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Type | Label | Description |
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Statement | ||
Theorem | dia1dimid 38201 | 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 38202 | Lemma for dia2dim 38215. 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 38203 | Lemma for dia2dim 38215. 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 38204 | Lemma for dia2dim 38215. 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 38205 | Lemma for dia2dim 38215. 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 38206 | Lemma for dia2dim 38215. 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 38207 | Lemma for dia2dim 38215. 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 38208 | Lemma for dia2dim 38215. 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 38209 | Lemma for dia2dim 38215. 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 38210 | Lemma for dia2dim 38215. 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 38211 | Lemma for dia2dim 38215. 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 38212 | Lemma for dia2dim 38215. 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 38213 | Lemma for dia2dim 38215. 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 38214 | Lemma for dia2dim 38215. 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 38215 | 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 38216 | Extend class notation with constructed full vector space H. |
class DVecH | ||
Definition | df-dvech 38217* | 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 38218* | 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 38219* | 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 38220 | 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 38221 | 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 38222* | 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 38223* | 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 38224 | 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 38225 | 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 38226 | 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 38227* | 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 38228* | 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 38229* | 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 38230 | 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 38231 | 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 38232* | The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 38231 and/or dvhfplusr 38222. (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 38233 | 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 38234 | 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 38235 | 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 38236* | 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 38237* | 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 38238* | 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 38239 | 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 38240 | 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 38241 | 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 38242* | 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 38121. (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 38243* | 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 38244* | 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 38245 | 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 38246 | Lemma for dvhlvec 38247. 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 38247 | 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 38248 | 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 38249* | 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 38250 | 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 38254 and dihpN 38474. (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 38251 | 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 38252* | 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 38253* | 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 38254* | 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 38255* | 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 38256* | 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)) | ||
Syntax | cocaN 38257 | Extend class notation with subspace orthocomplement for DVecA partial vector space. |
class ocA | ||
Definition | df-docaN 38258* | 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) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤))))) | ||
Theorem | docaffvalN 38259* | 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) ⇒ ⊢ (𝐾 ∈ 𝑉 → (ocA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤))))) | ||
Theorem | docafvalN 38260* | 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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))) | ||
Theorem | docavalN 38261* | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))) | ||
Theorem | docaclN 38262 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼) | ||
Theorem | diaocN 38263 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) = (𝑁‘(𝐼‘𝑋))) | ||
Theorem | doca2N 38264 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋)) | ||
Theorem | doca3N 38265 | 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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | ||
Theorem | dvadiaN 38266 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝑆 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) → 𝑋 ∈ ran 𝐼) | ||
Theorem | diarnN 38267* | 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 ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = {𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥}) | ||
Theorem | diaf1oN 38268* | 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 38173 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 ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→{𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥}) | ||
Syntax | cdjaN 38269 | Extend class notation with subspace join for DVecA partial vector space. |
class vA | ||
Definition | df-djaN 38270* | 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)))))) | ||
Theorem | djaffvalN 38271* | Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (vA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦)))))) | ||
Theorem | djafvalN 38272* | 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) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))) | ||
Theorem | djavalN 38273 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) | ||
Theorem | djaclN 38274 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼) | ||
Theorem | djajN 38275 | 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 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) | ||
Syntax | cdib 38276 | Extend class notation with isomorphism B. |
class DIsoB | ||
Definition | df-dib 38277* | 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)))})))) | ||
Theorem | dibffval 38278* | The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))})))) | ||
Theorem | dibfval 38279* | 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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 }))) | ||
Theorem | dibval 38280* | 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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) | ||
Theorem | dibopelvalN 38281* | 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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))) | ||
Theorem | dibval2 38282* | 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 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 })) | ||
Theorem | dibopelval2 38283* | Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 ))) | ||
Theorem | dibval3N 38284* | Value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ({𝑓 ∈ 𝑇 ∣ (𝑅‘𝑓) ≤ 𝑋} × { 0 })) | ||
Theorem | dibelval3 38285* | Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝑌 ∈ (𝐼‘𝑋) ↔ ∃𝑓 ∈ 𝑇 (𝑌 = 〈𝑓, 0 〉 ∧ (𝑅‘𝑓) ≤ 𝑋))) | ||
Theorem | dibopelval3 38286* | Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 0 = (𝑔 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (〈𝐹, 𝑆〉 ∈ (𝐼‘𝑋) ↔ ((𝐹 ∈ 𝑇 ∧ (𝑅‘𝐹) ≤ 𝑋) ∧ 𝑆 = 0 ))) | ||
Theorem | dibelval1st 38287 | Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ (𝐽‘𝑋)) | ||
Theorem | dibelval1st1 38288 | Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (1st ‘𝑌) ∈ 𝑇) | ||
Theorem | dibelval1st2N 38289 | Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (𝑅‘(1st ‘𝑌)) ≤ 𝑋) | ||
Theorem | dibelval2nd 38290* | Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ (𝐼‘𝑋)) → (2nd ‘𝑌) = 0 ) | ||
Theorem | dibn0 38291 | The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ≠ ∅) | ||
Theorem | dibfna 38292 | Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐽) | ||
Theorem | dibdiadm 38293 | Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = dom 𝐽) | ||
Theorem | dibfnN 38294* | Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) | ||
Theorem | dibdmN 38295* | Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → dom 𝐼 = {𝑥 ∈ 𝐵 ∣ 𝑥 ≤ 𝑊}) | ||
Theorem | dibeldmN 38296 | Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))) | ||
Theorem | dibord 38297 | The isomorphism B for a lattice 𝐾 is order-preserving in the region under co-atom 𝑊. (Contributed by NM, 24-Feb-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌)) | ||
Theorem | dib11N 38298 | The isomorphism B for a lattice 𝐾 is one-to-one in the region under co-atom 𝑊. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌)) | ||
Theorem | dibf11N 38299 | 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 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼) | ||
Theorem | dibclN 38300 | Closure of partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼) |
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