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Theorem List for Metamath Proof Explorer - 40201-40300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremtendo0pl 40201* Property of the additive identity endormorphism. (Contributed by NM, 12-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑂𝑃𝑆) = 𝑆)
 
Theoremtendo0plr 40202* Property of the additive identity endormorphism. (Contributed by NM, 21-Feb-2014.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑆𝑃𝑂) = 𝑆)
 
Theoremtendoicbv 40203* Define inverse function for trace-preserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    β‡’   πΌ = (𝑒 ∈ 𝐸 ↦ (𝑔 ∈ 𝑇 ↦ β—‘(π‘’β€˜π‘”)))
 
Theoremtendoi 40204* Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (𝑆 ∈ 𝐸 β†’ (πΌβ€˜π‘†) = (𝑔 ∈ 𝑇 ↦ β—‘(π‘†β€˜π‘”)))
 
Theoremtendoi2 40205* Value of additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐼 = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) β†’ ((πΌβ€˜π‘†)β€˜πΉ) = β—‘(π‘†β€˜πΉ))
 
Theoremtendoicl 40206* Closure of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (πΌβ€˜π‘†) ∈ 𝐸)
 
Theoremtendoipl 40207* Property of the additive inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π΅ = (Baseβ€˜πΎ)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ ((πΌβ€˜π‘†)𝑃𝑆) = 𝑂)
 
Theoremtendoipl2 40208* Property of the additive inverse endomorphism. (Contributed by NM, 29-Sep-2014.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   πΌ = (𝑠 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ β—‘(π‘ β€˜π‘“)))    &   π΅ = (Baseβ€˜πΎ)    &   π‘ƒ = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑆 ∈ 𝐸) β†’ (𝑆𝑃(πΌβ€˜π‘†)) = 𝑂)
 
Theoremerngfset 40209* The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (EDRingβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑠 ∘ 𝑑))⟩}))
 
Theoremerngset 40210* The division ring on trace-preserving endomorphisms for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑠 ∘ 𝑑))⟩})
 
Theoremerngbase 40211 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom π‘Š). TODO: the .t hypothesis isn't used. (Also look at others.) (Contributed by NM, 9-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &   πΆ = (Baseβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝐸)
 
Theoremerngfplus 40212* Ring addition operation. (Contributed by NM, 9-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
 
Theoremerngplus 40213* Ring addition operation. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((π‘ˆβ€˜π‘“) ∘ (π‘‰β€˜π‘“))))
 
Theoremerngplus2 40214 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ˆ + 𝑉)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
Theoremerngfmul 40215* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑠 ∘ 𝑑)))
 
Theoremerngmul 40216 Ring addition operation. (Contributed by NM, 10-Jun-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   (((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ Β· 𝑉) = (π‘ˆ ∘ 𝑉))
 
Theoremerngfset-rN 40217* The division rings on trace-preserving endomorphisms for a lattice 𝐾. (Contributed by NM, 8-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝑉 β†’ (EDRingRβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ {⟨(Baseβ€˜ndx), ((TEndoβ€˜πΎ)β€˜π‘€)⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ ((TEndoβ€˜πΎ)β€˜π‘€), 𝑑 ∈ ((TEndoβ€˜πΎ)β€˜π‘€) ↦ (𝑑 ∘ 𝑠))⟩}))
 
Theoremerngset-rN 40218* The division ring on trace-preserving endomorphisms for a fiducial co-atom π‘Š. (Contributed by NM, 5-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐷 = {⟨(Baseβ€˜ndx), 𝐸⟩, ⟨(+gβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“))))⟩, ⟨(.rβ€˜ndx), (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠))⟩})
 
Theoremerngbase-rN 40219 The base set of the division ring on trace-preserving endomorphisms is the set of all trace-preserving endomorphisms (for a fiducial co-atom π‘Š). (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &   πΆ = (Baseβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ 𝐢 = 𝐸)
 
Theoremerngfplus-rN 40220* Ring addition operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ + = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((π‘ β€˜π‘“) ∘ (π‘‘β€˜π‘“)))))
 
Theoremerngplus-rN 40221* Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ + 𝑉) = (𝑓 ∈ 𝑇 ↦ ((π‘ˆβ€˜π‘“) ∘ (π‘‰β€˜π‘“))))
 
Theoremerngplus2-rN 40222 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    + = (+gβ€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) β†’ ((π‘ˆ + 𝑉)β€˜πΉ) = ((π‘ˆβ€˜πΉ) ∘ (π‘‰β€˜πΉ)))
 
Theoremerngfmul-rN 40223* Ring multiplication operation. (Contributed by NM, 9-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   ((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) β†’ Β· = (𝑠 ∈ 𝐸, 𝑑 ∈ 𝐸 ↦ (𝑑 ∘ 𝑠)))
 
Theoremerngmul-rN 40224 Ring addition operation. (Contributed by NM, 10-Jun-2013.) (New usage is discouraged.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π· = ((EDRingRβ€˜πΎ)β€˜π‘Š)    &    Β· = (.rβ€˜π·)    β‡’   (((𝐾 ∈ 𝑋 ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸)) β†’ (π‘ˆ Β· 𝑉) = (𝑉 ∘ π‘ˆ))
 
Theoremcdlemh1 40225 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑄 ≀ (𝑃 ∨ (π‘…β€˜πΉ)) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))) = (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemh2 40226 Part of proof of Lemma H of [Crawley] p. 118. (Contributed by NM, 16-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    &    0 = (0.β€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∧ π‘Š) = 0 )
 
Theoremcdlemh 40227 Lemma H of [Crawley] p. 118. (Contributed by NM, 17-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ (𝑄 ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑄 ≀ (𝑃 ∨ (π‘…β€˜πΉ))) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š))
 
Theoremcdlemi1 40228 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (𝑃 ∨ (π‘…β€˜πΊ)))
 
Theoremcdlemi2 40229 Part of proof of Lemma I of [Crawley] p. 118. (Contributed by NM, 18-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (((π‘ˆβ€˜πΉ)β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemi 40230 Lemma I of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘† = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ (((π‘ˆβ€˜πΉ)β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (π‘ˆ ∈ 𝐸 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΉ) β‰  (π‘…β€˜πΊ))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) = 𝑆)
 
Theoremcdlemj1 40231 Part of proof of Lemma J of [Crawley] p. 118. (Contributed by NM, 19-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ (β„Ž β‰  ( I β†Ύ 𝐡) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΉ) β‰  (π‘…β€˜π‘”) ∧ (π‘…β€˜π‘”) β‰  (π‘…β€˜β„Ž) ∧ (𝑝 ∈ 𝐴 ∧ Β¬ 𝑝 ≀ π‘Š))) β†’ ((π‘ˆβ€˜β„Ž)β€˜π‘) = ((π‘‰β€˜β„Ž)β€˜π‘))
 
Theoremcdlemj2 40232 Part of proof of Lemma J of [Crawley] p. 118. Eliminate 𝑝. (Contributed by NM, 20-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ (β„Ž β‰  ( I β†Ύ 𝐡) ∧ 𝑔 ∈ 𝑇 ∧ 𝑔 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΉ) β‰  (π‘…β€˜π‘”) ∧ (π‘…β€˜π‘”) β‰  (π‘…β€˜β„Ž))) β†’ (π‘ˆβ€˜β„Ž) = (π‘‰β€˜β„Ž))
 
Theoremcdlemj3 40233 Part of proof of Lemma J of [Crawley] p. 118. Eliminate 𝑔. (Contributed by NM, 20-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ β„Ž ∈ 𝑇)) ∧ β„Ž β‰  ( I β†Ύ 𝐡)) β†’ (π‘ˆβ€˜β„Ž) = (π‘‰β€˜β„Ž))
 
Theoremtendocan 40234 Cancellation law: if the values of two trace-preserving endormorphisms are equal, so are the endormorphisms. Lemma J of [Crawley] p. 118. (Contributed by NM, 21-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ (π‘ˆβ€˜πΉ) = (π‘‰β€˜πΉ)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡))) β†’ π‘ˆ = 𝑉)
 
Theoremtendoid0 40235* A trace-preserving endomorphism is the additive identity iff at least one of its values (at a non-identity translation) is the identity translation. (Contributed by NM, 1-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡))) β†’ ((π‘ˆβ€˜πΉ) = ( I β†Ύ 𝐡) ↔ π‘ˆ = 𝑂))
 
Theoremtendo0mul 40236* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 1-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (𝑂 ∘ π‘ˆ) = 𝑂)
 
Theoremtendo0mulr 40237* Additive identity multiplied by a trace-preserving endomorphism. (Contributed by NM, 13-Feb-2014.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ π‘ˆ ∈ 𝐸) β†’ (π‘ˆ ∘ 𝑂) = 𝑂)
 
Theoremtendo1ne0 40238* The identity (unity) is not equal to the zero trace-preserving endomorphism. (Contributed by NM, 8-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ( I β†Ύ 𝑇) β‰  𝑂)
 
Theoremtendoconid 40239* The composition (product) of trace-preserving endormorphisms is nonzero when each argument is nonzero. (Contributed by NM, 8-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ π‘ˆ β‰  𝑂) ∧ (𝑉 ∈ 𝐸 ∧ 𝑉 β‰  𝑂)) β†’ (π‘ˆ ∘ 𝑉) β‰  𝑂)
 
Theoremtendotr 40240* The trace of the value of a nonzero trace-preserving endomorphism equals the trace of the argument. (Contributed by NM, 11-Aug-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   πΈ = ((TEndoβ€˜πΎ)β€˜π‘Š)    &   π‘‚ = (𝑓 ∈ 𝑇 ↦ ( I β†Ύ 𝐡))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘ˆ ∈ 𝐸 ∧ π‘ˆ β‰  𝑂) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜(π‘ˆβ€˜πΉ)) = (π‘…β€˜πΉ))
 
Theoremcdlemk1 40241 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (𝑃 ∨ (π‘β€˜π‘ƒ)) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)))
 
Theoremcdlemk2 40242 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 22-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))
 
Theoremcdlemk3 40243 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) = (πΉβ€˜π‘ƒ))
 
Theoremcdlemk4 40244 Part of proof of Lemma K of [Crawley] p. 118, last line. We use 𝑋 for their h, since 𝐻 is already used. (Contributed by NM, 24-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (πΉβ€˜π‘ƒ) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk5a 40245 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk5 40246 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((𝑃 ∨ (π‘β€˜π‘ƒ)) ∧ ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))
 
Theoremcdlemk6 40247 Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 39296. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ)))) β†’ ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))) ≀ ((((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹)))) ∨ (((π‘‹β€˜π‘ƒ) ∨ 𝑃) ∧ ((π‘…β€˜(𝑋 ∘ ◑𝐹)) ∨ (π‘β€˜π‘ƒ)))))
 
Theoremcdlemk8 40248 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) = ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk9 40249 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ π‘Š) = (π‘…β€˜(𝑋 ∘ ◑𝐺)))
 
Theoremcdlemk9bN 40250 Part of proof of Lemma K of [Crawley] p. 118. TODO: is this needed? If so, shorten with cdlemk9 40249 if that one is also needed. (Contributed by NM, 28-Jun-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ π‘Š) = (π‘…β€˜(𝐺 ∘ ◑𝑋)))
 
Theoremcdlemki 40251* Part of proof of Lemma K of [Crawley] p. 118. TODO: Eliminate and put into cdlemksel 40255. (Contributed by NM, 25-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   πΌ = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ 𝐼 ∈ 𝑇)
 
Theoremcdlemkvcl 40252 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ 𝑃 ∈ 𝐴) β†’ 𝑉 ∈ 𝐡)
 
Theoremcdlemk10 40253 Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ 𝑉 ≀ (π‘…β€˜(𝑋 ∘ ◑𝐺)))
 
Theoremcdlemksv 40254* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   (𝐺 ∈ 𝑇 β†’ (π‘†β€˜πΊ) = (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹))))))
 
Theoremcdlemksel 40255* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma(p) function to be a translation. TODO: combine cdlemki 40251? (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ (π‘†β€˜πΊ) ∈ 𝑇)
 
Theoremcdlemksat 40256* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ∈ 𝐴)
 
Theoremcdlemksv2 40257* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma(p) function 𝑆 at the fixed 𝑃 parameter. (Contributed by NM, 26-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐹)))))
 
Theoremcdlemk7 40258* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119. (Contributed by NM, 27-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ 𝑉))
 
Theoremcdlemk11 40259* Part of proof of Lemma K of [Crawley] p. 118. Eq. 3, line 8, p. 119. (Contributed by NM, 29-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐹)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐹))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk12 40260* Part of proof of Lemma K of [Crawley] p. 118. Eq. 4, line 10, p. 119. (Contributed by NM, 30-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &    ∧ = (meetβ€˜πΎ)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝑋 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΉ)) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π‘‹))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ (((π‘†β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺)))))
 
Theoremcdlemkoatnle 40261* Utility lemma. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ ((π‘‚β€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (π‘‚β€˜π‘ƒ) ≀ π‘Š))
 
Theoremcdlemk13 40262* Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. 𝑂, 𝐷 are k1, f1. (Contributed by NM, 1-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘‚β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π·)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐷 ∘ ◑𝐹)))))
 
Theoremcdlemkole 40263* Utility lemma. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘‚β€˜π‘ƒ) ≀ (𝑃 ∨ (π‘…β€˜π·)))
 
Theoremcdlemk14 40264* Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. 𝑂, 𝐷 are k1, f1. (Contributed by NM, 1-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘β€˜π‘ƒ) ≀ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐷))))
 
Theoremcdlemk15 40265* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. 𝑂, 𝐷 are k1, f1. (Contributed by NM, 1-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘β€˜π‘ƒ) ≀ ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐷)))))
 
Theoremcdlemk16a 40266* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))) ∈ 𝐴 ∧ Β¬ ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))) ≀ π‘Š))
 
Theoremcdlemk16 40267* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐷)))) ∈ 𝐴 ∧ Β¬ ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐷)))) ≀ π‘Š))
 
Theoremcdlemk17 40268* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. 𝑂, 𝐷 are k1, f1. (Contributed by NM, 1-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐷)))))
 
Theoremcdlemk1u 40269* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (𝑃 ∨ (π‘‚β€˜π‘ƒ)) ≀ ((π·β€˜π‘ƒ) ∨ (π‘…β€˜π·)))
 
Theoremcdlemk5auN 40270* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 3-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐷 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((π·β€˜π‘ƒ) ∨ (π‘…β€˜π·)) ∧ ((π·β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐷))))
 
Theoremcdlemk5u 40271* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜π·)))) β†’ ((𝑃 ∨ (π‘‚β€˜π‘ƒ)) ∧ ((πΊβ€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))) ≀ ((π‘‹β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐷))))
 
Theoremcdlemk6u 40272* Part of proof of Lemma K of [Crawley] p. 118. Apply dalaw 39296. (Contributed by NM, 4-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜π·)))) β†’ ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))) ≀ ((((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐷)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐷)))) ∨ (((π‘‹β€˜π‘ƒ) ∨ 𝑃) ∧ ((π‘…β€˜(𝑋 ∘ ◑𝐷)) ∨ (π‘‚β€˜π‘ƒ)))))
 
Theoremcdlemkj 40273* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ 𝑍 ∈ 𝑇)
 
TheoremcdlemkuvN 40274* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma1 (p) function π‘ˆ. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   (𝐺 ∈ 𝑇 β†’ (π‘ˆβ€˜πΊ) = (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷))))))
 
Theoremcdlemkuel 40275* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma1 (p) function to be a translation. TODO: combine cdlemkj 40273? (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘ˆβ€˜πΊ) ∈ 𝑇)
 
Theoremcdlemkuat 40276* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 4-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ∈ 𝐴)
 
Theoremcdlemkuv2 40277* Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 1, where sigma1 (p) is π‘ˆ, f1 is 𝐷, and k1 is 𝑂. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐷)))))
 
Theoremcdlemk18 40278* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. 𝑁, π‘ˆ, 𝑂, 𝐷 are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘β€˜π‘ƒ) = ((π‘ˆβ€˜πΉ)β€˜π‘ƒ))
 
Theoremcdlemk19 40279* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. 𝑁, π‘ˆ, 𝑂, 𝐷 are k, sigma1 (p), k1, f1. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ (𝑁 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜π·) β‰  (π‘…β€˜πΉ))) β†’ (π‘ˆβ€˜πΉ) = 𝑁)
 
Theoremcdlemk7u 40280* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma1 case. (Contributed by NM, 3-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐷)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐷))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ 𝑋 β‰  ( I β†Ύ 𝐡) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜π·)))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (((π‘ˆβ€˜π‘‹)β€˜π‘ƒ) ∨ 𝑉))
 
Theoremcdlemk11u 40281* Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma1 (π‘ˆ) case. (Contributed by NM, 4-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    &   π‘‰ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐷)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐷))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ 𝑋 β‰  ( I β†Ύ 𝐡) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜π·)))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) ≀ (((π‘ˆβ€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk12u 40282* Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma1 (π‘ˆ) case. (Contributed by NM, 4-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑋 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑋 β‰  ( I β†Ύ 𝐡) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π‘‹)) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜π·)))) β†’ ((π‘ˆβ€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ (((π‘ˆβ€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺)))))
 
Theoremcdlemk21N 40283* Part of proof of Lemma K of [Crawley] p. 118. Lines 26-27, p. 119 for i=0 and j=1. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜π·) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΉ)))) β†’ ((π‘†β€˜πΊ)β€˜π‘ƒ) = ((π‘ˆβ€˜πΊ)β€˜π‘ƒ))
 
Theoremcdlemk20 40284* Part of proof of Lemma K of [Crawley] p. 118. Line 22, p. 119 for the i=2, j=1 case. Note typo on line 22: f should be fi. Our 𝐷, 𝐢, 𝑂, 𝑄, π‘ˆ, 𝑉 represent their f1, f2, k1, k2, sigma1, sigma2. (Contributed by NM, 5-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘‚ = (π‘†β€˜π·)    &   π‘ˆ = (𝑒 ∈ 𝑇 ↦ (℩𝑗 ∈ 𝑇 (π‘—β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘’)) ∧ ((π‘‚β€˜π‘ƒ) ∨ (π‘…β€˜(𝑒 ∘ ◑𝐷))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐷 ∈ 𝑇) ∧ ((𝑁 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐷 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ ((π‘…β€˜π·) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΆ) β‰  (π‘…β€˜π·)))) β†’ ((π‘ˆβ€˜πΆ)β€˜π‘ƒ) = (π‘„β€˜π‘ƒ))
 
Theoremcdlemkoatnle-2N 40285* Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘„β€˜π‘ƒ) ∈ 𝐴 ∧ Β¬ (π‘„β€˜π‘ƒ) ≀ π‘Š))
 
Theoremcdlemk13-2N 40286* Part of proof of Lemma K of [Crawley] p. 118. Line 13 on p. 119. 𝑄, 𝐢 are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘„β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΆ)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝐢 ∘ ◑𝐹)))))
 
Theoremcdlemkole-2N 40287* Utility lemma. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘„β€˜π‘ƒ) ≀ (𝑃 ∨ (π‘…β€˜πΆ)))
 
Theoremcdlemk14-2N 40288* Part of proof of Lemma K of [Crawley] p. 118. Line 19 on p. 119. 𝑄, 𝐢 are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘β€˜π‘ƒ) ≀ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐢))))
 
Theoremcdlemk15-2N 40289* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. 𝑄, 𝐢 are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘β€˜π‘ƒ) ≀ ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐢)))))
 
Theoremcdlemk16-2N 40290* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐢)))) ∈ 𝐴 ∧ Β¬ ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐢)))) ≀ π‘Š))
 
Theoremcdlemk17-2N 40291* Part of proof of Lemma K of [Crawley] p. 118. Line 21 on p. 119. 𝑄, 𝐢 are k2, f2. (Contributed by NM, 1-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐹 ∘ ◑𝐢)))))
 
Theoremcdlemkj-2N 40292* Part of proof of Lemma K of [Crawley] p. 118. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘Œ = (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐢)))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΆ) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ π‘Œ ∈ 𝑇)
 
Theoremcdlemkuv-2N 40293* Part of proof of Lemma K of [Crawley] p. 118. Value of the sigma2 (p) function, given 𝑉. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   (𝐺 ∈ 𝑇 β†’ (π‘‰β€˜πΊ) = (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐢))))))
 
Theoremcdlemkuel-2N 40294* Part of proof of Lemma K of [Crawley] p. 118. Conditions for the sigma2 (p) function to be a translation. TODO: combine cdlemkj 40273? (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΆ) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘‰β€˜πΊ) ∈ 𝑇)
 
Theoremcdlemkuv2-2 40295* Part of proof of Lemma K of [Crawley] p. 118. Line 16 on p. 119 for i = 2, where sigma2 (p) is 𝑉, f2 is 𝐢, and k2 is 𝑄. (Contributed by NM, 2-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘) ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΆ) β‰  (π‘…β€˜πΊ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐺 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘‰β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜πΊ)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝐺 ∘ ◑𝐢)))))
 
Theoremcdlemk18-2N 40296* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. 𝑁, 𝑉, 𝑄, 𝐢 are k, sigma2 (p), k2, f2. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘β€˜π‘ƒ) = ((π‘‰β€˜πΉ)β€˜π‘ƒ))
 
Theoremcdlemk19-2N 40297* Part of proof of Lemma K of [Crawley] p. 118. Line 22 on p. 119. 𝑁, 𝑉, 𝑄, 𝐢 are k, sigma2 (p), k2, f2. (Contributed by NM, 2-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ (𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ ((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ (π‘‰β€˜πΉ) = 𝑁)
 
Theoremcdlemk7u-2N 40298* Part of proof of Lemma K of [Crawley] p. 118. Line 5, p. 119 for the sigma2 case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    &   π‘ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐢)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐢))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑋 β‰  ( I β†Ύ 𝐡))) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΆ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΆ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘‰β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘‰β€˜π‘‹)β€˜π‘ƒ) ∨ 𝑍))
 
Theoremcdlemk11u-2N 40299* Part of proof of Lemma K of [Crawley] p. 118. Line 17, p. 119, showing Eq. 3 (line 8, p. 119) for the sigma2 (𝑍) case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    &   π‘ = (((πΊβ€˜π‘ƒ) ∨ (π‘‹β€˜π‘ƒ)) ∧ ((π‘…β€˜(𝐺 ∘ ◑𝐢)) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐢))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑋 β‰  ( I β†Ύ 𝐡))) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΆ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΆ)) ∧ (𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘‰β€˜πΊ)β€˜π‘ƒ) ≀ (((π‘‰β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺))))
 
Theoremcdlemk12u-2N 40300* Part of proof of Lemma K of [Crawley] p. 118. Line 18, p. 119, showing Eq. 4 (line 10, p. 119) for the sigma2 (𝑉) case. (Contributed by NM, 5-Jul-2013.) (New usage is discouraged.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    &   π‘† = (𝑓 ∈ 𝑇 ↦ (℩𝑖 ∈ 𝑇 (π‘–β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘“)) ∧ ((π‘β€˜π‘ƒ) ∨ (π‘…β€˜(𝑓 ∘ ◑𝐹))))))    &   π‘„ = (π‘†β€˜πΆ)    &   π‘‰ = (𝑑 ∈ 𝑇 ↦ (β„©π‘˜ ∈ 𝑇 (π‘˜β€˜π‘ƒ) = ((𝑃 ∨ (π‘…β€˜π‘‘)) ∧ ((π‘„β€˜π‘ƒ) ∨ (π‘…β€˜(𝑑 ∘ ◑𝐢))))))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻 ∧ (π‘…β€˜πΉ) = (π‘…β€˜π‘)) ∧ ((𝐹 ∈ 𝑇 ∧ 𝐢 ∈ 𝑇 ∧ 𝑁 ∈ 𝑇) ∧ (𝐺 ∈ 𝑇 ∧ 𝐺 β‰  ( I β†Ύ 𝐡)) ∧ (𝑋 ∈ 𝑇 ∧ 𝑋 β‰  ( I β†Ύ 𝐡))) ∧ (((π‘…β€˜πΆ) β‰  (π‘…β€˜πΉ) ∧ (π‘…β€˜πΊ) β‰  (π‘…β€˜πΆ) ∧ (π‘…β€˜π‘‹) β‰  (π‘…β€˜πΆ)) ∧ ((π‘…β€˜πΊ) β‰  (π‘…β€˜π‘‹) ∧ 𝐹 β‰  ( I β†Ύ 𝐡) ∧ 𝐢 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š))) β†’ ((π‘‰β€˜πΊ)β€˜π‘ƒ) = ((𝑃 ∨ (πΊβ€˜π‘ƒ)) ∧ (((π‘‰β€˜π‘‹)β€˜π‘ƒ) ∨ (π‘…β€˜(𝑋 ∘ ◑𝐺)))))
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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 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