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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | tendospass 41001 | Associative law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇)) → ((𝑈 ∘ 𝑉)‘𝐹) = (𝑈‘(𝑉‘𝐹))) | ||
Theorem | tendospdi1 41002 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑈 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → (𝑈‘(𝐹 ∘ 𝐺)) = ((𝑈‘𝐹) ∘ (𝑈‘𝐺))) | ||
Theorem | tendocnv 41003 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ◡(𝑆‘𝐹) = (𝑆‘◡𝐹)) | ||
Theorem | tendospdi2 41004* | Reverse distributive law for endomorphism scalar product operation. (Contributed by NM, 10-Oct-2013.) |
⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑃 = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) ⇒ ⊢ ((𝑈 ∈ 𝐸 ∧ 𝑉 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇) → ((𝑈𝑃𝑉)‘𝐹) = ((𝑈‘𝐹) ∘ (𝑉‘𝐹))) | ||
Theorem | tendospcanN 41005* | Cancellation law for trace-preserving 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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) ∧ (𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇)) → ((𝑆‘𝐹) = (𝑆‘𝐺) ↔ 𝐹 = 𝐺)) | ||
Theorem | dvaabl 41006 | 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 ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Abel) | ||
Theorem | dvalveclem 41007 | Lemma for dvalvec 41008. (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) & ⊢ + = (+g‘𝑈) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ ⨣ = (+g‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec) | ||
Theorem | dvalvec 41008 | 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 ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec) | ||
Theorem | dva0g 41009 | 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 = (0g‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ( I ↾ 𝐵)) | ||
Syntax | cdia 41010 | Extend class notation with partial isomorphism A. |
class DIsoA | ||
Definition | df-disoa 41011* | 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}))) | ||
Theorem | diaffval 41012* | The partial isomorphism A for a lattice 𝐾. (Contributed by NM, 15-Oct-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (DIsoA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑤} ↦ {𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ∣ (((trL‘𝐾)‘𝑤)‘𝑓) ≤ 𝑥}))) | ||
Theorem | diafval 41013* | 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ {𝑦 ∈ 𝐵 ∣ 𝑦 ≤ 𝑊} ↦ {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑥})) | ||
Theorem | diaval 41014* | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = {𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋}) | ||
Theorem | diaelval 41015 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐹 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋))) | ||
Theorem | diafn 41016* | 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) | ||
Theorem | diadm 41017* | Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) | ||
Theorem | diaeldm 41018 | 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) | ||
Theorem | diadmclN 41019 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ 𝐵) | ||
Theorem | diadmleN 41020 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ≤ 𝑊) | ||
Theorem | dian0 41021 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | ||
Theorem | dia0eldmN 41022 | 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 ∧ 𝑊 ∈ 𝐻) → 0 ∈ dom 𝐼) | ||
Theorem | dia1eldmN 41023 | 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 ∧ 𝑊 ∈ 𝐻) → 𝑊 ∈ dom 𝐼) | ||
Theorem | diass 41024 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑇) | ||
Theorem | diael 41025 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ (𝐼‘𝑋)) → 𝐹 ∈ 𝑇) | ||
Theorem | diatrl 41026 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝐹 ∈ (𝐼‘𝑋)) → (𝑅‘𝐹) ≤ 𝑋) | ||
Theorem | diaelrnN 41027 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇) | ||
Theorem | dialss 41028 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆) | ||
Theorem | diaord 41029 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) | ||
Theorem | dia11N 41030 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | diaf11N 41031 | 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 ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) | ||
Theorem | diaclN 41032 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼) | ||
Theorem | diacnvclN 41033 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ dom 𝐼) | ||
Theorem | dia0 41034 | 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 ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {( I ↾ 𝐵)}) | ||
Theorem | dia1N 41035 | 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 ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇) | ||
Theorem | dia1elN 41036 | 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 ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼) | ||
Theorem | diaglbN 41037* | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) | ||
Theorem | diameetN 41038 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
Theorem | diainN 41039 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (𝑋 ∩ 𝑌) = (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)))) | ||
Theorem | diaintclN 41040 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼) | ||
Theorem | diasslssN 41041 | 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 ∧ 𝑊 ∈ 𝐻) → ran 𝐼 ⊆ 𝑆) | ||
Theorem | diassdvaN 41042 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑉) | ||
Theorem | dia1dim 41043* | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)}) | ||
Theorem | dia1dim2 41044 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{𝐹})) | ||
Theorem | dia1dimid 41045 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹))) | ||
Theorem | dia2dimlem1 41046 | Lemma for dia2dim 41059. 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) ⇒ ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) | ||
Theorem | dia2dimlem2 41047 | Lemma for dia2dim 41059. 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑇) & ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) ⇒ ⊢ (𝜑 → (𝑅‘𝐺) = 𝑈) | ||
Theorem | dia2dimlem3 41048 | Lemma for dia2dim 41059. 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑇) & ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) ⇒ ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉) | ||
Theorem | dia2dimlem4 41049 | Lemma for dia2dim 41059. 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → 𝐺 ∈ 𝑇) & ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) & ⊢ (𝜑 → 𝐷 ∈ 𝑇) & ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) ⇒ ⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹) | ||
Theorem | dia2dimlem5 41050 | Lemma for dia2dim 41059. The sum of vectors 𝐺 and 𝐷 belongs to the sum of the subspaces generated by them. Thus, 𝐹 = (𝐺 ∘ 𝐷) 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) & ⊢ (𝜑 → 𝐺 ∈ 𝑇) & ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄) & ⊢ (𝜑 → 𝐷 ∈ 𝑇) & ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃)) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem6 41051 | Lemma for dia2dim 41059. 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃)) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem7 41052 | Lemma for dia2dim 41059. Eliminate (𝐹‘𝑃) ≠ 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) ∧ ((𝐹‘𝑃) ∨ 𝑉)) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem8 41053 | Lemma for dia2dim 41059. 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈) & ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem9 41054 | Lemma for dia2dim 41059. Eliminate (𝑅‘𝐹) ≠ 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem10 41055 | Lemma for dia2dim 41059. 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉))) ⇒ ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉)) | ||
Theorem | dia2dimlem11 41056 | Lemma for dia2dim 41059. Convert ordering hypothesis on 𝑅‘𝐹 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → 𝐹 ∈ 𝑇) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) & ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉))) ⇒ ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem12 41057 | Lemma for dia2dim 41059. 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) & ⊢ (𝜑 → 𝑈 ≠ 𝑉) ⇒ ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dimlem13 41058 | Lemma for dia2dim 41059. Eliminate 𝑈 ≠ 𝑉 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Theorem | dia2dim 41059 | 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 ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊)) & ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉))) | ||
Syntax | cdvh 41060 | Extend class notation with constructed full vector space H. |
class DVecH | ||
Definition | df-dvech 41061* | 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)) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}))) | ||
Theorem | dvhfset 41062* | 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) ⇒ ⊢ (𝐾 ∈ 𝑉 → (DVecH‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({〈(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))〉, 〈(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉}))) | ||
Theorem | dvhset 41063* | 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) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({〈(Base‘ndx), (𝑇 × 𝐸)〉, 〈(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))〉)〉, 〈(Scalar‘ndx), 𝐷〉} ∪ {〈( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)〉})) | ||
Theorem | dvhsca 41064 | 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) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷) | ||
Theorem | dvhbase 41065 | 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) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸) | ||
Theorem | dvhfplusr 41066* | 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) & ⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) & ⊢ ✚ = (+g‘𝐹) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ✚ = + ) | ||
Theorem | dvhfmulr 41067* | 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) & ⊢ · = (.r‘𝐹) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡))) | ||
Theorem | dvhmulr 41068 | 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) & ⊢ · = (.r‘𝐹) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → (𝑅 · 𝑆) = (𝑅 ∘ 𝑆)) | ||
Theorem | dvhvbase 41069 | 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) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑉 = (𝑇 × 𝐸)) | ||
Theorem | dvhelvbasei 41070 | 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) ⇒ ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)) → 〈𝐹, 𝑆〉 ∈ 𝑉) | ||
Theorem | dvhvaddcbv 41071* | Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.) |
⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) ⇒ ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))〉) | ||
Theorem | dvhvaddval 41072* | The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) |
⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) ⇒ ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) | ||
Theorem | dvhfvadd 41073* | 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) & ⊢ ⨣ = (+g‘𝐷) & ⊢ ✚ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))〉) & ⊢ + = (+g‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ ) | ||
Theorem | dvhvadd 41074 | 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) & ⊢ + = (+g‘𝑈) & ⊢ ⨣ = (+g‘𝐷) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = 〈((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))〉) | ||
Theorem | dvhopvadd 41075 | 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) & ⊢ + = (+g‘𝑈) & ⊢ ⨣ = (+g‘𝐷) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (〈𝐹, 𝑄〉 + 〈𝐺, 𝑅〉) = 〈(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)〉) | ||
Theorem | dvhopvadd2 41076* | The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 41075 and/or dvhfplusr 41066. (Contributed by NM, 26-Sep-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓)))) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ✚ = (+g‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (〈𝐹, 𝑄〉 ✚ 〈𝐺, 𝑅〉) = 〈(𝐹 ∘ 𝐺), (𝑄 + 𝑅)〉) | ||
Theorem | dvhvaddcl 41077 | 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) & ⊢ ⨣ = (+g‘𝐷) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸)) | ||
Theorem | dvhvaddcomN 41078 | 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) & ⊢ ⨣ = (+g‘𝐷) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = (𝐺 + 𝐹)) | ||
Theorem | dvhvaddass 41079 | 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) & ⊢ ⨣ = (+g‘𝐷) & ⊢ + = (+g‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼))) | ||
Theorem | dvhvscacbv 41080* | Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.) |
⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) ⇒ ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))〉) | ||
Theorem | dvhvscaval 41081* | The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.) |
⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) ⇒ ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = 〈(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))〉) | ||
Theorem | dvhfvsca 41082* | 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) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉)) | ||
Theorem | dvhvsca 41083 | 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) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = 〈(𝑅‘(1st ‘𝐹)), (𝑅 ∘ (2nd ‘𝐹))〉) | ||
Theorem | dvhopvsca 41084 | 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) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · 〈𝐹, 𝑋〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑋)〉) | ||
Theorem | dvhvscacl 41085 | 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) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸)) | ||
Theorem | tendoinvcl 41086* | Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 40965. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝑁 = (invr‘𝐹) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂)) | ||
Theorem | tendolinv 41087* | Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝑁 = (invr‘𝐹) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∘ 𝑆) = ( I ↾ 𝑇)) | ||
Theorem | tendorinv 41088* | Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝑁 = (invr‘𝐹) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑆 ∘ (𝑁‘𝑆)) = ( I ↾ 𝑇)) | ||
Theorem | dvhgrp 41089 | The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝐷) & ⊢ + = (+g‘𝑈) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐼 = (invg‘𝐷) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp) | ||
Theorem | dvhlveclem 41090 | Lemma for dvhlvec 41091. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does 𝜑 → method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ ⨣ = (+g‘𝐷) & ⊢ + = (+g‘𝑈) & ⊢ 0 = (0g‘𝐷) & ⊢ 𝐼 = (invg‘𝐷) & ⊢ × = (.r‘𝐷) & ⊢ · = ( ·𝑠 ‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec) | ||
Theorem | dvhlvec 41091 | The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a left module. (Contributed by NM, 23-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ LVec) | ||
Theorem | dvhlmod 41092 | The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a left module. (Contributed by NM, 23-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝑈 ∈ LMod) | ||
Theorem | dvh0g 41093* | The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = 〈( I ↾ 𝐵), 𝑂〉) | ||
Theorem | dvheveccl 41094 | Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 41098 and dihpN 41318. (Contributed by NM, 27-Mar-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐸 = 〈( I ↾ 𝐵), ( I ↾ 𝑇)〉 & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 })) | ||
Theorem | dvhopclN 41095 | Closure of a DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.) |
⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → 〈𝐹, 𝑈〉 ∈ (𝑇 × 𝐸)) | ||
Theorem | dvhopaddN 41096* | Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.) |
⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) ⇒ ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (〈𝐹, 𝑈〉𝐴〈𝐺, 𝑉〉) = 〈(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)〉) | ||
Theorem | dvhopspN 41097* | Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.) |
⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) ⇒ ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆〈𝐹, 𝑈〉) = 〈(𝑅‘𝐹), (𝑅 ∘ 𝑈)〉) | ||
Theorem | dvhopN 41098* | Decompose a DVecH vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of DVecA and the other from the one-dimensional vector subspace 𝐸. Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by 〈( I ↾ 𝐵), ( I ↾ 𝑇)〉, 𝑈, 〈𝐹, 𝑂〉. We swapped the order of vector sum (their juxtaposition i.e. composition) to show 〈𝐹, 𝑂〉 first. Note that 𝑂 and ( I ↾ 𝑇) are the zero and one of the division ring 𝐸, and ( I ↾ 𝐵) is the zero of the translation group. 𝑆 is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐)))) & ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ 〈((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))〉) & ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ 〈(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))〉) & ⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → 〈𝐹, 𝑈〉 = (〈𝐹, 𝑂〉𝐴(𝑈𝑆〈( I ↾ 𝐵), ( I ↾ 𝑇)〉))) | ||
Theorem | dvhopellsm 41099* | Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ + = (+g‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (〈𝐹, 𝑇〉 ∈ (𝑋 ⊕ 𝑌) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((〈𝑥, 𝑦〉 ∈ 𝑋 ∧ 〈𝑧, 𝑤〉 ∈ 𝑌) ∧ 〈𝐹, 𝑇〉 = (〈𝑥, 𝑦〉 + 〈𝑧, 𝑤〉)))) | ||
Theorem | cdlemm10N 41100* | The image of the map 𝐺 is the entire one-dimensional subspace (𝐼‘𝑉). Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.) |
⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐶 = {𝑟 ∈ 𝐴 ∣ (𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊)} & ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠) & ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉)) |
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