P. 418 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 41701-41800   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	cdlemg17jq 41701*	cdlemg17j 41696 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘(𝐹‘𝑄)) = (𝐹‘(𝐺‘𝑄)))
Theorem	cdlemg17 41702*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → (𝐺‘((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄))))) = ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))))
Theorem	cdlemg18a 41703	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → (𝑃 ∨ (𝐹‘𝑄)) ≠ (𝑄 ∨ (𝐹‘𝑃)))
Theorem	cdlemg18b 41704	Lemma for cdlemg18c 41705. 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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → ¬ 𝑃 ≤ (𝑈 ∨ (𝐹‘𝑄)))
Theorem	cdlemg18c 41705	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 ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑄 ∧ ((𝐹‘𝑄) ∨ (𝐹‘𝑃)) ≠ (𝑃 ∨ 𝑄))) → ((𝑃 ∨ (𝐹‘𝑄)) ∧ (𝑄 ∨ (𝐹‘𝑃))) ∈ 𝐴)
Theorem	cdlemg18d 41706*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ∈ 𝐴)
Theorem	cdlemg18 41707*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) ≤ 𝑊)
Theorem	cdlemg19a 41708*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ (𝑄 ∨ (𝐹‘(𝐺‘𝑄)))) = ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊))
Theorem	cdlemg19 41709*	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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg20 41710*	Show two lines intersect at an atom. TODO: fix comment. (Contributed by NM, 23-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg21 41711*	Version of cdlemg19 with (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) instead of (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑄) as a condition. (Contributed by NM, 23-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄 ∧ (𝐹‘𝑃) ≠ 𝑃) ∧ ((𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg22 41712*	cdlemg21 41711 with (𝐹‘𝑃) ≠ 𝑃 condition removed. (Contributed by NM, 23-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ ((𝑅‘𝐹) ≤ (𝑃 ∨ 𝑄) ∧ ((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg24 41713*	Combine cdlemg16z 41684 and cdlemg22 41712. TODO: Fix comment. (Contributed by NM, 24-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘(𝐺‘𝑃)) ∨ (𝐹‘(𝐺‘𝑄))) ≠ (𝑃 ∨ 𝑄) ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg37 41714*	Use cdlemg8 41656 to eliminate the ≠ (𝑃 ∨ 𝑄) condition of cdlemg24 41713. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg25zz 41715	cdlemg16zz 41685 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑧) ∧ ¬ (𝑅‘𝐺) ≤ (𝑃 ∨ 𝑧))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑧 ∨ (𝐹‘(𝐺‘𝑧))) ∧ 𝑊))
Theorem	cdlemg26zz 41716	cdlemg16zz 41685 restated for easier studying. TODO: Discard this after everything is figured out. (Contributed by NM, 26-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ ¬ (𝑅‘𝐹) ≤ (𝑄 ∨ 𝑧) ∧ ¬ (𝑅‘𝐺) ≤ (𝑄 ∨ 𝑧))) → ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊) = ((𝑧 ∨ (𝐹‘(𝐺‘𝑧))) ∧ 𝑊))
Theorem	cdlemg27a 41717	For use with case when (𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)) or (𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)) is zero, letting us establish ¬ 𝑧 ≤ 𝑊 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣) via 4atex 41101. TODO: Fix comment. (Contributed by NM, 28-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝑧 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑧 ≤ (𝑃 ∨ 𝑣) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ (𝑅‘𝐹) ≤ (𝑃 ∨ 𝑧))
Theorem	cdlemg28a 41718	Part of proof of Lemma G of [Crawley] p. 116. First equality of the equation of line 14 on p. 117. (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ ((𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺)) ∧ 𝑧 ≤ (𝑃 ∨ 𝑣) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑧 ∨ (𝐹‘(𝐺‘𝑧))) ∧ 𝑊))
Theorem	cdlemg31b0N 41719	TODO: Fix comment. (Contributed by NM, 30-May-2013.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ∧ 𝐹 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃)) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))
Theorem	cdlemg31b0a 41720	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹))) → (𝑁 ∈ 𝐴 ∨ 𝑁 = (0.‘𝐾)))
Theorem	cdlemg27b 41721	TODO: Fix comment. (Contributed by NM, 28-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝑧 ∈ 𝐴 ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝑧 ≠ 𝑁)) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑧 ≤ (𝑃 ∨ 𝑣) ∧ (𝐹‘𝑃) ≠ 𝑃)) → ¬ (𝑅‘𝐹) ≤ (𝑄 ∨ 𝑧))
Theorem	cdlemg31a 41722	TODO: fix comment. (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇)) → 𝑁 ≤ (𝑃 ∨ 𝑣))
Theorem	cdlemg31b 41723	TODO: fix comment. (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑣 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇)) → 𝑁 ≤ (𝑄 ∨ (𝑅‘𝐹)))
Theorem	cdlemg31c 41724	Show that when 𝑁 is an atom, it is not under 𝑊. TODO: Is there a shorter direct proof? TODO: should we eliminate (𝐹‘𝑃) ≠ 𝑃 here? (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ (𝐹‘𝑃) ≠ 𝑃 ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑁 ≤ 𝑊)
Theorem	cdlemg31d 41725	Eliminate (𝐹‘𝑃) ≠ 𝑃 from cdlemg31c 41724. TODO: Prove directly. TODO: do we need to eliminate (𝐹‘𝑃) ≠ 𝑃? It might be better to do this all at once at the end. See also cdlemg29 41730 versus cdlemg28 41729. (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ 𝑁 ∈ 𝐴)) → ¬ 𝑁 ≤ 𝑊)
Theorem	cdlemg33b0 41726*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝑁 ∈ 𝐴 ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33c0 41727*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)))
Theorem	cdlemg28b 41728*	Part of proof of Lemma G of [Crawley] p. 116. Second equality of the equation of line 14 on p. 117. Note that ¬ 𝑧 ≤ 𝑊 is redundant here (but simplifies cdlemg28 41729.) (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺)) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃))) → ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊) = ((𝑧 ∨ (𝐹‘(𝐺‘𝑧))) ∧ 𝑊))
Theorem	cdlemg28 41729*	Part of proof of Lemma G of [Crawley] p. 116. Chain the equalities of line 14 on p. 117. TODO: rearrange hypotheses in the order of cdlemg29 41730 (and maybe leading up to this too)? (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣)) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺)) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg29 41730*	Eliminate (𝐹‘𝑃) ≠ 𝑃 and (𝐺‘𝑃) ≠ 𝑃 from cdlemg28 41729. TODO: would it be better to do this later? (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂) ∧ 𝑧 ≤ (𝑃 ∨ 𝑣) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg33a 41731*	TODO: Fix comment. (Contributed by NM, 29-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ≠ 𝑄 ∧ 𝑁 ≠ 𝑂) ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33b 41732*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33c 41733*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 ∈ 𝐴 ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33d 41734*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 ∈ 𝐴) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33e 41735*	TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝑁 = (0.‘𝐾) ∧ 𝑂 = (0.‘𝐾)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ (𝑃 ≠ 𝑄 ∧ 𝑣 ≠ (𝑅‘𝐹) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg33 41736*	Combine cdlemg33b 41732, cdlemg33c 41733, cdlemg33d 41734, cdlemg33e 41735. TODO: Fix comment. (Contributed by NM, 30-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ∃𝑧 ∈ 𝐴 (¬ 𝑧 ≤ 𝑊 ∧ (𝑧 ≠ 𝑁 ∧ 𝑧 ≠ 𝑂 ∧ 𝑧 ≤ (𝑃 ∨ 𝑣))))
Theorem	cdlemg34 41737*	Use cdlemg33 to eliminate 𝑧 from cdlemg29 41730. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐹)))    &   ⊢ 𝑂 = ((𝑃 ∨ 𝑣) ∧ (𝑄 ∨ (𝑅‘𝐺)))    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ ((𝑣 ∈ 𝐴 ∧ 𝑣 ≤ 𝑊) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑃 ≠ 𝑄) ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg35 41738*	TODO: Fix comment. TODO: should we have a more general version of hlsupr 40411 to avoid the ≠ conditions? (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ∃𝑣 ∈ 𝐴 (𝑣 ≤ 𝑊 ∧ (𝑣 ≠ (𝑅‘𝐹) ∧ 𝑣 ≠ (𝑅‘𝐺))))
Theorem	cdlemg36 41739*	Use cdlemg35 to eliminate 𝑣 from cdlemg34 41737. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ ∃𝑟 ∈ 𝐴 (¬ 𝑟 ≤ 𝑊 ∧ (𝑃 ∨ 𝑟) = (𝑄 ∨ 𝑟)))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg38 41740	Use cdlemg37 41714 to eliminate ∃𝑟 ∈ 𝐴 from cdlemg36 41739. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄) ∧ (((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg39 41741	Eliminate ≠ conditions from cdlemg38 41740. TODO: Would this better be done at cdlemg35 41738? TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg40 41742	Eliminate 𝑃 ≠ 𝑄 conditions from cdlemg39 41741. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ (𝐹‘(𝐺‘𝑃))) ∧ 𝑊) = ((𝑄 ∨ (𝐹‘(𝐺‘𝑄))) ∧ 𝑊))
Theorem	cdlemg41 41743	Convert cdlemg40 41742 to function composition. TODO: Fix comment. (Contributed by NM, 31-May-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑃 ∨ ((𝐹 ∘ 𝐺)‘𝑃)) ∧ 𝑊) = ((𝑄 ∨ ((𝐹 ∘ 𝐺)‘𝑄)) ∧ 𝑊))
Theorem	ltrnco 41744	The composition of two translations is a translation. Part of proof of Lemma G of [Crawley] p. 116, line 15 on p. 117. (Contributed by NM, 31-May-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) ∈ 𝑇)
Theorem	trlcocnv 41745	Swap the arguments of the trace of a composition with converse. (Contributed by NM, 1-Jul-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ ◡𝐺)) = (𝑅‘(𝐺 ∘ ◡𝐹)))
Theorem	trlcoabs 41746	Absorption into a composition by joining with trace. (Contributed by NM, 22-Jul-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (((𝐹 ∘ 𝐺)‘𝑃) ∨ (𝑅‘𝐹)) = ((𝐺‘𝑃) ∨ (𝑅‘𝐹)))
Theorem	trlcoabs2N 41747	Absorption of the trace of a composition. (Contributed by NM, 29-Jul-2013.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → ((𝐹‘𝑃) ∨ (𝑅‘(𝐺 ∘ ◡𝐹))) = ((𝐹‘𝑃) ∨ (𝐺‘𝑃)))
Theorem	trlcoat 41748	The trace of a composition of two translations is an atom if their traces are different. (Contributed by NM, 15-Jun-2013.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ 𝐺)) ∈ 𝐴)
Theorem	trlcocnvat 41749	Commonly used special case of trlcoat 41748. (Contributed by NM, 1-Jul-2013.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝑅‘(𝐹 ∘ ◡𝐺)) ∈ 𝐴)
Theorem	trlconid 41750	The composition of two different translations is not the identity translation. (Contributed by NM, 22-Jul-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝐹 ∘ 𝐺) ≠ ( I ↾ 𝐵))
Theorem	trlcolem 41751	Lemma for trlco 41752. (Contributed by NM, 1-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑅‘(𝐹 ∘ 𝐺)) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
Theorem	trlco 41752	The trace of a composition of translations is less than or equal to the join of their traces. Part of proof of Lemma G of [Crawley] p. 116, second paragraph on p. 117. (Contributed by NM, 2-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝑅‘(𝐹 ∘ 𝐺)) ≤ ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
Theorem	trlcone 41753	If two translations have different traces, the trace of their composition is also different. (Contributed by NM, 14-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑅‘𝐹) ≠ (𝑅‘𝐺) ∧ 𝐺 ≠ ( I ↾ 𝐵))) → (𝑅‘𝐹) ≠ (𝑅‘(𝐹 ∘ 𝐺)))
Theorem	cdlemg42 41754	Part of proof of Lemma G of [Crawley] p. 116, first line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → ¬ (𝐺‘𝑃) ≤ (𝑃 ∨ (𝐹‘𝑃)))
Theorem	cdlemg43 41755	Part of proof of Lemma G of [Crawley] p. 116, third line of third paragraph on p. 117. (Contributed by NM, 3-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ ∧ = (meet‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐹‘(𝐺‘𝑃)) = (((𝐺‘𝑃) ∨ (𝑅‘𝐹)) ∧ ((𝐹‘𝑃) ∨ (𝑅‘𝐺))))
Theorem	cdlemg44a 41756	Part of proof of Lemma G of [Crawley] p. 116, fourth line of third paragraph on p. 117: "so fg(p) = gf(p)." (Contributed by NM, 3-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ ((𝐹‘𝑃) ≠ 𝑃 ∧ (𝐺‘𝑃) ≠ 𝑃 ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺))) → (𝐹‘(𝐺‘𝑃)) = (𝐺‘(𝐹‘𝑃)))
Theorem	cdlemg44b 41757	Eliminate (𝐹‘𝑃) ≠ 𝑃, (𝐺‘𝑃) ≠ 𝑃 from cdlemg44a 41756. (Contributed by NM, 3-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝐹‘(𝐺‘𝑃)) = (𝐺‘(𝐹‘𝑃)))
Theorem	cdlemg44 41758	Part of proof of Lemma G of [Crawley] p. 116, fifth line of third paragraph on p. 117: "and hence fg = gf." (Contributed by NM, 3-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑅‘𝐹) ≠ (𝑅‘𝐺)) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
Theorem	cdlemg47a 41759	TODO: fix comment. TODO: Use this above in place of (𝐹‘𝑃) = 𝑃 antecedents? (Contributed by NM, 5-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝐹 = ( I ↾ 𝐵)) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
Theorem	cdlemg46 41760*	Part of proof of Lemma G of [Crawley] p. 116, seventh line of third paragraph on p. 117: "hf and f have different traces." (Contributed by NM, 5-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ ℎ ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ ℎ ≠ ( I ↾ 𝐵) ∧ (𝑅‘ℎ) ≠ (𝑅‘𝐹))) → (𝑅‘(ℎ ∘ 𝐹)) ≠ (𝑅‘𝐹))
Theorem	cdlemg47 41761*	Part of proof of Lemma G of [Crawley] p. 116, ninth line of third paragraph on p. 117: "we conclude that gf = fg." (Contributed by NM, 5-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (ℎ ∈ 𝑇 ∧ (𝑅‘𝐹) = (𝑅‘𝐺)) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ ℎ ≠ ( I ↾ 𝐵) ∧ (𝑅‘ℎ) ≠ (𝑅‘𝐹))) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
Theorem	cdlemg48 41762	Eliminate ℎ from cdlemg47 41761. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ≠ ( I ↾ 𝐵) ∧ (𝑅‘𝐹) = (𝑅‘𝐺))) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
Theorem	ltrncom 41763	Composition is commutative for translations. Part of proof of Lemma G of [Crawley] p. 116. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → (𝐹 ∘ 𝐺) = (𝐺 ∘ 𝐹))
Theorem	ltrnco4 41764	Rearrange a composition of 4 translations, analogous to an4 669. (Contributed by NM, 10-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐸 ∈ 𝑇 ∧ 𝐹 ∈ 𝑇) → ((𝐷 ∘ 𝐸) ∘ (𝐹 ∘ 𝐺)) = ((𝐷 ∘ 𝐹) ∘ (𝐸 ∘ 𝐺)))
Theorem	trljco 41765	Trace joined with trace of composition. (Contributed by NM, 15-Jun-2013.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐹) ∨ (𝑅‘𝐺)))
Theorem	trljco2 41766	Trace joined with trace of composition. (Contributed by NM, 16-Jun-2013.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) → ((𝑅‘𝐹) ∨ (𝑅‘(𝐹 ∘ 𝐺))) = ((𝑅‘𝐺) ∨ (𝑅‘(𝐹 ∘ 𝐺))))
Syntax	ctgrp 41767	Extend class notation with translation group.
class TGrp
Definition	df-tgrp 41768*	Define the class of all translation groups. 𝑘 is normally a member of HL. Each base set is the set of all lattice translations with respect to a hyperplane 𝑤, and the operation is function composition. Similar to definition of G in [Crawley] p. 116, third paragraph (which defines this for geomodular lattices). (Contributed by NM, 5-Jun-2013.)
⊢ TGrp = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((LTrn‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝑘)‘𝑤), 𝑔 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
Theorem	tgrpfset 41769*	The translation group maps for a lattice 𝐾. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (TGrp‘𝐾) = (𝑤 ∈ 𝐻 ↦ {⟨(Base‘ndx), ((LTrn‘𝐾)‘𝑤)⟩, ⟨(+g‘ndx), (𝑓 ∈ ((LTrn‘𝐾)‘𝑤), 𝑔 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑓 ∘ 𝑔))⟩}))
Theorem	tgrpset 41770*	The translation group for a fiducial co-atom 𝑊. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐺 = {⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔))⟩})
Theorem	tgrpbase 41771	The base set of the translation group is the set of all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    &   ⊢ 𝐶 = (Base‘𝐺)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝑇)
Theorem	tgrpopr 41772*	The group operation of the translation group is function composition. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → + = (𝑓 ∈ 𝑇, 𝑔 ∈ 𝑇 ↦ (𝑓 ∘ 𝑔)))
Theorem	tgrpov 41773	The group operation value of the translation group is the composition of translations. (Contributed by NM, 5-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ (𝑋 ∈ 𝑇 ∧ 𝑌 ∈ 𝑇)) → (𝑋 + 𝑌) = (𝑋 ∘ 𝑌))
Theorem	tgrpgrplem 41774	Lemma for tgrpgrp 41775. (Contributed by NM, 6-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐵 = (Base‘𝐾)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Grp)
Theorem	tgrpgrp 41775	The translation group is a group. (Contributed by NM, 6-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Grp)
Theorem	tgrpabl 41776	The translation group is an Abelian group. Lemma G of [Crawley] p. 116. (Contributed by NM, 6-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐺 = ((TGrp‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐺 ∈ Abel)
Syntax	ctendo 41777	Extend class notation with translation group endomorphisms.
class TEndo
Syntax	cedring 41778	Extend class notation with division ring on trace-preserving endomorphisms.
class EDRing
Syntax	cedring-rN 41779	Extend class notation with division ring on trace-preserving endomorphisms, with multiplication reversed. TODO: remove EDRingR theorems if not used.
class EDRingR
Definition	df-tendo 41780*	Define trace-preserving endomorphisms on the set of translations. (Contributed by NM, 8-Jun-2013.)
⊢ TEndo = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {𝑓 ∣ (𝑓:((LTrn‘𝑘)‘𝑤)⟶((LTrn‘𝑘)‘𝑤) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)∀𝑦 ∈ ((LTrn‘𝑘)‘𝑤)(𝑓‘(𝑥 ∘ 𝑦)) = ((𝑓‘𝑥) ∘ (𝑓‘𝑦)) ∧ ∀𝑥 ∈ ((LTrn‘𝑘)‘𝑤)(((trL‘𝑘)‘𝑤)‘(𝑓‘𝑥))(le‘𝑘)(((trL‘𝑘)‘𝑤)‘𝑥))}))
Definition	df-edring-rN 41781*	Define division ring on trace-preserving endomorphisms. Definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
⊢ EDRingR = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑡 ∘ 𝑠))⟩}))
Definition	df-edring 41782*	Define division ring on trace-preserving endomorphisms. The multiplication operation is reversed composition, per the definition of E of [Crawley] p. 117, 4th line from bottom. (Contributed by NM, 8-Jun-2013.)
⊢ EDRing = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ {⟨(Base‘ndx), ((TEndo‘𝑘)‘𝑤)⟩, ⟨(+g‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))⟩, ⟨(.r‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑡 ∈ ((TEndo‘𝑘)‘𝑤) ↦ (𝑠 ∘ 𝑡))⟩}))
Theorem	tendofset 41783*	The set of all trace-preserving endomorphisms on the set of translations for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (TEndo‘𝐾) = (𝑤 ∈ 𝐻 ↦ {𝑠 ∣ (𝑠:((LTrn‘𝐾)‘𝑤)⟶((LTrn‘𝐾)‘𝑤) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)∀𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ ((LTrn‘𝐾)‘𝑤)(((trL‘𝐾)‘𝑤)‘(𝑠‘𝑓)) ≤ (((trL‘𝐾)‘𝑤)‘𝑓))}))
Theorem	tendoset 41784*	The set of trace-preserving endomorphisms on the set of translations for a fiducial co-atom 𝑊. (Contributed by NM, 8-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐸 = {𝑠 ∣ (𝑠:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑠‘(𝑓 ∘ 𝑔)) = ((𝑠‘𝑓) ∘ (𝑠‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑠‘𝑓)) ≤ (𝑅‘𝑓))})
Theorem	istendo 41785*	The predicate "is a trace-preserving endomorphism". Similar to definition of trace-preserving endomorphism in [Crawley] p. 117, penultimate line. (Contributed by NM, 8-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑆 ∈ 𝐸 ↔ (𝑆:𝑇⟶𝑇 ∧ ∀𝑓 ∈ 𝑇 ∀𝑔 ∈ 𝑇 (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)) ∧ ∀𝑓 ∈ 𝑇 (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))))
Theorem	tendotp 41786	Trace-preserving property of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑅‘(𝑆‘𝐹)) ≤ (𝑅‘𝐹))
Theorem	istendod 41787*	Deduce the predicate "is a trace-preserving endomorphism". (Contributed by NM, 9-Jun-2013.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑆:𝑇⟶𝑇)    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇 ∧ 𝑔 ∈ 𝑇) → (𝑆‘(𝑓 ∘ 𝑔)) = ((𝑆‘𝑓) ∘ (𝑆‘𝑔)))    &   ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑇) → (𝑅‘(𝑆‘𝑓)) ≤ (𝑅‘𝑓))    ⇒   ⊢ (𝜑 → 𝑆 ∈ 𝐸)
Theorem	tendof 41788	Functionality of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → 𝑆:𝑇⟶𝑇)
Theorem	tendoeq1 41789*	Condition determining equality of two trace-preserving endomorphisms. (Contributed by NM, 11-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑈‘𝑓) = (𝑉‘𝑓)) → 𝑈 = 𝑉)
Theorem	tendovalco 41790	Value of composition of translations in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ∧ 𝑆 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑆‘(𝐹 ∘ 𝐺)) = ((𝑆‘𝐹) ∘ (𝑆‘𝐺)))
Theorem	tendocoval 41791	Value of composition of endomorphisms in a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ 𝐹 ∈ 𝑇) → ((𝑈 ∘ 𝑉)‘𝐹) = (𝑈‘(𝑉‘𝐹)))
Theorem	tendocl 41792	Closure of a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → (𝑆‘𝐹) ∈ 𝑇)
Theorem	tendoco2 41793	Distribution of compositions in preparation for endomorphism sum definition. (Contributed by NM, 10-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑈‘(𝐹 ∘ 𝐺)) ∘ (𝑉‘(𝐹 ∘ 𝐺))) = (((𝑈‘𝐹) ∘ (𝑉‘𝐹)) ∘ ((𝑈‘𝐺) ∘ (𝑉‘𝐺))))
Theorem	tendoidcl 41794	The identity is a trace-preserving endomorphism. (Contributed by NM, 30-Jul-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( I ↾ 𝑇) ∈ 𝐸)
Theorem	tendo1mul 41795	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (( I ↾ 𝑇) ∘ 𝑈) = 𝑈)
Theorem	tendo1mulr 41796	Multiplicative identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 20-Nov-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑈 ∈ 𝐸) → (𝑈 ∘ ( I ↾ 𝑇)) = 𝑈)
Theorem	tendococl 41797	The composition of two trace-preserving endomorphisms (multiplication in the endormorphism ring) is a trace-preserving endomorphism. (Contributed by NM, 9-Jun-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑇 ∈ 𝐸) → (𝑆 ∘ 𝑇) ∈ 𝐸)
Theorem	tendoid 41798	The identity value of a trace-preserving endomorphism. (Contributed by NM, 21-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) → (𝑆‘( I ↾ 𝐵)) = ( I ↾ 𝐵))
Theorem	tendoeq2 41799*	Condition determining equality of two trace-preserving endomorphisms, showing it is unnecessary to consider the identity translation. In tendocan 41849, we show that we only need to consider a single non-identity translation. (Contributed by NM, 21-Jun-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸) ∧ ∀𝑓 ∈ 𝑇 (𝑓 ≠ ( I ↾ 𝐵) → (𝑈‘𝑓) = (𝑉‘𝑓))) → 𝑈 = 𝑉)
Theorem	tendoplcbv 41800*	Define sum operation for trace-preserving endomorphisms. Change bound variables to isolate them later. (Contributed by NM, 11-Jun-2013.)
⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))    ⇒   ⊢ 𝑃 = (𝑢 ∈ 𝐸, 𝑣 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ ((𝑢‘𝑔) ∘ (𝑣‘𝑔))))