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Theorem List for Metamath Proof Explorer - 36201-36300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremltrncvr 36201 Covering property of a lattice translation. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐶 = ( ⋖ ‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑋𝐵𝑌𝐵)) → (𝑋𝐶𝑌 ↔ (𝐹𝑋)𝐶(𝐹𝑌)))

Theoremltrnval1 36202 Value of a lattice translation under its co-atom. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹𝑋) = 𝑋)

Theoremltrnid 36203* A lattice translation is the identity function iff all atoms not under the fiducial co-atom 𝑊 are equal to their values. (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (∀𝑝𝐴𝑝 𝑊 → (𝐹𝑝) = 𝑝) ↔ 𝐹 = ( I ↾ 𝐵)))

Theoremltrnnid 36204* If a lattice translation is not the identity, then there is an atom not under the fiducial co-atom 𝑊 and not equal to its translation. (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) → ∃𝑝𝐴𝑝 𝑊 ∧ (𝐹𝑝) ≠ 𝑝))

Theoremltrnatb 36205 The lattice translation of an atom is an atom. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐵) → (𝑃𝐴 ↔ (𝐹𝑃) ∈ 𝐴))

Theoremltrncnvatb 36206 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 2-Jun-2012.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐵) → (𝑃𝐴 ↔ (𝐹𝑃) ∈ 𝐴))

Theoremltrnel 36207 The lattice translation of an atom not under the fiducial co-atom is also an atom not under the fiducial co-atom. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))

Theoremltrnat 36208 The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel 36207 uses. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)

Theoremltrncnvat 36209 The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝑃𝐴) → (𝐹𝑃) ∈ 𝐴)

Theoremltrncnvel 36210 The converse of the lattice translation of an atom not under the fiducial co-atom. (Contributed by NM, 10-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) ∈ 𝐴 ∧ ¬ (𝐹𝑃) 𝑊))

TheoremltrncoelN 36211 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 36207 uses. (Contributed by NM, 1-May-2013.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹‘(𝐺𝑃)) ∈ 𝐴 ∧ ¬ (𝐹‘(𝐺𝑃)) 𝑊))

Theoremltrncoat 36212 Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel 36207, ltrnat 36208 uses. (Contributed by NM, 1-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑃𝐴) → (𝐹‘(𝐺𝑃)) ∈ 𝐴)

Theoremltrncoval 36213 Two ways to express value of translation composition. (Contributed by NM, 31-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑃𝐴) → ((𝐹𝐺)‘𝑃) = (𝐹‘(𝐺𝑃)))

Theoremltrncnv 36214 The converse of a lattice translation is a lattice translation. (Contributed by NM, 10-May-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹𝑇)

Theoremltrn11at 36215 Frequently used one-to-one property of lattice translation atoms. (Contributed by NM, 5-May-2013.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴𝑄𝐴𝑃𝑄)) → (𝐹𝑃) ≠ (𝐹𝑄))

Theoremltrneq2 36216* The equality of two translations is determined by their equality at atoms. (Contributed by NM, 2-Mar-2014.)
𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴 (𝐹𝑝) = (𝐺𝑝) ↔ 𝐹 = 𝐺))

Theoremltrneq 36217* The equality of two translations is determined by their equality at atoms not under co-atom 𝑊. (Contributed by NM, 20-Jun-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) → (∀𝑝𝐴𝑝 𝑊 → (𝐹𝑝) = (𝐺𝑝)) ↔ 𝐹 = 𝐺))

Theoremidltrn 36218 The identity function is a lattice translation. Remark below Lemma B in [Crawley] p. 112. (Contributed by NM, 18-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( I ↾ 𝐵) ∈ 𝑇)

Theoremltrnmw 36219 Property of lattice translation value. Remark below Lemma B in [Crawley] p. 112. TODO: Can this be used in more places? (Contributed by NM, 20-May-2012.) (Proof shortened by OpenAI, 25-Mar-2020.)
= (le‘𝐾)    &    = (meet‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) 𝑊) = 0 )

TheoremdilfsetN 36220* The mapping from fiducial atom to set of dilations. (Contributed by NM, 30-Jan-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)       (𝐾𝐵𝐿 = (𝑑𝐴 ↦ {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝑑) → (𝑓𝑥) = 𝑥)}))

TheoremdilsetN 36221* The set of dilations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝐿𝐷) = {𝑓𝑀 ∣ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝑓𝑥) = 𝑥)})

TheoremisdilN 36222* The predicate "is a dilation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝐿𝐷) ↔ (𝐹𝑀 ∧ ∀𝑥𝑆 (𝑥 ⊆ (𝑊𝐷) → (𝐹𝑥) = 𝑥))))

TheoremtrnfsetN 36223* The mapping from fiducial atom to set of translations. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)    &   𝑇 = (Trn‘𝐾)       (𝐾𝐶𝑇 = (𝑑𝐴 ↦ {𝑓 ∈ (𝐿𝑑) ∣ ∀𝑞 ∈ (𝑊𝑑)∀𝑟 ∈ (𝑊𝑑)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝑑})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝑑}))}))

TheoremtrnsetN 36224* The set of translations for a fiducial atom 𝐷. (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)    &   𝑇 = (Trn‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝑇𝐷) = {𝑓 ∈ (𝐿𝐷) ∣ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝑓𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝑓𝑟)) ∩ ( ‘{𝐷}))})

TheoremistrnN 36225* The predicate "is a translation". (Contributed by NM, 4-Feb-2012.) (New usage is discouraged.)
𝐴 = (Atoms‘𝐾)    &   𝑆 = (PSubSp‘𝐾)    &    + = (+𝑃𝐾)    &    = (⊥𝑃𝐾)    &   𝑊 = (WAtoms‘𝐾)    &   𝑀 = (PAut‘𝐾)    &   𝐿 = (Dil‘𝐾)    &   𝑇 = (Trn‘𝐾)       ((𝐾𝐵𝐷𝐴) → (𝐹 ∈ (𝑇𝐷) ↔ (𝐹 ∈ (𝐿𝐷) ∧ ∀𝑞 ∈ (𝑊𝐷)∀𝑟 ∈ (𝑊𝐷)((𝑞 + (𝐹𝑞)) ∩ ( ‘{𝐷})) = ((𝑟 + (𝐹𝑟)) ∩ ( ‘{𝐷})))))

Syntaxctrl 36226 Extend class notation with set of all traces of lattice translations.
class trL

Definitiondf-trl 36227* Define trace of a lattice translation. (Contributed by NM, 20-May-2012.)
trL = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ (𝑥 ∈ (Base‘𝑘)∀𝑝 ∈ (Atoms‘𝑘)(¬ 𝑝(le‘𝑘)𝑤𝑥 = ((𝑝(join‘𝑘)(𝑓𝑝))(meet‘𝑘)𝑤))))))

Theoremtrlfset 36228* The set of all traces of lattice translations for a lattice 𝐾. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝐶 → (trL‘𝐾) = (𝑤𝐻 ↦ (𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ (𝑥𝐵𝑝𝐴𝑝 𝑤𝑥 = ((𝑝 (𝑓𝑝)) 𝑤))))))

Theoremtrlset 36229* The set of traces of lattice translations for a fiducial co-atom 𝑊. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((𝐾𝐶𝑊𝐻) → 𝑅 = (𝑓𝑇 ↦ (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝑓𝑝)) 𝑊)))))

Theoremtrlval 36230* The value of the trace of a lattice translation. (Contributed by NM, 20-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑥𝐵𝑝𝐴𝑝 𝑊𝑥 = ((𝑝 (𝐹𝑝)) 𝑊))))

Theoremtrlval2 36231 The value of the trace of a lattice translation, given any atom 𝑃 not under the fiducial co-atom 𝑊. Note: this requires only the weaker assumption 𝐾 ∈ Lat; we use 𝐾 ∈ HL for convenience. (Contributed by NM, 20-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) 𝑊))

Theoremtrlcl 36232 Closure of the trace of a lattice translation. (Contributed by NM, 22-May-2012.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) ∈ 𝐵)

Theoremtrlcnv 36233 The trace of the converse of a lattice translation. (Contributed by NM, 10-May-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) = (𝑅𝐹))

Theoremtrljat1 36234 The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 35932? (Contributed by NM, 22-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = (𝑃 (𝐹𝑃)))

Theoremtrljat2 36235 The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) (𝑅𝐹)) = (𝑃 (𝐹𝑃)))

Theoremtrljat3 36236 The value of a translation of an atom 𝑃 not under the fiducial co-atom 𝑊, joined with trace. Equation above Lemma C in [Crawley] p. 112. (Contributed by NM, 22-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 (𝑅𝐹)) = ((𝐹𝑃) (𝑅𝐹)))

Theoremtrlat 36237 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.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃)) → (𝑅𝐹) ∈ 𝐴)

Theoremtrl0 36238 If an atom not under the fiducial co-atom 𝑊 equals its lattice translation, the trace of the translation is zero. (Contributed by NM, 24-May-2012.)
= (le‘𝐾)    &    0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑇 ∧ (𝐹𝑃) = 𝑃)) → (𝑅𝐹) = 0 )

Theoremtrlator0 36239 The trace of a lattice translation is an atom or zero. (Contributed by NM, 5-May-2013.)
0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑅𝐹) ∈ 𝐴 ∨ (𝑅𝐹) = 0 ))

Theoremtrlatn0 36240 The trace of a lattice translation is an atom iff it is nonzero. (Contributed by NM, 14-Jun-2013.)
0 = (0.‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → ((𝑅𝐹) ∈ 𝐴 ↔ (𝑅𝐹) ≠ 0 ))

Theoremtrlnidat 36241 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.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) → (𝑅𝐹) ∈ 𝐴)

Theoremltrnnidn 36242 If a lattice translation is not the identity, then the translation of any atom not under the fiducial co-atom 𝑊 is different from the atom. Remark above Lemma C in [Crawley] p. 112. (Contributed by NM, 24-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐹 ≠ ( I ↾ 𝐵)) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹𝑃) ≠ 𝑃)

Theoremltrnideq 36243 Property of the identity lattice translation. (Contributed by NM, 27-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝐹𝑃) = 𝑃))

Theoremtrlid0 36244 The trace of the identity translation is zero. (Contributed by NM, 11-Jun-2013.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑅 = ((trL‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑅‘( I ↾ 𝐵)) = 0 )

Theoremtrlnidatb 36245 A lattice translation is not the identity iff its trace is an atom. TODO: Can proofs be reorganized so this goes with trlnidat 36241? Why do both this and ltrnideq 36243 need trlnidat 36241? (Contributed by NM, 4-Jun-2013.)
𝐵 = (Base‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐹 ≠ ( I ↾ 𝐵) ↔ (𝑅𝐹) ∈ 𝐴))

Theoremtrlid0b 36246 A lattice translation is the identity iff its trace is zero. (Contributed by NM, 14-Jun-2013.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐹 = ( I ↾ 𝐵) ↔ (𝑅𝐹) = 0 ))

Theoremtrlnid 36247 Different translations with the same trace cannot be the identity. (Contributed by NM, 26-Jul-2013.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝐹𝐺 ∧ (𝑅𝐹) = (𝑅𝐺))) → 𝐹 ≠ ( I ↾ 𝐵))

Theoremltrn2ateq 36248 Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) = 𝑃 ↔ (𝐹𝑄) = 𝑄))

Theoremltrnateq 36249 If any atom (under 𝑊) is not equal to its translation, so is any other atom. (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = 𝑄)

Theoremltrnatneq 36250 If any atom (under 𝑊) is not equal to its translation, so is any other atom. TODO: ¬ 𝑃 𝑊 isn't needed to prove this. Will removing it shorten (and not lengthen) proofs using it? (Contributed by NM, 6-May-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) ≠ 𝑃) → (𝐹𝑄) ≠ 𝑄)

Theoremltrnatlw 36251 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.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ ((𝐹𝑃) ≠ 𝑃 ∧ (𝐹𝑄) = 𝑄)) → 𝑄 𝑊)

Theoremtrlle 36252 The trace of a lattice translation is less than the fiducial co-atom 𝑊. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝑅𝐹) 𝑊)

Theoremtrlne 36253 The trace of a lattice translation is not equal to any atom not under the fiducial co-atom 𝑊. Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → 𝑃 ≠ (𝑅𝐹))

Theoremtrlnle 36254 The atom not under the fiducial co-atom 𝑊 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.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ¬ 𝑃 (𝑅𝐹))

Theoremtrlval3 36255 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.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃 (𝐹𝑃)) ≠ (𝑄 (𝐹𝑄)))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))

Theoremtrlval4 36256 The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝑃𝑄 ∧ ¬ (𝑅𝐹) (𝑃 𝑄))) → (𝑅𝐹) = ((𝑃 (𝐹𝑃)) (𝑄 (𝐹𝑄))))

Theoremtrlval5 36257 The value of the trace of a lattice translation in terms of itself. (Contributed by NM, 19-Jul-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑅𝐹) = ((𝑃 (𝑅𝐹)) 𝑊))

Theoremarglem1N 36258 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, 𝑈, 𝐶, 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.)
= (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐹 = ((𝑃 𝑄) (𝑆 𝑇))    &   𝐺 = ((𝑃 𝑆) (𝑄 𝑇))       ((((𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴) ∧ (𝑆𝐴𝑇𝐴𝑃𝑄) ∧ (𝑃𝑆𝑄𝑇𝑆𝑇)) ∧ 𝐺𝐴) → 𝐹𝐴)

Theoremcdlemc1 36259 Part of proof of Lemma C in [Crawley] p. 112. TODO: shorten with atmod3i1 35932? (Contributed by NM, 29-May-2012.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → (𝑃 ((𝑃 𝑋) 𝑊)) = (𝑃 𝑋))

Theoremcdlemc2 36260 Part of proof of Lemma C in [Crawley] p. 112. (Contributed by NM, 25-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇 ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝐹𝑄) ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))

Theoremcdlemc3 36261 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → ((𝐹𝑃) (𝑄 (𝑅𝐹)) → 𝑄 (𝑃 (𝐹𝑃))))

Theoremcdlemc4 36262 Part of proof of Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝑄 (𝑅𝐹)) ≠ ((𝐹𝑃) ((𝑃 𝑄) 𝑊)))

Theoremcdlemc5 36263 Lemma for cdlemc 36265. (Contributed by NM, 26-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (¬ 𝑄 (𝑃 (𝐹𝑃)) ∧ (𝐹𝑃) ≠ 𝑃)) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))

Theoremcdlemc6 36264 Lemma for cdlemc 36265. (Contributed by NM, 26-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ (𝐹𝑃) = 𝑃) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))

Theoremcdlemc 36265 Lemma C in [Crawley] p. 113. (Contributed by NM, 26-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇 ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) → (𝐹𝑄) = ((𝑄 (𝑅𝐹)) ((𝐹𝑃) ((𝑃 𝑄) 𝑊))))

Theoremcdlemd1 36266 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑅 = ((𝑃 ((𝑃 𝑅) 𝑊)) (𝑄 ((𝑄 𝑅) 𝑊))))

Theoremcdlemd2 36267 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 29-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd3 36268 Part of proof of Lemma D in [Crawley] p. 113. The 𝑅𝑃 requirement is not mentioned in their proof. (Contributed by NM, 29-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ (𝑅𝐴𝑆𝐴 ∧ ¬ 𝑆 (𝑃 𝑄))) → ¬ 𝑅 (𝑃 𝑆))

Theoremcdlemd4 36269 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ 𝑅𝑃)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd5 36270 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 30-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑃𝑄) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑄) = (𝐺𝑄))) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd6 36271 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 31-May-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃))) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑄) = (𝐺𝑄))

Theoremcdlemd7 36272 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ ¬ 𝑄 (𝑃 (𝐹𝑃)))) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd8 36273 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 1-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ ((𝐹𝑃) = (𝐺𝑃) ∧ (𝐹𝑃) = 𝑃)) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd9 36274 Part of proof of Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ 𝑅𝐴) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑃) = (𝐺𝑃)) → (𝐹𝑅) = (𝐺𝑅))

Theoremcdlemd 36275 If two translations agree at any atom not under the fiducial co-atom 𝑊, then they are equal. Lemma D in [Crawley] p. 113. (Contributed by NM, 2-Jun-2012.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝐹𝑃) = (𝐺𝑃)) → 𝐹 = 𝐺)

Theoremltrneq3 36276 Two translations agree at any atom not under the fiducial co-atom 𝑊 iff they are equal. (Contributed by NM, 25-Jul-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊)) → ((𝐹𝑃) = (𝐺𝑃) ↔ 𝐹 = 𝐺))

Theoremcdleme00a 36277 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)       ((𝐾 ∈ HL ∧ (𝑃𝐴𝑄𝐴𝑅𝐴) ∧ ¬ 𝑅 (𝑃 𝑄)) → 𝑅𝑃)

Theoremcdleme0aa 36278 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑃𝐴𝑄𝐴) → 𝑈𝐵)

Theoremcdleme0a 36279 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄)) → 𝑈𝐴)

Theoremcdleme0b 36280 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) → 𝑈𝑃)

Theoremcdleme0c 36281 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 12-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → 𝑈𝑅)

Theoremcdleme0cp 36282 Part of proof of Lemma E in [Crawley] p. 113. TODO: Reformat as in cdlemg3a 36665- swap consequent equality; make antecedent use df-3an 1113. (Contributed by NM, 13-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴)) → (𝑃 𝑈) = (𝑃 𝑄))

Theoremcdleme0cq 36283 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 25-Apr-2013.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊))) → (𝑄 𝑈) = (𝑃 𝑄))

Theoremcdleme0dN 36284 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑅) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑅𝐴𝑃𝑅)) → 𝑉𝐴)

Theoremcdleme0e 36285 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 13-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑅) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄 ∧ ¬ 𝑅 (𝑃 𝑄))) → 𝑈𝑉)

Theoremcdleme0fN 36286 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑅) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑅𝐴)) → 𝑉𝑃)

Theoremcdleme0gN 36287 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 14-Jun-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝑉 = ((𝑃 𝑅) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑅𝐴) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → 𝑉𝑄)

Theoremcdlemeulpq 36288 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Dec-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴)) → 𝑈 (𝑃 𝑄))

Theoremcdleme01N 36289 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 5-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ((𝑈𝑃𝑈𝑄𝑈 (𝑃 𝑄)) ∧ 𝑈 𝑊))

Theoremcdleme02N 36290 Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ((𝑃 𝑈) = (𝑄 𝑈) ∧ 𝑈 𝑊))

Theoremcdleme0ex1N 36291* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴) ∧ 𝑃𝑄) → ∃𝑢𝐴 (𝑢 (𝑃 𝑄) ∧ 𝑢 𝑊))

Theoremcdleme0ex2N 36292* Part of proof of Lemma E in [Crawley] p. 113. Note that (𝑃 𝑢) = (𝑄 𝑢) is a shorter way to express 𝑢𝑃𝑢𝑄𝑢 (𝑃 𝑄). (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) ∧ 𝑃𝑄) → ∃𝑢𝐴 ((𝑃 𝑢) = (𝑄 𝑢) ∧ 𝑢 𝑊))

Theoremcdleme0moN 36293* Part of proof of Lemma E in [Crawley] p. 113. (Contributed by NM, 9-Nov-2012.) (New usage is discouraged.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑃𝑄𝑅 (𝑃 𝑄) ∧ ∃*𝑟(𝑟𝐴 ∧ (𝑃 𝑟) = (𝑄 𝑟)))) → (𝑅 = 𝑃𝑅 = 𝑄))

Theoremcdleme1b 36294 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.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &   𝐵 = (Base‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴𝑅𝐴)) → 𝐹𝐵)

Theoremcdleme1 36295 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.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → (𝑅 𝐹) = (𝑅 𝑈))

Theoremcdleme2 36296 Part of proof of Lemma E in [Crawley] p. 113. 𝐹 represents f(r). 𝑊 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.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → ((𝑅 𝐹) 𝑊) = 𝑈)

Theoremcdleme3b 36297 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 36304 and cdleme3 36305. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐹𝑅)

Theoremcdleme3c 36298 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 36304 and cdleme3 36305. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &    0 = (0.‘𝐾)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑄𝐴𝑃𝑄) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊))) → 𝐹0 )

Theoremcdleme3d 36299 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 36304 and cdleme3 36305. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &   𝑉 = ((𝑃 𝑅) 𝑊)       𝐹 = ((𝑅 𝑈) (𝑄 𝑉))

Theoremcdleme3e 36300 Part of proof of Lemma E in [Crawley] p. 113. Lemma leading to cdleme3fa 36304 and cdleme3 36305. (Contributed by NM, 6-Jun-2012.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((𝑃 𝑄) 𝑊)    &   𝐹 = ((𝑅 𝑈) (𝑄 ((𝑃 𝑅) 𝑊)))    &   𝑉 = ((𝑃 𝑅) 𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ 𝑄𝐴 ∧ (𝑅𝐴 ∧ ¬ 𝑅 (𝑃 𝑄)))) → 𝑉𝐴)

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