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Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdiaelrnN 37201 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 37202 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 37203 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 37204 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 37205 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 37206 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 37207 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 37208 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 37209 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 37210 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 37211* 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 37212 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 37213 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 37214 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 37215 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 37216 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 37217* 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 37218 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 37219 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 37220 Lemma for dia2dim 37233. 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 37221 Lemma for dia2dim 37233. 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 37222 Lemma for dia2dim 37233. 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 37223 Lemma for dia2dim 37233. 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 37224 Lemma for dia2dim 37233. 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 37225 Lemma for dia2dim 37233. 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 37226 Lemma for dia2dim 37233. 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 37227 Lemma for dia2dim 37233. 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 37228 Lemma for dia2dim 37233. 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 37229 Lemma for dia2dim 37233. 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 37230 Lemma for dia2dim 37233. 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 37231 Lemma for dia2dim 37233. 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 37232 Lemma for dia2dim 37233. 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 37233 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 37234 Extend class notation with constructed full vector space H.
class DVecH

Definitiondf-dvech 37235* 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 37236* 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 37237* 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 37238 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 37239 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 37240* 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 37241* 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 37242 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 37243 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 37244 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 37245* 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 37246* 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 37247* 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 37248 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 37249 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 37250* The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 37249 and/or dvhfplusr 37240. (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 37251 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 37252 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 37253 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 37254* 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 37255* 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 37256* 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 37257 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 37258 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 37259 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))

Theoremtendoinvcl 37260* 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 37139. (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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → ((𝑁𝑆) ∈ 𝐸 ∧ (𝑁𝑆) ≠ 𝑂))

Theoremtendolinv 37261* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → ((𝑁𝑆) ∘ 𝑆) = ( I ↾ 𝑇))

Theoremtendorinv 37262* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → (𝑆 ∘ (𝑁𝑆)) = ( I ↾ 𝑇))

Theoremdvhgrp 37263 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)    &    = (+g𝐷)    &    + = (+g𝑈)    &    0 = (0g𝐷)    &   𝐼 = (invg𝐷)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)

Theoremdvhlveclem 37264 Lemma for dvhlvec 37265. 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)    &    = (+g𝐷)    &    + = (+g𝑈)    &    0 = (0g𝐷)    &   𝐼 = (invg𝐷)    &    × = (.r𝐷)    &    · = ( ·𝑠𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)

Theoremdvhlvec 37265 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LVec)

Theoremdvhlmod 37266 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LMod)

Theoremdvh0g 37267* 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 = (0g𝑈)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)

Theoremdvheveccl 37268 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 37272 and dihpN 37492. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐸 = ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐸 ∈ (𝑉 ∖ { 0 }))

TheoremdvhopclN 37269 Closure of a DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))

TheoremdvhopaddN 37270* 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) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)       (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)

TheoremdvhopspN 37271* Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)       ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)

TheoremdvhopN 37272* 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 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑐𝑇 ↦ ((𝑎𝑐) ∘ (𝑏𝑐))))    &   𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)    &   𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)    &   𝑂 = (𝑐𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Theoremdvhopellsm 37273* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑆𝑌𝑆) → (⟨𝐹, 𝑇⟩ ∈ (𝑋 𝑌) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ 𝑋 ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑌) ∧ ⟨𝐹, 𝑇⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩))))

Theoremcdlemm10N 37274* The image of the map 𝐺 is the entire one-dimensional subspace (𝐼𝑉). 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 = {𝑟𝐴 ∣ (𝑟 (𝑃 𝑉) ∧ ¬ 𝑟 𝑊)}    &   𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑠)    &   𝐺 = (𝑞𝐶 ↦ (𝑓𝑇 (𝑓𝑃) = 𝑞))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ran 𝐺 = (𝐼𝑉))

SyntaxcocaN 37275 Extend class notation with subspace orthocomplement for DVecA partial vector space.
class ocA

Definitiondf-docaN 37276* Define subspace orthocomplement for DVecA partial vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 6-Dec-2013.)
ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(((DIsoA‘𝑘)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))

TheoremdocaffvalN 37277* Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))

TheoremdocafvalN 37278* Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑁 = ((ocA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))

TheoremdocavalN 37279* Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑁 = ((ocA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))

TheoremdocaclN 37280 Closure of subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)

TheoremdiaocN 37281 Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom 𝑊). (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
= (join‘𝐾)    &    = (meet‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝑁 = ((ocA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((( 𝑋) ( 𝑊)) 𝑊)) = (𝑁‘(𝐼𝑋)))

Theoremdoca2N 37282 Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ‘( ‘(𝐼𝑋))) = (𝐼𝑋))

Theoremdoca3N 37283 Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ‘( 𝑋)) = 𝑋)

TheoremdvadiaN 37284 Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed 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)    &    = ((ocA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑆 ∧ ( ‘( 𝑋)) = 𝑋)) → 𝑋 ∈ ran 𝐼)

TheoremdiarnN 37285* Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ran 𝐼 = {𝑥𝑆 ∣ ( ‘( 𝑥)) = 𝑥})

Theoremdiaf1oN 37286* The partial isomorphism A for a lattice 𝐾 is a one-to-one, onto function. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. See diadm 37191 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→{𝑥𝑆 ∣ ( ‘( 𝑥)) = 𝑥})

SyntaxcdjaN 37287 Extend class notation with subspace join for DVecA partial vector space.
class vA

Definitiondf-djaN 37288* Define (closed) subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.)
vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))))

TheoremdjaffvalN 37289* Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))

TheoremdjafvalN 37290* Subspace join for DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)    &   𝐽 = ((vA‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))

TheoremdjavalN 37291 Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &    = ((ocA‘𝐾)‘𝑊)    &   𝐽 = ((vA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))

TheoremdjaclN 37292 Closure of subspace join for DVecA partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝐽 = ((vA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼)

TheoremdjajN 37293 Transfer lattice join to DVecA partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
= (join‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   𝐽 = ((vA‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋)𝐽(𝐼𝑌)))

Syntaxcdib 37294 Extend class notation with isomorphism B.
class DIsoB

Definitiondf-dib 37295* Isomorphism B is isomorphism A extended with an extra dimension set to the zero vector component i.e. the zero endormorphism. Its domain is lattice elements less than or equal to the fiducial co-atom 𝑤. (Contributed by NM, 8-Dec-2013.)
DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))

Theoremdibffval 37296* The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))

Theoremdibfval 37297* The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))

Theoremdibval 37298* The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))

TheoremdibopelvalN 37299* Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))

Theoremdibval2 37300* Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))

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