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Theorem List for Metamath Proof Explorer - 38501-38600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
TheoremdilfsetN 38501* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐡 β†’ 𝐿 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π‘‘) β†’ (π‘“β€˜π‘₯) = π‘₯)}))
 
TheoremdilsetN 38502* The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (πΏβ€˜π·) = {𝑓 ∈ 𝑀 ∣ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (π‘“β€˜π‘₯) = π‘₯)})
 
TheoremisdilN 38503* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (πΏβ€˜π·) ↔ (𝐹 ∈ 𝑀 ∧ βˆ€π‘₯ ∈ 𝑆 (π‘₯ βŠ† (π‘Šβ€˜π·) β†’ (πΉβ€˜π‘₯) = π‘₯))))
 
TheoremtrnfsetN 38504* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    &   π‘‡ = (Trnβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐢 β†’ 𝑇 = (𝑑 ∈ 𝐴 ↦ {𝑓 ∈ (πΏβ€˜π‘‘) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π‘‘)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π‘‘)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝑑})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝑑}))}))
 
TheoremtrnsetN 38505* The set of translations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    &   π‘‡ = (Trnβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (π‘‡β€˜π·) = {𝑓 ∈ (πΏβ€˜π·) ∣ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (π‘“β€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (π‘“β€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷}))})
 
TheoremistrnN 38506* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atomsβ€˜πΎ)    &   π‘† = (PSubSpβ€˜πΎ)    &    + = (+π‘ƒβ€˜πΎ)    &    βŠ₯ = (βŠ₯π‘ƒβ€˜πΎ)    &   π‘Š = (WAtomsβ€˜πΎ)    &   π‘€ = (PAutβ€˜πΎ)    &   πΏ = (Dilβ€˜πΎ)    &   π‘‡ = (Trnβ€˜πΎ)    β‡’   ((𝐾 ∈ 𝐡 ∧ 𝐷 ∈ 𝐴) β†’ (𝐹 ∈ (π‘‡β€˜π·) ↔ (𝐹 ∈ (πΏβ€˜π·) ∧ βˆ€π‘ž ∈ (π‘Šβ€˜π·)βˆ€π‘Ÿ ∈ (π‘Šβ€˜π·)((π‘ž + (πΉβ€˜π‘ž)) ∩ ( βŠ₯ β€˜{𝐷})) = ((π‘Ÿ + (πΉβ€˜π‘Ÿ)) ∩ ( βŠ₯ β€˜{𝐷})))))
 
Syntaxctrl 38507 Extend class notation with set of all traces of lattice translations.
class trL
 
Definitiondf-trl 38508* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)
trL = (π‘˜ ∈ V ↦ (𝑀 ∈ (LHypβ€˜π‘˜) ↦ (𝑓 ∈ ((LTrnβ€˜π‘˜)β€˜π‘€) ↦ (β„©π‘₯ ∈ (Baseβ€˜π‘˜)βˆ€π‘ ∈ (Atomsβ€˜π‘˜)(Β¬ 𝑝(leβ€˜π‘˜)𝑀 β†’ π‘₯ = ((𝑝(joinβ€˜π‘˜)(π‘“β€˜π‘))(meetβ€˜π‘˜)𝑀))))))
 
Theoremtrlfset 38509* The set of all traces of lattice translations for a lattice 𝐾. (Contributed by NM, 20-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (𝐾 ∈ 𝐢 β†’ (trLβ€˜πΎ) = (𝑀 ∈ 𝐻 ↦ (𝑓 ∈ ((LTrnβ€˜πΎ)β€˜π‘€) ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ 𝑀 β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ 𝑀))))))
 
Theoremtrlset 38510* The set of traces of lattice translations for a fiducial co-atom π‘Š. (Contributed by NM, 20-May-2012.)
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)    β‡’   ((𝐾 ∈ 𝐢 ∧ π‘Š ∈ 𝐻) β†’ 𝑅 = (𝑓 ∈ 𝑇 ↦ (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (π‘“β€˜π‘)) ∧ π‘Š)))))
 
Theoremtrlval 38511* The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
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)    β‡’   (((𝐾 ∈ 𝑉 ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) = (β„©π‘₯ ∈ 𝐡 βˆ€π‘ ∈ 𝐴 (Β¬ 𝑝 ≀ π‘Š β†’ π‘₯ = ((𝑝 ∨ (πΉβ€˜π‘)) ∧ π‘Š))))
 
Theoremtrlval2 38512 The value of the trace of a lattice translation, given any atom 𝑃 not under the fiducial co-atom π‘Š. Note: this requires only the weaker assumption 𝐾 ∈ Lat; we use 𝐾 ∈ HL for convenience. (Contributed by NM, 20-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜πΉ) = ((𝑃 ∨ (πΉβ€˜π‘ƒ)) ∧ π‘Š))
 
Theoremtrlcl 38513 Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) ∈ 𝐡)
 
Theoremtrlcnv 38514 The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
𝐻 = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜β—‘πΉ) = (π‘…β€˜πΉ))
 
Theoremtrljat1 38515 The value of a translation of an atom 𝑃 not under the fiducial co-atom π‘Š, joined with trace. Equation above Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 38213? (Contributed by NM, 22-May-2012.)
≀ = (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 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∨ (π‘…β€˜πΉ)) = (𝑃 ∨ (πΉβ€˜π‘ƒ)))
 
Theoremtrljat2 38516 The value of a translation of an atom 𝑃 not under the fiducial co-atom π‘Š, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
≀ = (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 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)) = (𝑃 ∨ (πΉβ€˜π‘ƒ)))
 
Theoremtrljat3 38517 The value of a translation of an atom 𝑃 not under the fiducial co-atom π‘Š, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
≀ = (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 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∨ (π‘…β€˜πΉ)) = ((πΉβ€˜π‘ƒ) ∨ (π‘…β€˜πΉ)))
 
Theoremtrlat 38518 If an atom differs from its translation, the trace is an atom. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ (π‘…β€˜πΉ) ∈ 𝐴)
 
Theoremtrl0 38519 If an atom not under the fiducial co-atom π‘Š equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
≀ = (leβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝐹 ∈ 𝑇 ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (π‘…β€˜πΉ) = 0 )
 
Theoremtrlator0 38520 The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)
0 = (0.β€˜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) β†’ ((π‘…β€˜πΉ) ∈ 𝐴 ∨ (π‘…β€˜πΉ) = 0 ))
 
Theoremtrlatn0 38521 The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)
0 = (0.β€˜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) β†’ ((π‘…β€˜πΉ) ∈ 𝐴 ↔ (π‘…β€˜πΉ) β‰  0 ))
 
Theoremtrlnidat 38522 The trace of a lattice translation other than the identity is an atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 23-May-2012.)
𝐡 = (Baseβ€˜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 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) β†’ (π‘…β€˜πΉ) ∈ 𝐴)
 
Theoremltrnnidn 38523 If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom π‘Š is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐹 β‰  ( I β†Ύ 𝐡)) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (πΉβ€˜π‘ƒ) β‰  𝑃)
 
Theoremltrnideq 38524 Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ (πΉβ€˜π‘ƒ) = 𝑃))
 
Theoremtrlid0 38525 The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘…β€˜( I β†Ύ 𝐡)) = 0 )
 
Theoremtrlnidatb 38526 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 38522? Why do both this and ltrnideq 38524 need trlnidat 38522? (Contributed by NM, 4-Jun-2013.)
𝐡 = (Baseβ€˜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) β†’ (𝐹 β‰  ( I β†Ύ 𝐡) ↔ (π‘…β€˜πΉ) ∈ 𝐴))
 
Theoremtrlid0b 38527 A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)
𝐡 = (Baseβ€˜πΎ)    &    0 = (0.β€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (𝐹 = ( I β†Ύ 𝐡) ↔ (π‘…β€˜πΉ) = 0 ))
 
Theoremtrlnid 38528 Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)
𝐡 = (Baseβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝐹 β‰  𝐺 ∧ (π‘…β€˜πΉ) = (π‘…β€˜πΊ))) β†’ 𝐹 β‰  ( I β†Ύ 𝐡))
 
Theoremltrn2ateq 38529 Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ ((πΉβ€˜π‘ƒ) = 𝑃 ↔ (πΉβ€˜π‘„) = 𝑄))
 
Theoremltrnateq 38530 If any atom (under π‘Š) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = 𝑄)
 
Theoremltrnatneq 38531 If any atom (under π‘Š) is not equal to its translation, so is any other atom. TODO: Β¬ 𝑃 ≀ π‘Š isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃) β†’ (πΉβ€˜π‘„) β‰  𝑄)
 
Theoremltrnatlw 38532 If the value of an atom equals the atom in a non-identity translation, the atom is under the fiducial hyperplane. (Contributed by NM, 15-May-2013.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴) ∧ ((πΉβ€˜π‘ƒ) β‰  𝑃 ∧ (πΉβ€˜π‘„) = 𝑄)) β†’ 𝑄 ≀ π‘Š)
 
Theoremtrlle 38533 The trace of a lattice translation is less than the fiducial co-atom π‘Š. (Contributed by NM, 25-May-2012.)
≀ = (leβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) β†’ (π‘…β€˜πΉ) ≀ π‘Š)
 
Theoremtrlne 38534 The trace of a lattice translation is not equal to any atom not under the fiducial co-atom π‘Š. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
≀ = (leβ€˜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 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ 𝑃 β‰  (π‘…β€˜πΉ))
 
Theoremtrlnle 38535 The atom not under the fiducial co-atom π‘Š is not less than the trace of a lattice translation. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜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 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ Β¬ 𝑃 ≀ (π‘…β€˜πΉ))
 
Theoremtrlval3 38536 The value of the trace of a lattice translation in terms of 2 atoms. TODO: Try to shorten proof. (Contributed by NM, 3-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 ∨ (πΉβ€˜π‘ƒ)) β‰  (𝑄 ∨ (πΉβ€˜π‘„)))) β†’ (π‘…β€˜πΉ) = ((𝑃 ∨ (πΉβ€˜π‘ƒ)) ∧ (𝑄 ∨ (πΉβ€˜π‘„))))
 
Theoremtrlval4 38537 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ (π‘…β€˜πΉ) ≀ (𝑃 ∨ 𝑄))) β†’ (π‘…β€˜πΉ) = ((𝑃 ∨ (πΉβ€˜π‘ƒ)) ∧ (𝑄 ∨ (πΉβ€˜π‘„))))
 
Theoremtrlval5 38538 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (π‘…β€˜πΉ) = ((𝑃 ∨ (π‘…β€˜πΉ)) ∧ π‘Š))
 
Theoremarglem1N 38539 Lemma for Desargues's law. Theorem 13.3 of [Crawley] p. 110, third and fourth lines from bottom. In these lemmas, 𝑃, 𝑄, 𝑅, 𝑆, 𝑇, π‘ˆ, 𝐢, 𝐷, 𝐸, 𝐹, and 𝐺 represent Crawley's a0, a1, a2, b0, b1, b2, c, z0, z1, z2, and p respectively. (Contributed by NM, 28-Jun-2012.) (New usage is discouraged.)
∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   πΉ = ((𝑃 ∨ 𝑄) ∧ (𝑆 ∨ 𝑇))    &   πΊ = ((𝑃 ∨ 𝑆) ∧ (𝑄 ∨ 𝑇))    β‡’   ((((𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑃 β‰  𝑆 ∧ 𝑄 β‰  𝑇 ∧ 𝑆 β‰  𝑇)) ∧ 𝐺 ∈ 𝐴) β†’ 𝐹 ∈ 𝐴)
 
Theoremcdlemc1 38540 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 38213? (Contributed by NM, 29-May-2012.)
𝐡 = (Baseβ€˜πΎ)    &    ≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑋 ∈ 𝐡 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ (𝑃 ∨ ((𝑃 ∨ 𝑋) ∧ π‘Š)) = (𝑃 ∨ 𝑋))
 
Theoremcdlemc2 38541 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (πΉβ€˜π‘„) ≀ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)))
 
Theoremcdlemc3 38542 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ ((πΉβ€˜π‘ƒ) ≀ (𝑄 ∨ (π‘…β€˜πΉ)) β†’ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))))
 
Theoremcdlemc4 38543 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) β†’ (𝑄 ∨ (π‘…β€˜πΉ)) β‰  ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š)))
 
Theoremcdlemc5 38544 Lemma for cdlemc 38546. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ)) ∧ (πΉβ€˜π‘ƒ) β‰  𝑃)) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
 
Theoremcdlemc6 38545 Lemma for cdlemc 38546. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ (πΉβ€˜π‘ƒ) = 𝑃) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
 
Theoremcdlemc 38546 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    &   π‘… = ((trLβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) β†’ (πΉβ€˜π‘„) = ((𝑄 ∨ (π‘…β€˜πΉ)) ∧ ((πΉβ€˜π‘ƒ) ∨ ((𝑃 ∨ 𝑄) ∧ π‘Š))))
 
Theoremcdlemd1 38547 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑅 = ((𝑃 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)) ∧ (𝑄 ∨ ((𝑄 ∨ 𝑅) ∧ π‘Š))))
 
Theoremcdlemd2 38548 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) ∧ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ∧ (πΉβ€˜π‘„) = (πΊβ€˜π‘„))) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd3 38549 Part of proof of Lemma D in [Crawley] p. 113. The 𝑅 β‰  𝑃 requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ (𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑆))
 
Theoremcdlemd4 38550 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 β‰  𝑃)) ∧ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ∧ (πΉβ€˜π‘„) = (πΊβ€˜π‘„))) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd5 38551 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ 𝑃 β‰  𝑄) ∧ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ∧ (πΉβ€˜π‘„) = (πΊβ€˜π‘„))) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd6 38552 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ))) ∧ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ)) β†’ (πΉβ€˜π‘„) = (πΊβ€˜π‘„))
 
Theoremcdlemd7 38553 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ∧ Β¬ 𝑄 ≀ (𝑃 ∨ (πΉβ€˜π‘ƒ)))) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd8 38554 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ∧ (πΉβ€˜π‘ƒ) = 𝑃)) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd9 38555 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ 𝑅 ∈ 𝐴) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ)) β†’ (πΉβ€˜π‘…) = (πΊβ€˜π‘…))
 
Theoremcdlemd 38556 If two translations agree at any atom not under the fiducial co-atom π‘Š, then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ)) β†’ 𝐹 = 𝐺)
 
Theoremltrneq3 38557 Two translations agree at any atom not under the fiducial co-atom π‘Š iff they are equal. (Contributed by NM, 25-Jul-2013.)
≀ = (leβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘‡ = ((LTrnβ€˜πΎ)β€˜π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š)) β†’ ((πΉβ€˜π‘ƒ) = (πΊβ€˜π‘ƒ) ↔ 𝐹 = 𝐺))
 
Theoremcdleme00a 38558 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    β‡’   ((𝐾 ∈ HL ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)) β†’ 𝑅 β‰  𝑃)
 
Theoremcdleme0aa 38559 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   π΅ = (Baseβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) β†’ π‘ˆ ∈ 𝐡)
 
Theoremcdleme0a 38560 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄)) β†’ π‘ˆ ∈ 𝐴)
 
Theoremcdleme0b 38561 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴) β†’ π‘ˆ β‰  𝑃)
 
Theoremcdleme0c 38562 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) β†’ π‘ˆ β‰  𝑅)
 
Theoremcdleme0cp 38563 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 38946- swap consequent equality; make antecedent use df-3an 1090. (Contributed by NM, 13-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴)) β†’ (𝑃 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))
 
Theoremcdleme0cq 38564 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š))) β†’ (𝑄 ∨ π‘ˆ) = (𝑃 ∨ 𝑄))
 
Theoremcdleme0dN 38565 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ 𝑃 β‰  𝑅)) β†’ 𝑉 ∈ 𝐴)
 
Theoremcdleme0e 38566 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ π‘ˆ β‰  𝑉)
 
Theoremcdleme0fN 38567 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝑉 β‰  𝑃)
 
Theoremcdleme0gN 38568 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) β†’ 𝑉 β‰  𝑄)
 
Theoremcdlemeulpq 38569 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴)) β†’ π‘ˆ ≀ (𝑃 ∨ 𝑄))
 
Theoremcdleme01N 38570 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄) β†’ ((π‘ˆ β‰  𝑃 ∧ π‘ˆ β‰  𝑄 ∧ π‘ˆ ≀ (𝑃 ∨ 𝑄)) ∧ π‘ˆ ≀ π‘Š))
 
Theoremcdleme02N 38571 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄) β†’ ((𝑃 ∨ π‘ˆ) = (𝑄 ∨ π‘ˆ) ∧ π‘ˆ ≀ π‘Š))
 
Theoremcdleme0ex1N 38572* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘’ ∈ 𝐴 (𝑒 ≀ (𝑃 ∨ 𝑄) ∧ 𝑒 ≀ π‘Š))
 
Theoremcdleme0ex2N 38573* Part of proof of Lemma E in [Crawley] p. 113. Note that (𝑃 ∨ 𝑒) = (𝑄 ∨ 𝑒) is a shorter way to express 𝑒 β‰  𝑃 ∧ 𝑒 β‰  𝑄 ∧ 𝑒 ≀ (𝑃 ∨ 𝑄). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ 𝑃 β‰  𝑄) β†’ βˆƒπ‘’ ∈ 𝐴 ((𝑃 ∨ 𝑒) = (𝑄 ∨ 𝑒) ∧ 𝑒 ≀ π‘Š))
 
Theoremcdleme0moN 38574* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ βˆƒ*π‘Ÿ(π‘Ÿ ∈ 𝐴 ∧ (𝑃 ∨ π‘Ÿ) = (𝑄 ∨ π‘Ÿ)))) β†’ (𝑅 = 𝑃 ∨ 𝑅 = 𝑄))
 
Theoremcdleme1b 38575 Part of proof of Lemma E in [Crawley] p. 113. Utility lemma showing 𝐹 is a lattice element. 𝐹 represents their f(r). (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π΅ = (Baseβ€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴)) β†’ 𝐹 ∈ 𝐡)
 
Theoremcdleme1 38576 Part of proof of Lemma E in [Crawley] p. 113. 𝐹 represents their f(r). Here we show r ∨ f(r) = r ∨ u (7th through 5th lines from bottom on p. 113). (Contributed by NM, 4-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ (𝑅 ∨ 𝐹) = (𝑅 ∨ π‘ˆ))
 
Theoremcdleme2 38577 Part of proof of Lemma E in [Crawley] p. 113. 𝐹 represents f(r). π‘Š is the fiducial co-atom (hyperplane) w. Here we show that (r ∨ f(r)) ∧ w = u in their notation (4th line from bottom on p. 113). (Contributed by NM, 5-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ ((𝑅 ∨ 𝐹) ∧ π‘Š) = π‘ˆ)
 
Theoremcdleme3b 38578 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝐹 β‰  𝑅)
 
Theoremcdleme3c 38579 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &    0 = (0.β€˜πΎ)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ 𝑃 β‰  𝑄) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š))) β†’ 𝐹 β‰  0 )
 
Theoremcdleme3d 38580 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ 𝑉))
 
Theoremcdleme3e 38581 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄)))) β†’ 𝑉 ∈ 𝐴)
 
Theoremcdleme3fN 38582 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. TODO: Delete - duplicates cdleme0e 38566. (Contributed by NM, 6-Jun-2012.) (New usage is discouraged.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ π‘ˆ β‰  𝑉)
 
Theoremcdleme3g 38583 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 β‰  π‘ˆ)
 
Theoremcdleme3h 38584 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 38585 and cdleme3 38586. (Contributed by NM, 6-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    &   π‘‰ = ((𝑃 ∨ 𝑅) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 ∈ 𝐴)
 
Theoremcdleme3fa 38585 Part of proof of Lemma E in [Crawley] p. 113. See cdleme3 38586. (Contributed by NM, 6-Oct-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐹 ∈ 𝐴)
 
Theoremcdleme3 38586 Part of proof of Lemma E in [Crawley] p. 113. 𝐹 represents f(r). π‘Š is the fiducial co-atom (hyperplane) w. Here and in cdleme3fa 38585 above, we show that f(r) ∈ W (4th line from bottom on p. 113), meaning it is an atom and not under w, which in our notation is expressed as 𝐹 ∈ 𝐴 ∧ Β¬ 𝐹 ≀ π‘Š. Their proof provides no details of our lemmas cdleme3b 38578 through cdleme3 38586, so there may be a simpler proof that we have overlooked. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑅 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑅) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ ((𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ Β¬ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐹 ≀ π‘Š)
 
Theoremcdleme4 38587 Part of proof of Lemma E in [Crawley] p. 113. 𝐹 and 𝐺 represent f(s) and fs(r). Here show p ∨ q = r ∨ u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄)) β†’ (𝑃 ∨ 𝑄) = (𝑅 ∨ π‘ˆ))
 
Theoremcdleme4a 38588 Part of proof of Lemma E in [Crawley] p. 114 top. 𝐺 represents fs(r). Auxiliary lemma derived from cdleme5 38589. We show fs(r) ≀ p ∨ q. (Contributed by NM, 10-Nov-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴) ∧ 𝑆 ∈ 𝐴) β†’ 𝐺 ≀ (𝑃 ∨ 𝑄))
 
Theoremcdleme5 38589 Part of proof of Lemma E in [Crawley] p. 113. 𝐺 represents fs(r). We show r ∨ fs(r)) = p ∨ q at the top of p. 114. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ ((𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ (𝑅 ∨ 𝐺) = (𝑃 ∨ 𝑄))
 
Theoremcdleme6 38590 Part of proof of Lemma E in [Crawley] p. 113. This expresses (r ∨ fs(r)) ∧ w = u at the top of p. 114. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š)) ∧ ((𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ ((𝑅 ∨ 𝐺) ∧ π‘Š) = π‘ˆ)
 
Theoremcdleme7aa 38591 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝑅 ≀ (π‘ˆ ∨ 𝑆))
 
Theoremcdleme7a 38592 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    &   π‘‰ = ((𝑅 ∨ 𝑆) ∧ π‘Š)    β‡’   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ 𝑉))
 
Theoremcdleme7b 38593 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    &   π‘‰ = ((𝑅 ∨ 𝑆) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄) ∧ 𝑅 ≀ (𝑃 ∨ 𝑄))) β†’ 𝑉 ∈ 𝐴)
 
Theoremcdleme7c 38594 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 7-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    &   π‘‰ = ((𝑅 ∨ 𝑆) ∧ π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑄 ∈ 𝐴) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ π‘ˆ β‰  𝑉)
 
Theoremcdleme7d 38595 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 8-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    &   π‘‰ = ((𝑅 ∨ 𝑆) ∧ π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐺 β‰  π‘ˆ)
 
Theoremcdleme7e 38596 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme7ga 38597 and cdleme7 38598. (Contributed by NM, 8-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    &   π‘‰ = ((𝑅 ∨ 𝑆) ∧ π‘Š)    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐺 β‰  (0.β€˜πΎ))
 
Theoremcdleme7ga 38597 Part of proof of Lemma E in [Crawley] p. 113. See cdleme7 38598. (Contributed by NM, 8-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ 𝐺 ∈ 𝐴)
 
Theoremcdleme7 38598 Part of proof of Lemma E in [Crawley] p. 113. 𝐺 and 𝐹 represent fs(r) and f(s) respectively. π‘Š is the fiducial co-atom (hyperplane) that they call w. Here and in cdleme7ga 38597 above, we show that fs(r) ∈ W (top of p. 114), meaning it is an atom and not under w, which in our notation is expressed as 𝐺 ∈ 𝐴 ∧ Β¬ 𝐺 ≀ π‘Š. (Note that we do not have a symbol for their W.) Their proof provides no details of our cdleme7aa 38591 through cdleme7 38598, so there may be a simpler proof that we have overlooked. (Contributed by NM, 9-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   π‘ˆ = ((𝑃 ∨ 𝑄) ∧ π‘Š)    &   πΉ = ((𝑆 ∨ π‘ˆ) ∧ (𝑄 ∨ ((𝑃 ∨ 𝑆) ∧ π‘Š)))    &   πΊ = ((𝑃 ∨ 𝑄) ∧ (𝐹 ∨ ((𝑅 ∨ 𝑆) ∧ π‘Š)))    β‡’   ((((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑄 ∈ 𝐴 ∧ Β¬ 𝑄 ≀ π‘Š)) ∧ ((𝑅 ∈ 𝐴 ∧ Β¬ 𝑅 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ Β¬ 𝑆 ≀ π‘Š)) ∧ (𝑃 β‰  𝑄 ∧ 𝑅 ≀ (𝑃 ∨ 𝑄) ∧ Β¬ 𝑆 ≀ (𝑃 ∨ 𝑄))) β†’ Β¬ 𝐺 ≀ π‘Š)
 
Theoremcdleme8 38599 Part of proof of Lemma E in [Crawley] p. 113, 2nd paragraph on p. 114. 𝐢 represents s1. In their notation, we prove p ∨ s1 = p ∨ s. (Contributed by NM, 9-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΆ = ((𝑃 ∨ 𝑆) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ 𝑆 ∈ 𝐴) β†’ (𝑃 ∨ 𝐢) = (𝑃 ∨ 𝑆))
 
Theoremcdleme9a 38600 Part of proof of Lemma E in [Crawley] p. 113. 𝐢 represents s1, which we prove is an atom. (Contributed by NM, 10-Jun-2012.)
≀ = (leβ€˜πΎ)    &    ∨ = (joinβ€˜πΎ)    &    ∧ = (meetβ€˜πΎ)    &   π΄ = (Atomsβ€˜πΎ)    &   π» = (LHypβ€˜πΎ)    &   πΆ = ((𝑃 ∨ 𝑆) ∧ π‘Š)    β‡’   (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ Β¬ 𝑃 ≀ π‘Š) ∧ (𝑆 ∈ 𝐴 ∧ 𝑃 β‰  𝑆)) β†’ 𝐢 ∈ 𝐴)
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