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Theorem List for Metamath Proof Explorer - 35201-35300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdvaset 35201* The constructed partial vector space A for a lattice 𝐾. (Contributed by NM, 8-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), 𝑇⟩, ⟨(+g‘ndx), (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔))⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓))⟩}))
 
Theoremdvasca 35202 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom 𝑊). (Contributed by NM, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)       ((𝐾𝑋𝑊𝐻) → 𝐹 = 𝐷)
 
Theoremdvabase 35203 The ring base set of the constructed partial vector space A are all translation group endomorphisms (for a fiducial co-atom 𝑊). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐶 = (Base‘𝐹)       ((𝐾𝑋𝑊𝐻) → 𝐶 = 𝐸)
 
Theoremdvafplusg 35204* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (+g𝐹)       ((𝐾𝑉𝑊𝐻) → + = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓)))))
 
Theoremdvaplusg 35205* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (+g𝐹)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝑆𝐸)) → (𝑅 + 𝑆) = (𝑓𝑇 ↦ ((𝑅𝑓) ∘ (𝑆𝑓))))
 
Theoremdvaplusgv 35206 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (+g𝐹)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝑆𝐸𝐺𝑇)) → ((𝑅 + 𝑆)‘𝐺) = ((𝑅𝐺) ∘ (𝑆𝐺)))
 
Theoremdvafmulr 35207* Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    · = (.r𝐹)       ((𝐾𝑉𝑊𝐻) → · = (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡)))
 
Theoremdvamulr 35208 Ring multiplication operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    · = (.r𝐹)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝑆𝐸)) → (𝑅 · 𝑆) = (𝑅𝑆))
 
Theoremdvavbase 35209 The vectors (vector base set) of the constructed partial vector space A are all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       ((𝐾𝑋𝑊𝐻) → 𝑉 = 𝑇)
 
Theoremdvafvadd 35210* The vector sum operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &    + = (+g𝑈)       ((𝐾𝑋𝑊𝐻) → + = (𝑓𝑇, 𝑔𝑇 ↦ (𝑓𝑔)))
 
Theoremdvavadd 35211 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &    + = (+g𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝐹𝑇𝐺𝑇)) → (𝐹 + 𝐺) = (𝐹𝐺))
 
Theoremdvafvsca 35212* Ring addition operation for the constructed partial vector space A. (Contributed by NM, 9-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       ((𝐾𝑉𝑊𝐻) → · = (𝑠𝐸, 𝑓𝑇 ↦ (𝑠𝑓)))
 
Theoremdvavsca 35213 Ring addition operation for the constructed partial vector space A. (Contributed by NM, 11-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝐹𝑇)) → (𝑅 · 𝐹) = (𝑅𝐹))
 
Theoremtendospid 35214 Identity property of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
(𝐹𝑇 → (( I ↾ 𝑇)‘𝐹) = 𝐹)
 
Theoremtendospcl 35215 Closure of endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑈𝐸𝐹𝑇) → (𝑈𝐹) ∈ 𝑇)
 
Theoremtendospass 35216 Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾𝑋𝑊𝐻) ∧ (𝑈𝐸𝑉𝐸𝐹𝑇)) → ((𝑈𝑉)‘𝐹) = (𝑈‘(𝑉𝐹)))
 
Theoremtendospdi1 35217 Forward distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑈𝐸𝐹𝑇𝐺𝑇)) → (𝑈‘(𝐹𝐺)) = ((𝑈𝐹) ∘ (𝑈𝐺)))
 
Theoremtendocnv 35218 Converse of a trace-preserving endomorphism value. (Contributed by NM, 7-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝐹𝑇) → (𝑆𝐹) = (𝑆𝐹))
 
Theoremtendospdi2 35219* Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.)
𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑃 = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))       ((𝑈𝐸𝑉𝐸𝐹𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈𝐹) ∘ (𝑉𝐹)))
 
TheoremtendospcanN 35220* Cancellation law for trace-perserving endomorphism values (used as scalar product). (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆𝐸𝑆𝑂) ∧ (𝐹𝑇𝐺𝑇)) → ((𝑆𝐹) = (𝑆𝐺) ↔ 𝐹 = 𝐺))
 
Theoremdvaabl 35221 The constructed partial vector space A for a lattice 𝐾 is an abelian group. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Abel)
 
Theoremdvalveclem 35222 Lemma for dvalvec 35223. (Contributed by NM, 11-Oct-2013.) (Proof shortened by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &   𝐵 = (Base‘𝐾)    &    = (+g𝐷)    &    × = (.r𝐷)    &    · = ( ·𝑠𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)
 
Theoremdvalvec 35223 The constructed partial vector space A for a lattice 𝐾 is a left vector space. (Contributed by NM, 11-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)
 
Theoremdva0g 35224 The zero vector of partial vector space A. (Contributed by NM, 9-Sep-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &    0 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ( I ↾ 𝐵))
 
Syntaxcdia 35225 Extend class notation with partial isomorphism A.
class DIsoA
 
Definitiondf-disoa 35226* Define partial isomorphism A. (Contributed by NM, 15-Oct-2013.)
DIsoA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ {𝑦 ∈ (Base‘𝑘) ∣ 𝑦(le‘𝑘)𝑤} ↦ {𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ∣ (((trL‘𝑘)‘𝑤)‘𝑓)(le‘𝑘)𝑥})))
 
Theoremdiaffval 35227* The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ {𝑦𝐵𝑦 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) 𝑥})))
 
Theoremdiafval 35228* The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ {𝑦𝐵𝑦 𝑊} ↦ {𝑓𝑇 ∣ (𝑅𝑓) 𝑥}))
 
Theoremdiaval 35229* The partial isomorphism A for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 120 line 24. (Contributed by NM, 15-Oct-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = {𝑓𝑇 ∣ (𝑅𝑓) 𝑋})
 
Theoremdiaelval 35230 Member of the partial isomorphism A for a lattice 𝐾. (Contributed by NM, 3-Dec-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐹 ∈ (𝐼𝑋) ↔ (𝐹𝑇 ∧ (𝑅𝐹) 𝑋)))
 
Theoremdiafn 35231* Functionality and domain of the partial isomorphism A. (Contributed by NM, 26-Nov-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
 
Theoremdiadm 35232* Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
 
Theoremdiaeldm 35233 Member of domain of the partial isomorphism A. (Contributed by NM, 4-Dec-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))
 
TheoremdiadmclN 35234 A member of domain of the partial isomorphism A is a lattice element. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋𝐵)
 
TheoremdiadmleN 35235 A member of domain of the partial isomorphism A is under the fiducial hyperplane. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 𝑊)
 
Theoremdian0 35236 The value of the partial isomorphism A is not empty. (Contributed by NM, 17-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
 
Theoremdia0eldmN 35237 The lattice zero belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 ∈ dom 𝐼)
 
Theoremdia1eldmN 35238 The fiducial hyperplane (the largest allowed lattice element) belongs to the domain of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑊 ∈ dom 𝐼)
 
Theoremdiass 35239 The value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑇)
 
Theoremdiael 35240 A member of the value of the partial isomorphism A is a translation i.e. a vector. (Contributed by NM, 17-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝐹 ∈ (𝐼𝑋)) → 𝐹𝑇)
 
Theoremdiatrl 35241 Trace of a member of the partial isomorphism A. (Contributed by NM, 17-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝐹 ∈ (𝐼𝑋)) → (𝑅𝐹) 𝑋)
 
TheoremdiaelrnN 35242 Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆𝑇)
 
Theoremdialss 35243 The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
 
Theoremdiaord 35244 The partial isomorphism A for a lattice 𝐾 is order-preserving in the region under co-atom 𝑊. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))
 
Theoremdia11N 35245 The partial isomorphism A for a lattice 𝐾 is one-to-one in the region under co-atom 𝑊. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → ((𝐼𝑋) = (𝐼𝑌) ↔ 𝑋 = 𝑌))
 
Theoremdiaf11N 35246 The partial isomorphism A for a lattice 𝐾 is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
 
TheoremdiaclN 35247 Closure of partial isomorphism A for a lattice 𝐾. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ ran 𝐼)
 
TheoremdiacnvclN 35248 Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝐼𝑋) ∈ dom 𝐼)
 
Theoremdia0 35249 The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
𝐵 = (Base‘𝐾)    &    0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼0 ) = {( I ↾ 𝐵)})
 
Theoremdia1N 35250 The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼𝑊) = 𝑇)
 
Theoremdia1elN 35251 The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑇 ∈ ran 𝐼)
 
TheoremdiaglbN 35252* Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆 ⊆ dom 𝐼𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))
 
TheoremdiameetN 35253 Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
= (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋) ∩ (𝐼𝑌)))
 
TheoremdiainN 35254 Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
= (meet‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ ran 𝐼𝑌 ∈ ran 𝐼)) → (𝑋𝑌) = (𝐼‘((𝐼𝑋) (𝐼𝑌))))
 
TheoremdiaintclN 35255 The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆 ⊆ ran 𝐼𝑆 ≠ ∅)) → 𝑆 ∈ ran 𝐼)
 
TheoremdiasslssN 35256 The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ran 𝐼𝑆)
 
TheoremdiassdvaN 35257 The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       (((𝐾𝑌𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑉)
 
Theoremdia1dim 35258* Two expressions for the 1-dimensional subspaces of partial vector space A (when 𝐹 is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = {𝑔 ∣ ∃𝑠𝐸 𝑔 = (𝑠𝐹)})
 
Theoremdia1dim2 35259 Two expressions for a 1-dimensional subspace of partial vector space A (when 𝐹 is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = (𝑁‘{𝐹}))
 
Theoremdia1dimid 35260 A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → 𝐹 ∈ (𝐼‘(𝑅𝐹)))
 
Theoremdia2dimlem1 35261 Lemma for dia2dim 35274. Show properties of the auxiliary atom 𝑄. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)       (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))
 
Theoremdia2dimlem2 35262 Lemma for dia2dim 35274. Define a translation 𝐺 whose trace is atom 𝑈. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)    &   (𝜑𝐺𝑇)    &   (𝜑 → (𝐺𝑃) = 𝑄)       (𝜑 → (𝑅𝐺) = 𝑈)
 
Theoremdia2dimlem3 35263 Lemma for dia2dim 35274. Define a translation 𝐷 whose trace is atom 𝑉. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)    &   (𝜑𝐷𝑇)    &   (𝜑 → (𝐷𝑄) = (𝐹𝑃))       (𝜑 → (𝑅𝐷) = 𝑉)
 
Theoremdia2dimlem4 35264 Lemma for dia2dim 35274. Show that the composition (sum) of translations (vectors) 𝐺 and 𝐷 equals 𝐹. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑𝐺𝑇)    &   (𝜑 → (𝐺𝑃) = 𝑄)    &   (𝜑𝐷𝑇)    &   (𝜑 → (𝐷𝑄) = (𝐹𝑃))       (𝜑 → (𝐷𝐺) = 𝐹)
 
Theoremdia2dimlem5 35265 Lemma for dia2dim 35274. The sum of vectors 𝐺 and 𝐷 belongs to the sum of the subspaces generated by them. Thus, 𝐹 = (𝐺𝐷) belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)    &   (𝜑𝐺𝑇)    &   (𝜑 → (𝐺𝑃) = 𝑄)    &   (𝜑𝐷𝑇)    &   (𝜑 → (𝐷𝑄) = (𝐹𝑃))       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem6 35266 Lemma for dia2dim 35274. Eliminate auxiliary translations 𝐺 and 𝐷. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝐹𝑇 ∧ (𝐹𝑃) ≠ 𝑃))    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem7 35267 Lemma for dia2dim 35274. Eliminate (𝐹𝑃) ≠ 𝑃 condition. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑄 = ((𝑃 𝑈) ((𝐹𝑃) 𝑉))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem8 35268 Lemma for dia2dim 35274. Eliminate no-longer used auxiliary atoms 𝑃 and 𝑄. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)    &   (𝜑 → (𝑅𝐹) ≠ 𝑈)    &   (𝜑 → (𝑅𝐹) ≠ 𝑉)       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem9 35269 Lemma for dia2dim 35274. Eliminate (𝑅𝐹) ≠ 𝑈, 𝑉 conditions. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑 → (𝑅𝐹) (𝑈 𝑉))    &   (𝜑𝑈𝑉)       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem10 35270 Lemma for dia2dim 35274. Convert membership in closed subspace (𝐼‘(𝑈 𝑉)) to a lattice ordering. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑𝐹 ∈ (𝐼‘(𝑈 𝑉)))       (𝜑 → (𝑅𝐹) (𝑈 𝑉))
 
Theoremdia2dimlem11 35271 Lemma for dia2dim 35274. Convert ordering hypothesis on 𝑅𝐹 to subspace membership 𝐹 ∈ (𝐼‘(𝑈 𝑉)). (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑𝐹𝑇)    &   (𝜑𝑈𝑉)    &   (𝜑𝐹 ∈ (𝐼‘(𝑈 𝑉)))       (𝜑𝐹 ∈ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem12 35272 Lemma for dia2dim 35274. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))    &   (𝜑𝑈𝑉)       (𝜑 → (𝐼‘(𝑈 𝑉)) ⊆ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dimlem13 35273 Lemma for dia2dim 35274. Eliminate 𝑈𝑉 condition. (Contributed by NM, 8-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑌)    &    = (LSSum‘𝑌)    &   𝑁 = (LSpan‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))       (𝜑 → (𝐼‘(𝑈 𝑉)) ⊆ ((𝐼𝑈) (𝐼𝑉)))
 
Theoremdia2dim 35274 A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)
= (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑌 = ((DVecA‘𝐾)‘𝑊)    &    = (LSSum‘𝑌)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑈𝐴𝑈 𝑊))    &   (𝜑 → (𝑉𝐴𝑉 𝑊))       (𝜑 → (𝐼‘(𝑈 𝑉)) ⊆ ((𝐼𝑈) (𝐼𝑉)))
 
Syntaxcdvh 35275 Extend class notation with constructed full vector space H.
class DVecH
 
Definitiondf-dvech 35276* Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
DVecH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
 
Theoremdvhfset 35277* The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DVecH‘𝐾) = (𝑤𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ( ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩})))
 
Theoremdvhset 35278* The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), (𝑇 ↦ (((2nd𝑓)‘) ∘ ((2nd𝑔)‘)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)⟩}))
 
Theoremdvhsca 35279 The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐷 = ((EDRing‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)       ((𝐾𝑋𝑊𝐻) → 𝐹 = 𝐷)
 
Theoremdvhbase 35280 The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝐶 = (Base‘𝐹)       ((𝐾𝑋𝑊𝐻) → 𝐶 = 𝐸)
 
Theoremdvhfplusr 35281* Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    + = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))    &    = (+g𝐹)       ((𝐾𝑉𝑊𝐻) → = + )
 
Theoremdvhfmulr 35282* Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    · = (.r𝐹)       ((𝐾𝑉𝑊𝐻) → · = (𝑠𝐸, 𝑡𝐸 ↦ (𝑠𝑡)))
 
Theoremdvhmulr 35283 Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &    · = (.r𝐹)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝑆𝐸)) → (𝑅 · 𝑆) = (𝑅𝑆))
 
Theoremdvhvbase 35284 The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       ((𝐾𝑋𝑊𝐻) → 𝑉 = (𝑇 × 𝐸))
 
Theoremdvhelvbasei 35285 Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       (((𝐾𝑋𝑊𝐻) ∧ (𝐹𝑇𝑆𝐸)) → ⟨𝐹, 𝑆⟩ ∈ 𝑉)
 
Theoremdvhvaddcbv 35286* Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
+ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)        + = ( ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st) ∘ (1st𝑖)), ((2nd) (2nd𝑖))⟩)
 
Theoremdvhvaddval 35287* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
+ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)       ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
 
Theoremdvhfvadd 35288* The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    = (+g𝐷)    &    = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓) (2nd𝑔))⟩)    &    + = (+g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → + = )
 
Theoremdvhvadd 35289 The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    + = (+g𝑈)    &    = (+g𝐷)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st𝐹) ∘ (1st𝐺)), ((2nd𝐹) (2nd𝐺))⟩)
 
Theoremdvhopvadd 35290 The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    + = (+g𝑈)    &    = (+g𝐷)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (⟨𝐹, 𝑄+𝐺, 𝑅⟩) = ⟨(𝐹𝐺), (𝑄 𝑅)⟩)
 
Theoremdvhopvadd2 35291* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 35290 and/or dvhfplusr 35281. (Contributed by NM, 26-Sep-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (𝑠𝐸, 𝑡𝐸 ↦ (𝑓𝑇 ↦ ((𝑠𝑓) ∘ (𝑡𝑓))))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑄𝐸) ∧ (𝐺𝑇𝑅𝐸)) → (⟨𝐹, 𝑄𝐺, 𝑅⟩) = ⟨(𝐹𝐺), (𝑄 + 𝑅)⟩)
 
Theoremdvhvaddcl 35292 Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    = (+g𝐷)    &    + = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
 
TheoremdvhvaddcomN 35293 Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    = (+g𝐷)    &    + = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = (𝐺 + 𝐹))
 
Theoremdvhvaddass 35294 Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐷 = (Scalar‘𝑈)    &    = (+g𝐷)    &    + = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))
 
Theoremdvhvscacbv 35295* Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
· = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)        · = (𝑡𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st𝑔)), (𝑡 ∘ (2nd𝑔))⟩)
 
Theoremdvhvscaval 35296* The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
· = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)       ((𝑈𝐸𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st𝐹)), (𝑈 ∘ (2nd𝐹))⟩)
 
Theoremdvhfvsca 35297* Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       ((𝐾𝑉𝑊𝐻) → · = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩))
 
Theoremdvhvsca 35298 Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1st𝐹)), (𝑅 ∘ (2nd𝐹))⟩)
 
Theoremdvhopvsca 35299 Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝑅𝐸𝐹𝑇𝑋𝐸)) → (𝑅 ·𝐹, 𝑋⟩) = ⟨(𝑅𝐹), (𝑅𝑋)⟩)
 
Theoremdvhvscacl 35300 Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑅𝐸𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸))
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