 Home Metamath Proof ExplorerTheorem List (p. 353 of 425) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-26947) Hilbert Space Explorer (26948-28472) Users' Mathboxes (28473-42426)

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))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
 Copyright terms: Public domain < Previous  Next >