
 
Theoremcdlemk29-3 35107* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 14-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑌 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → 𝑧 = (𝑏𝑌𝐺)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → 𝑋𝑇)
 
Theoremcdlemk35 35108* Part of proof of Lemma K of [Crawley] p. 118. cdlemk29-3 35107 with shorter hypotheses. (Contributed by NM, 18-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → 𝑋𝑇)
 
Theoremcdlemk36 35109* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → (𝑋𝑃) = 𝑌)
 
Theoremcdlemk37 35110* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 18-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → (𝑋𝑃) (𝑃 (𝑅𝐺)))
 
Theoremcdlemk38 35111* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. TODO: derive more directly with r19.23 2908? (Contributed by NM, 19-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → (𝑋𝑃) (𝑃 (𝑅𝐺)))
 
Theoremcdlemk39 35112* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of tau, represented by 𝑋. (Contributed by NM, 19-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → (𝑅𝑋) (𝑅𝐺))
 
Theoremcdlemk40 35113* TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
𝑋 = (𝑧𝑇 𝜑)    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       (𝐺𝑇 → (𝑈𝐺) = if(𝐹 = 𝑁, 𝐺, 𝐺 / 𝑔𝑋))
 
Theoremcdlemk40t 35114* TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
𝑋 = (𝑧𝑇 𝜑)    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((𝐹 = 𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺)
 
Theoremcdlemk40f 35115* TODO: fix comment. (Contributed by NM, 31-Jul-2013.)
𝑋 = (𝑧𝑇 𝜑)    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((𝐹𝑁𝐺𝑇) → (𝑈𝐺) = 𝐺 / 𝑔𝑋)
 
Theoremcdlemk41 35116* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 19-Jul-2013.)
𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))       (𝐺𝑇𝐺 / 𝑔𝑌 = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
 
Theoremcdlemkfid1N 35117 Lemma for cdlemkfid3N 35121. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝐺𝑇) ∧ ((𝑅𝐺) ≠ (𝑅𝐹) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊))) → ((𝑃 (𝑅𝐺)) ((𝐹𝑃) (𝑅‘(𝐺𝐹)))) = (𝐺𝑃))
 
Theoremcdlemkid1 35118 Lemma for cdlemkid 35132. (Contributed by NM, 24-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑏𝑇𝑏 ≠ ( I ↾ 𝐵)))) → (𝑍 (𝑅𝑏)) = (𝑃 (𝑅𝑏)))
 
Theoremcdlemkfid2N 35119 Lemma for cdlemkfid3N 35121. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 = 𝑁) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑏𝑇) ∧ ((𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊))) → 𝑍 = (𝑏𝑃))
 
Theoremcdlemkid2 35120* Lemma for cdlemkid 35132. (Contributed by NM, 24-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺 = ( I ↾ 𝐵) ∧ (𝑏𝑇𝑏 ≠ ( I ↾ 𝐵)))) → 𝐺 / 𝑔𝑌 = 𝑃)
 
Theoremcdlemkfid3N 35121* TODO: is this useful or should it be deleted? (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹 = 𝑁) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝐺𝑇 ∧ (𝑏𝑇𝑏 ≠ ( I ↾ 𝐵))) ∧ ((𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊))) → 𝐺 / 𝑔𝑌 = (𝐺𝑃))
 
Theoremcdlemky 35122* Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up (𝑏𝑌𝐺) stuff. 𝑉 represents 𝑌 in cdlemk31 35092. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑉 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → 𝐺 / 𝑔𝑌 = ((𝑏𝑉𝐺)‘𝑃))
 
Theoremcdlemkyu 35123* Convert between function and explicit forms. 𝐶 represents 𝑍 in cdlemkuu 35091. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑉 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))    &   𝑄 = (𝑆𝑏)    &   𝐶 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) ((𝑄𝑃) (𝑅‘(𝑒𝑏))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → 𝐺 / 𝑔𝑌 = ((𝐶𝐺)‘𝑃))
 
Theoremcdlemkyuu 35124* cdlemkyu 35123 with some hypotheses eliminated. TODO: Clean all this up. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝐶 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑏)‘𝑃) (𝑅‘(𝑒𝑏))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → 𝐺 / 𝑔𝑌 = ((𝐶𝐺)‘𝑃))
 
Theoremcdlemk11ta 35125* Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. 𝐺, 𝐼 stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝐶 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑏)‘𝑃) (𝑅‘(𝑒𝑏))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐼)))) → 𝐺 / 𝑔𝑌 (𝐼 / 𝑔𝑌 (𝑅‘(𝐼𝐺))))
 
Theoremcdlemk19ylem 35126* Lemma for cdlemk19y 35128. (Contributed by NM, 30-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝐶 = (𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑏)‘𝑃) (𝑅‘(𝑒𝑏))))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹)))) → 𝐹 / 𝑔𝑌 = (𝑁𝑃))
 
Theoremcdlemk11tb 35127* Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. 𝐺, 𝐼 stand for g, h. cdlemk11ta 35125 with hypotheses removed. TODO: Can this be proved directly with no quantification? (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐼)))) → 𝐺 / 𝑔𝑌 (𝐼 / 𝑔𝑌 (𝑅‘(𝐼𝐺))))
 
Theoremcdlemk19y 35128* cdlemk19 35065 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹)))) → 𝐹 / 𝑔𝑌 = (𝑁𝑃))
 
Theoremcdlemkid3N 35129* Lemma for cdlemkid 35132. (Contributed by NM, 25-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → 𝐺 / 𝑔𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → (𝑧𝑃) = 𝑃)))
 
Theoremcdlemkid4 35130* Lemma for cdlemkid 35132. (Contributed by NM, 25-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → 𝐺 / 𝑔𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) → 𝑧 = ( I ↾ 𝐵))))
 
Theoremcdlemkid5 35131* Lemma for cdlemkid 35132. (Contributed by NM, 25-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → 𝐺 / 𝑔𝑋𝑇)
 
Theoremcdlemkid 35132* The value of the tau function (in Lemma K of [Crawley] p. 118) on the identity relation. (Contributed by NM, 25-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝐺 = ( I ↾ 𝐵))) → 𝐺 / 𝑔𝑋 = ( I ↾ 𝐵))
 
Theoremcdlemk35s 35133* Substitution version of cdlemk35 35108. (Contributed by NM, 22-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → 𝐺 / 𝑔𝑋𝑇)
 
Theoremcdlemk35s-id 35134* Substitution version of cdlemk35 35108. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝐺𝑇𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → 𝐺 / 𝑔𝑋𝑇)
 
Theoremcdlemk39s 35135* Substitution version of cdlemk39 35112. TODO: Can any commonality with cdlemk35s 35133 be exploited? (Contributed by NM, 23-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ 𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → (𝑅𝐺 / 𝑔𝑋) (𝑅𝐺))
 
Theoremcdlemk39s-id 35136* Substitution version of cdlemk39 35112 with non-identity requirement on 𝐺 removed. TODO: Can any commonality with cdlemk35s 35133 be exploited? (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ 𝐺𝑇𝑁𝑇) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁))) → (𝑅𝐺 / 𝑔𝑋) (𝑅𝐺))
 
Theoremcdlemk42 35137* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → (𝐺 / 𝑔𝑋𝑃) = 𝐺 / 𝑔𝑌)
 
Theoremcdlemk19xlem 35138* Lemma for cdlemk19x 35139. (Contributed by NM, 30-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹)))) → (𝐹 / 𝑔𝑋𝑃) = (𝑁𝑃))
 
Theoremcdlemk19x 35139* cdlemk19 35065 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 30-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹 / 𝑔𝑋𝑃) = (𝑁𝑃))
 
Theoremcdlemk42yN 35140* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 20-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → (𝐺 / 𝑔𝑋𝑃) = ((𝑃 (𝑅𝐺)) (𝑍 (𝑅‘(𝐺𝑏)))))
 
Theoremcdlemk11tc 35141* Part of proof of Lemma K of [Crawley] p. 118. Lemma for Eq. 5, p. 119. 𝐺, 𝐼 stand for g, h. TODO: fix comment. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐼)))) → (𝐺 / 𝑔𝑋𝑃) ((𝐼 / 𝑔𝑋𝑃) (𝑅‘(𝐼𝐺))))
 
Theoremcdlemk11t 35142* Part of proof of Lemma K of [Crawley] p. 118. Eq. 5, line 36, p. 119. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 21-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵))) → (𝐺 / 𝑔𝑋𝑃) ((𝐼 / 𝑔𝑋𝑃) (𝑅‘(𝐼𝐺))))
 
Theoremcdlemk45 35143* Part of proof of Lemma K of [Crawley] p. 118. Line 37, p. 119. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. They do not explicitly mention the requirement (𝐺𝐼) ≠ ( I ∣ ‘𝐵). (Contributed by NM, 22-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝐺𝐼) ≠ ( I ↾ 𝐵))) → ((𝐺𝐼) / 𝑔𝑋𝑃) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺)))
 
Theoremcdlemk46 35144* Part of proof of Lemma K of [Crawley] p. 118. Line 38 (last line), p. 119. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 22-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝐺𝐼) ≠ ( I ↾ 𝐵))) → ((𝐺𝐼) / 𝑔𝑋𝑃) ((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼)))
 
Theoremcdlemk47 35145* Part of proof of Lemma K of [Crawley] p. 118. Line 2, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 22-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝐺) ≠ (𝑅𝐼))) → ((𝐺𝐼) / 𝑔𝑋𝑃) = (((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼)) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺))))
 
Theoremcdlemk48 35146* Part of proof of Lemma K of [Crawley] p. 118. Line 4, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 22-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵))) → ((𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋)‘𝑃) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺 / 𝑔𝑋)))
 
Theoremcdlemk49 35147* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 23-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵))) → ((𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋)‘𝑃) ((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼 / 𝑔𝑋)))
 
Theoremcdlemk50 35148* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. TODO: Combine into cdlemk52 35150? (Contributed by NM, 23-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵))) → ((𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋)‘𝑃) (((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼 / 𝑔𝑋)) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺 / 𝑔𝑋))))
 
Theoremcdlemk51 35149* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. TODO: Combine into cdlemk52 35150? (Contributed by NM, 23-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵))) → (((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼 / 𝑔𝑋)) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺 / 𝑔𝑋))) (((𝐺 / 𝑔𝑋𝑃) (𝑅𝐼)) ((𝐼 / 𝑔𝑋𝑃) (𝑅𝐺))))
 
Theoremcdlemk52 35150* Part of proof of Lemma K of [Crawley] p. 118. Line 6, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 23-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝐺) ≠ (𝑅𝐼))) → ((𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋)‘𝑃) = ((𝐺𝐼) / 𝑔𝑋𝑃))
 
Theoremcdlemk53a 35151* Lemma for cdlemk53 35153. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵))) ∧ (𝑁𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝐺) ≠ (𝑅𝐼))) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
Theoremcdlemk53b 35152* Lemma for cdlemk53 35153. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝐼𝑇𝐼 ≠ ( I ↾ 𝐵) ∧ (𝑅𝐺) ≠ (𝑅𝐼))) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
Theoremcdlemk53 35153* Part of proof of Lemma K of [Crawley] p. 118. Line 7, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝐼𝑇 ∧ (𝑅𝐺) ≠ (𝑅𝐼))) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
Theoremcdlemk54 35154* Part of proof of Lemma K of [Crawley] p. 118. Line 10, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ((𝐼𝑇 ∧ (𝑅𝐺) = (𝑅𝐼)) ∧ 𝑗𝑇 ∧ (𝑗 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑗) ≠ (𝑅𝐺) ∧ (𝑅𝑗) ≠ (𝑅‘(𝐺𝐼))))) → ((𝐺𝐼) / 𝑔𝑋𝑗 / 𝑔𝑋) = ((𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋) ∘ 𝑗 / 𝑔𝑋))
 
Theoremcdlemk55a 35155* Lemma for cdlemk55 35157. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ ((𝐼𝑇 ∧ (𝑅𝐺) = (𝑅𝐼)) ∧ 𝑗𝑇 ∧ (𝑗 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑗) ≠ (𝑅𝐺) ∧ (𝑅𝑗) ≠ (𝑅‘(𝐺𝐼))))) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
Theoremcdlemk55b 35156* Lemma for cdlemk55 35157. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝐼𝑇 ∧ (𝑅𝐺) = (𝑅𝐼))) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
Theoremcdlemk55 35157* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ 𝐺𝑇𝐼𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐺𝐼) / 𝑔𝑋 = (𝐺 / 𝑔𝑋𝐼 / 𝑔𝑋))
 
TheoremcdlemkyyN 35158* Part of proof of Lemma K of [Crawley] p. 118. TODO: clean up (𝑏𝑌𝐺) stuff. (Contributed by NM, 21-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑆 = (𝑓𝑇 ↦ (𝑖𝑇 (𝑖𝑃) = ((𝑃 (𝑅𝑓)) ((𝑁𝑃) (𝑅‘(𝑓𝐹))))))    &   𝑉 = (𝑑𝑇, 𝑒𝑇 ↦ (𝑗𝑇 (𝑗𝑃) = ((𝑃 (𝑅𝑒)) (((𝑆𝑑)‘𝑃) (𝑅‘(𝑒𝑑))))))       (((𝐾 ∈ HL ∧ 𝑊𝐻 ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝐹 ≠ ( I ↾ 𝐵) ∧ 𝑁𝑇) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → (𝐺 / 𝑔𝑋𝑃) = ((𝑏𝑉𝐺)‘𝑃))
 
Theoremcdlemk43N 35159* Part of proof of Lemma K of [Crawley] p. 118. TODO: fix comment. (Contributed by NM, 31-Jul-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ ((𝐹𝑇𝑁𝑇𝐹𝑁) ∧ (𝐺𝑇𝐺 ≠ ( I ↾ 𝐵)) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) ∧ (𝑏𝑇 ∧ (𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝐺)))) → ((𝑈𝐺)‘𝑃) = 𝐺 / 𝑔𝑌)
 
Theoremcdlemk35u 35160* Substitution version of cdlemk35 35108. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐹𝑇𝑁𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈𝐺) ∈ 𝑇)
 
Theoremcdlemk55u1 35161* Lemma for cdlemk55u 35162. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (((𝑅𝐹) = (𝑅𝑁) ∧ 𝐹𝑁) ∧ 𝐺𝑇𝐼𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈‘(𝐺𝐼)) = ((𝑈𝐺) ∘ (𝑈𝐼)))
 
Theoremcdlemk55u 35162* Part of proof of Lemma K of [Crawley] p. 118. Line 11, p. 120. 𝐺, 𝐼 stand for g, h. 𝑋 represents tau. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝐺𝑇𝐼𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈‘(𝐺𝐼)) = ((𝑈𝐺) ∘ (𝑈𝐼)))
 
Theoremcdlemk39u1 35163* Lemma for cdlemk39u 35164. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝐹𝑁𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅‘(𝑈𝐺)) (𝑅𝐺))
 
Theoremcdlemk39u 35164* Part of proof of Lemma K of [Crawley] p. 118. Line 31, p. 119. Trace-preserving property of the value of tau, represented by (𝑈𝐺). (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ ((𝑅𝐹) = (𝑅𝑁) ∧ 𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅‘(𝑈𝐺)) (𝑅𝐺))
 
Theoremcdlemk19u1 35165* cdlemk19 35065 with simpler hypotheses. TODO: Clean all this up. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐹𝑇𝐹𝑁𝑁𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝑈𝐹)‘𝑃) = (𝑁𝑃))
 
Theoremcdlemk19u 35166* Part of Lemma K of [Crawley] p. 118. Line 12, p. 120, "f (exponent) tau = k". We represent f, k, tau with 𝐹, 𝑁, 𝑈. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐹) = (𝑅𝑁)) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑈𝐹) = 𝑁)
 
Theoremcdlemk56 35167* Part of Lemma K of [Crawley] p. 118. Line 11, p. 120, "tau is in Delta" i.e. 𝑈 is a trace-preserving endormorphism. (Contributed by NM, 31-Jul-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑈𝐸)
 
Theoremcdlemk19w 35168* Use a fixed element to eliminate 𝑃 in cdlemk19u 35166. (Contributed by NM, 1-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑃 = ( 𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → (𝑈𝐹) = 𝑁)
 
Theoremcdlemk56w 35169* Use a fixed element to eliminate 𝑃 in cdlemk56 35167. (Contributed by NM, 1-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑃 = ( 𝑊)    &   𝑍 = ((𝑃 (𝑅𝑏)) ((𝑁𝑃) (𝑅‘(𝑏𝐹))))    &   𝑌 = ((𝑃 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅𝐹) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑃) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if(𝐹 = 𝑁, 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → (𝑈𝐸 ∧ (𝑈𝐹) = 𝑁))
 
Theoremcdlemk 35170* Lemma K of [Crawley] p. 118. Final result, lines 11 and 12 on p. 120: given two translations f and k with the same trace, there exists a trace-preserving endomorphism tau whose value at f is k. We use 𝐹, 𝑁, and 𝑢 to represent f, k, and tau. (Contributed by NM, 1-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝐹) = (𝑅𝑁)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
 
Theoremtendoex 35171* Generalization of Lemma K of [Crawley] p. 118, cdlemk 35170. TODO: can this be used to shorten uses of cdlemk 35170? (Contributed by NM, 15-Oct-2013.)
= (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑁𝑇) ∧ (𝑅𝑁) (𝑅𝐹)) → ∃𝑢𝐸 (𝑢𝐹) = 𝑁)
 
Theoremcdleml1N 35172 Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸𝑓𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉𝑓) ≠ ( I ↾ 𝐵))) → (𝑅‘(𝑈𝑓)) = (𝑅‘(𝑉𝑓)))
 
Theoremcdleml2N 35173* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸𝑓𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝑈𝑓) ≠ ( I ↾ 𝐵) ∧ (𝑉𝑓) ≠ ( I ↾ 𝐵))) → ∃𝑠𝐸 (𝑠‘(𝑈𝑓)) = (𝑉𝑓))
 
Theoremcdleml3N 35174* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸𝑓𝑇) ∧ (𝑓 ≠ ( I ↾ 𝐵) ∧ 𝑈0𝑉0 )) → ∃𝑠𝐸 (𝑠𝑈) = 𝑉)
 
Theoremcdleml4N 35175* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ (𝑈0𝑉0 )) → ∃𝑠𝐸 (𝑠𝑈) = 𝑉)
 
Theoremcdleml5N 35176* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 1-Aug-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸) ∧ 𝑈0 ) → ∃𝑠𝐸 (𝑠𝑈) = 𝑉)
 
Theoremcdleml6 35177* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇 ∧ (𝑠𝐸𝑠0 )) → (𝑈𝐸 ∧ (𝑈‘(𝑠)) = ))
 
Theoremcdleml7 35178* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑇 ∧ (𝑠𝐸𝑠0 )) → ((𝑈𝑠)‘) = (( I ↾ 𝑇)‘))
 
Theoremcdleml8 35179* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇 ≠ ( I ↾ 𝐵)) ∧ (𝑠𝐸𝑠0 )) → (𝑈𝑠) = ( I ↾ 𝑇))
 
Theoremcdleml9 35180* Part of proof of Lemma L of [Crawley] p. 120. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐵 = (Base‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇 ≠ ( I ↾ 𝐵)) ∧ (𝑠𝐸𝑠0 )) → 𝑈0 )
 
Theoremdva1dim 35181* Two expressions for the 1-dimensional subspaces of partial vector space A. Remark in [Crawley] p. 120 line 21, but using a non-identity translation (nonzero vector) 𝐹 whose trace is 𝑃 rather than 𝑃 itself; 𝐹 exists by cdlemf 34759. 𝐸 is the division ring base by erngdv 35189, and 𝑠𝐹 is the scalar product by dvavsca 35213. 𝐹 must be a non-identity translation for the expression to be a 1-dimensional subspace, although the theorem doesn't require it. (Contributed by NM, 14-Oct-2013.)
= (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)} = {𝑔𝑇 ∣ (𝑅𝑔) (𝑅𝐹)})
 
Theoremdvhb1dimN 35182* Two expressions for the 1-dimensional subspaces of vector space H, in the isomorphism B case where the 2nd vector component is zero. (Contributed by NM, 23-Feb-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    0 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 0 ⟩} = {𝑔 ∈ (𝑇 × 𝐸) ∣ ((𝑅‘(1st𝑔)) (𝑅𝐹) ∧ (2nd𝑔) = 0 )})
 
Theoremerng1lem 35183 Value of the endomorphism division ring unit. (Contributed by NM, 12-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (1r𝐷) = ( I ↾ 𝑇))
 
Theoremerngdvlem1 35184* Lemma for eringring 35188. (Contributed by NM, 4-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
 
Theoremerngdvlem2N 35185* Lemma for eringring 35188. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Abel)
 
Theoremerngdvlem3 35186* Lemma for eringring 35188. (Contributed by NM, 6-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))    &    + = (𝑎𝐸, 𝑏𝐸 ↦ (𝑎𝑏))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)
 
Theoremerngdvlem4 35187* Lemma for erngdv 35189. (Contributed by NM, 11-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))    &    + = (𝑎𝐸, 𝑏𝐸 ↦ (𝑎𝑏))    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇 ≠ ( I ↾ 𝐵))) → 𝐷 ∈ DivRing)
 
Theoremeringring 35188 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)
 
Theoremerngdv 35189 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
 
Theoremerng0g 35190* The division ring zero of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &    0 = (0g𝐷)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = 𝑂)
 
Theoremerng1r 35191 The division ring unit of an endomorphism ring. (Contributed by NM, 5-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &    1 = (1r𝐷)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 1 = ( I ↾ 𝑇))
 
Theoremerngdvlem1-rN 35192* Lemma for eringring 35188. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Grp)
 
Theoremerngdvlem2-rN 35193* Lemma for eringring 35188. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Abel)
 
Theoremerngdvlem3-rN 35194* Lemma for eringring 35188. (Contributed by NM, 6-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))    &   𝑀 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑏𝑎))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)
 
Theoremerngdvlem4-rN 35195* Lemma for erngdv 35189. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑃 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑓𝑇 ↦ ((𝑎𝑓) ∘ (𝑏𝑓))))    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = (𝑎𝐸 ↦ (𝑓𝑇(𝑎𝑓)))    &   𝑀 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑏𝑎))    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((oc‘𝐾)‘𝑊)    &   𝑍 = ((𝑄 (𝑅𝑏)) ((𝑄) (𝑅‘(𝑏(𝑠)))))    &   𝑌 = ((𝑄 (𝑅𝑔)) (𝑍 (𝑅‘(𝑔𝑏))))    &   𝑋 = (𝑧𝑇𝑏𝑇 ((𝑏 ≠ ( I ↾ 𝐵) ∧ (𝑅𝑏) ≠ (𝑅‘(𝑠)) ∧ (𝑅𝑏) ≠ (𝑅𝑔)) → (𝑧𝑄) = 𝑌))    &   𝑈 = (𝑔𝑇 ↦ if((𝑠) = , 𝑔, 𝑋))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑇 ≠ ( I ↾ 𝐵))) → 𝐷 ∈ DivRing)
 
Theoremerngring-rN 35196 An endomorphism ring is a ring. TODO: fix comment. (Contributed by NM, 4-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ Ring)
 
Theoremerngdv-rN 35197 An endomorphism ring is a division ring. TODO: fix comment. (Contributed by NM, 11-Aug-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRingR𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐷 ∈ DivRing)
 
Syntaxcdveca 35198 Extend class notation with constructed vector space A.
class DVecA
 
Definitiondf-dveca 35199* Define constructed partial vector space A. (Contributed by NM, 8-Oct-2013.)
DVecA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑠𝑓))⟩})))
 
Theoremdvafset 35200* The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DVecA‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑠𝑓))⟩})))
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