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| Type | Label | Description |
|---|---|---|
| Statement | ||
| Theorem | dihmeetlem2N 41301 | Isomorphism H of a conjunction. (Contributed by NM, 22-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑞) & ⊢ 0 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihglbcpreN 41302* | Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 20-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝐹 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑞) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ ¬ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) | ||
| Theorem | dihglbcN 41303* | Isomorphism H of a lattice glb when the glb is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ≤ = (le‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅) ∧ ¬ (𝐺‘𝑆) ≤ 𝑊) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) | ||
| Theorem | dihmeetcN 41304 | Isomorphism H of a lattice meet when the meet is not under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ¬ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetbN 41305 | Isomorphism H of a lattice meet when one element is under the fiducial hyperplane 𝑊. (Contributed by NM, 26-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetbclemN 41306 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊) → (𝐼‘(𝑋 ∧ 𝑌)) = (((𝐼‘𝑋) ∩ (𝐼‘𝑌)) ∩ (𝐼‘𝑊))) | ||
| Theorem | dihmeetlem3N 41307 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ∨ (𝑋 ∧ 𝑊)) = 𝑋) ∧ ((𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊) ∧ (𝑅 ∨ (𝑌 ∧ 𝑊)) = 𝑌)) → 𝑄 ≠ 𝑅) | ||
| Theorem | dihmeetlem4preN 41308* | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐺 = (℩𝑔 ∈ 𝑇 (𝑔‘𝑃) = 𝑄) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 }) | ||
| Theorem | dihmeetlem4N 41309 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 30-Mar-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) → ((𝐼‘𝑄) ∩ (𝐼‘(𝑋 ∧ 𝑊))) = { 0 }) | ||
| Theorem | dihmeetlem5 41310 | Part of proof that isomorphism H is order-preserving . (Contributed by NM, 6-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑋)) → (𝑋 ∧ (𝑌 ∨ 𝑄)) = ((𝑋 ∧ 𝑌) ∨ 𝑄)) | ||
| Theorem | dihmeetlem6 41311 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ 𝑄 ≤ 𝑋)) → ¬ (𝑋 ∧ (𝑌 ∨ 𝑄)) ≤ 𝑊) | ||
| Theorem | dihmeetlem7N 41312 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑌)) → (((𝑋 ∧ 𝑌) ∨ 𝑝) ∧ 𝑌) = (𝑋 ∧ 𝑌)) | ||
| Theorem | dihjatc1 41313 | Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change ∨ order of (𝑋 ∧ 𝑌) ∨ 𝑄 here and down? (Contributed by NM, 6-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑄)) = ((𝐼‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑌)))) | ||
| Theorem | dihjatc2N 41314 | Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑄 ∨ (𝑋 ∧ 𝑌))) = ((𝐼‘𝑄) ⊕ (𝐼‘(𝑋 ∧ 𝑌)))) | ||
| Theorem | dihjatc3 41315 | Isomorphism H of join with an atom. (Contributed by NM, 26-Aug-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑄 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑄)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ (𝐼‘𝑄))) | ||
| Theorem | dihmeetlem8N 41316 | Lemma for isomorphism H of a lattice meet. TODO: shorter proof if we change ∨ order of (𝑋 ∧ 𝑌) ∨ 𝑝 here and down? (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌)))) | ||
| Theorem | dihmeetlem9N 41317 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑝 ∈ 𝐴) → (((𝐼‘𝑝) ⊕ (𝐼‘(𝑋 ∧ 𝑌))) ∩ (𝐼‘𝑌)) = ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌)))) | ||
| Theorem | dihmeetlem10N 41318 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → (𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) = ((𝐼‘𝑋) ∩ (𝐼‘(𝑌 ∨ 𝑝)))) | ||
| Theorem | dihmeetlem11N 41319 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋)) → ((𝐼‘((𝑋 ∧ 𝑌) ∨ 𝑝)) ∩ (𝐼‘𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetlem12N 41320 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 6-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ 𝑝 ≤ 𝑋 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → ((𝐼‘(𝑋 ∧ 𝑌)) ⊕ ((𝐼‘𝑝) ∩ (𝐼‘𝑌))) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetlem13N 41321* | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑄) & ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = 𝑅) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ((𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊) ∧ (𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊)) ∧ 𝑄 ≠ 𝑅) → ((𝐼‘𝑄) ∩ (𝐼‘𝑅)) = { 0 }) | ||
| Theorem | dihmeetlem14N 41322 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ 𝑝 ∈ 𝐵) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘(𝑌 ∧ 𝑝)) ⊕ ((𝐼‘𝑟) ∩ (𝐼‘𝑝))) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝))) | ||
| Theorem | dihmeetlem15N 41323 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → ((𝐼‘𝑟) ∩ (𝐼‘𝑝)) = { 0 }) | ||
| Theorem | dihmeetlem16N 41324 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ 𝐵 ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ ((𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ 𝑟 ≤ 𝑌 ∧ (𝑌 ∧ 𝑝) ≤ 𝑊)) → (𝐼‘(𝑌 ∧ 𝑝)) = ((𝐼‘𝑌) ∩ (𝐼‘𝑝))) | ||
| Theorem | dihmeetlem17N 41325 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 0 = (0.‘𝐾) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ (𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊)) ∧ (𝑌 ∈ 𝐵 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊 ∧ 𝑝 ≤ 𝑋)) → (𝑌 ∧ 𝑝) = 0 ) | ||
| Theorem | dihmeetlem18N 41326 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))) → ((𝐼‘𝑌) ∩ (𝐼‘𝑝)) = { 0 }) | ||
| Theorem | dihmeetlem19N 41327 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ ((((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ 𝑌 ∈ 𝐵) ∧ ((𝑝 ∈ 𝐴 ∧ ¬ 𝑝 ≤ 𝑊) ∧ (𝑟 ∈ 𝐴 ∧ ¬ 𝑟 ≤ 𝑊) ∧ (𝑝 ≤ 𝑋 ∧ 𝑟 ≤ 𝑌 ∧ (𝑋 ∧ 𝑌) ≤ 𝑊))) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetlem20N 41328 | Lemma for isomorphism H of a lattice meet. (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊) ∧ ((𝑌 ∈ 𝐵 ∧ ¬ 𝑌 ≤ 𝑊) ∧ (𝑋 ∧ 𝑌) ≤ 𝑊)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihmeetALTN 41329 | Isomorphism H of a lattice meet. This version does not depend on the atomisticity of the constructed vector space. TODO: Delete? (Contributed by NM, 7-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dih1dimatlem0 41330* | Lemma for dih1dimat 41332. (Contributed by NM, 11-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = (Atoms‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐽 = (invr‘𝐹) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = (((𝐽‘𝑠)‘𝑓)‘𝑃)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑓 ∈ 𝑇 ∧ 𝑠 ∈ 𝐸) ∧ 𝑠 ≠ 𝑂) → ((𝑖 = (𝑝‘𝐺) ∧ 𝑝 ∈ 𝐸) ↔ ((𝑖 ∈ 𝑇 ∧ 𝑝 ∈ 𝐸) ∧ ∃𝑡 ∈ 𝐸 (𝑖 = (𝑡‘𝑓) ∧ 𝑝 = (𝑡 ∘ 𝑠))))) | ||
| Theorem | dih1dimatlem 41331* | Lemma for dih1dimat 41332. (Contributed by NM, 10-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐶 = (Atoms‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐹 = (Scalar‘𝑈) & ⊢ 𝐽 = (invr‘𝐹) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐺 = (℩ℎ ∈ 𝑇 (ℎ‘𝑃) = (((𝐽‘𝑠)‘𝑓)‘𝑃)) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐷 ∈ 𝐴) → 𝐷 ∈ ran 𝐼) | ||
| Theorem | dih1dimat 41332 | Any 1-dimensional subspace is a value of isomorphism H. (Contributed by NM, 11-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑃 ∈ 𝐴) → 𝑃 ∈ ran 𝐼) | ||
| Theorem | dihlsprn 41333 | The span of a vector belongs to the range of isomorphism H. (Contributed by NM, 27-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran 𝐼) | ||
| Theorem | dihlspsnssN 41334 | A subspace included in a 1-dim subspace belongs to the range of isomorphism H. (Contributed by NM, 26-Apr-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑇 ⊆ (𝑁‘{𝑋})) → (𝑇 ∈ 𝑆 ↔ 𝑇 ∈ ran 𝐼)) | ||
| Theorem | dihlspsnat 41335 | The inverse isomorphism H of the span of a singleton is a Hilbert lattice atom. (Contributed by NM, 27-Apr-2014.) |
| ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 ) → (◡𝐼‘(𝑁‘{𝑋})) ∈ 𝐴) | ||
| Theorem | dihatlat 41336 | The isomorphism H of an atom is a 1-dim subspace. (Contributed by NM, 28-Apr-2014.) |
| ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSAtoms‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐴) → (𝐼‘𝑄) ∈ 𝐿) | ||
| Theorem | dihat 41337 | There exists at least one atom in the subspaces of vector space H. (Contributed by NM, 12-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝐼‘𝑃) ∈ 𝐴) | ||
| Theorem | dihpN 41338* | The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector 〈0, 𝑆〉 (the zero translation ltrnid 40137 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unity ( I ↾ 𝑇). (Contributed by NM, 26-Aug-2014.) (New usage is discouraged.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑃 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂)) ⇒ ⊢ (𝜑 → (𝐼‘𝑃) = (𝑁‘{〈( I ↾ 𝐵), 𝑆〉})) | ||
| Theorem | dihlatat 41339 | The reverse isomorphism H of a 1-dim subspace is an atom. (Contributed by NM, 28-Apr-2014.) |
| ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐿 = (LSAtoms‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑄 ∈ 𝐿) → (◡𝐼‘𝑄) ∈ 𝐴) | ||
| Theorem | dihatexv 41340* | There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥}))) | ||
| Theorem | dihatexv2 41341* | There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.) |
| ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥})))) | ||
| Theorem | dihglblem6 41342* | Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑃 = (LSubSp‘𝑈) & ⊢ 𝐷 = (LSAtoms‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) | ||
| Theorem | dihglb 41343* | Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥)) | ||
| Theorem | dihglb2 41344* | Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝑉) → (𝐼‘(𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑆 ⊆ (𝐼‘𝑥)})) = ∩ {𝑦 ∈ ran 𝐼 ∣ 𝑆 ⊆ 𝑦}) | ||
| Theorem | dihmeet 41345 | Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌))) | ||
| Theorem | dihintcl 41346 | The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼) | ||
| Theorem | dihmeetcl 41347 | Closure of closed subspace meet for DVecH vector space. (Contributed by NM, 5-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (𝑋 ∩ 𝑌) ∈ ran 𝐼) | ||
| Theorem | dihmeet2 41348 | Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.) |
| ⊢ ∧ = (meet‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))) | ||
| Syntax | coch 41349 | Extend class notation with subspace orthocomplement for DVecH vector space. |
| class ocH | ||
| Definition | df-doch 41350* | Define subspace orthocomplement for DVecH 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, 14-Mar-2014.) |
| ⊢ ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)})))))) | ||
| Theorem | dochffval 41351* | Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑉 → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)})))))) | ||
| Theorem | dochfval 41352* | Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)}))))) | ||
| Theorem | dochval 41353* | Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐺 = (glb‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)})))) | ||
| Theorem | dochval2 41354* | Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Apr-2014.) |
| ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})))) | ||
| Theorem | dochcl 41355 | Closure of subspace orthocomplement for DVecH vector space. (Contributed by NM, 9-Mar-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran 𝐼) | ||
| Theorem | dochlss 41356 | A subspace orthocomplement is a subspace of the DVecH vector space. (Contributed by NM, 22-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆) | ||
| Theorem | dochssv 41357 | A subspace orthocomplement belongs to the DVecH vector space. (Contributed by NM, 22-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉) | ||
| Theorem | dochfN 41358 | Domain and codomain of the subspace orthocomplement for the DVecH vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶ran 𝐼) | ||
| Theorem | dochvalr 41359 | Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.) |
| ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) | ||
| Theorem | doch0 41360 | Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉) | ||
| Theorem | doch1 41361 | Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) ⇒ ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 }) | ||
| Theorem | dochoc0 41362 | The zero subspace is closed. (Contributed by NM, 16-Feb-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{ 0 })) = { 0 }) | ||
| Theorem | dochoc1 41363 | The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉) | ||
| Theorem | dochvalr2 41364 | Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.) |
| ⊢ 𝐵 = (Base‘𝐾) & ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐼‘𝑋)) = (𝐼‘( ⊥ ‘𝑋))) | ||
| Theorem | dochvalr3 41365 | Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.) |
| ⊢ ⊥ = (oc‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋))) | ||
| Theorem | doch2val2 41366* | Double orthocomplement for DVecH vector space. (Contributed by NM, 26-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}) | ||
| Theorem | dochss 41367 | Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) | ||
| Theorem | dochocss 41368 | Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) | ||
| Theorem | dochoc 41369 | Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) | ||
| Theorem | dochsscl 41370 | If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌)) | ||
| Theorem | dochoccl 41371 | A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ ran 𝐼 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) | ||
| Theorem | dochord 41372 | Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) | ||
| Theorem | dochord2N 41373 | Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (( ⊥ ‘𝑋) ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ 𝑋)) | ||
| Theorem | dochord3 41374 | Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋))) | ||
| Theorem | doch11 41375 | Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) ↔ 𝑋 = 𝑌)) | ||
| Theorem | dochsordN 41376 | Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘𝑌) ⊊ ( ⊥ ‘𝑋))) | ||
| Theorem | dochn0nv 41377 | An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (( ⊥ ‘𝑋) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑉)) | ||
| Theorem | dihoml4c 41378 | Version of dihoml4 41379 with closed subspaces. (Contributed by NM, 15-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋) | ||
| Theorem | dihoml4 41379 | Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 39955 analog.) (Contributed by NM, 15-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋))) | ||
| Theorem | dochspss 41380 | The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋))) | ||
| Theorem | dochocsp 41381 | The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → ( ⊥ ‘(𝑁‘𝑋)) = ( ⊥ ‘𝑋)) | ||
| Theorem | dochspocN 41382 | The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝑁‘𝑋))) | ||
| Theorem | dochocsn 41383 | The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = (𝑁‘{𝑋})) | ||
| Theorem | dochsncom 41384 | Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑌}) ↔ 𝑌 ∈ ( ⊥ ‘{𝑋}))) | ||
| Theorem | dochsat 41385 | The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴 ↔ 𝑄 ∈ 𝐴)) | ||
| Theorem | dochshpncl 41386 | If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉)) | ||
| Theorem | dochlkr 41387 | Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) | ||
| Theorem | dochkrshp 41388 | The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌)) | ||
| Theorem | dochkrshp2 41389 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌))) | ||
| Theorem | dochkrshp3 41390 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ≠ 𝑉))) | ||
| Theorem | dochkrshp4 41391 | Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ∨ (𝐿‘𝐺) = 𝑉))) | ||
| Theorem | dochdmj1 41392 | De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))) | ||
| Theorem | dochnoncon 41393 | Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 }) | ||
| Theorem | dochnel2 41394 | A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑇 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ (𝑇 ∖ { 0 })) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘𝑇)) | ||
| Theorem | dochnel 41395 | A nonzero vector doesn't belong to the orthocomplement of its singleton. (Contributed by NM, 27-Oct-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋})) | ||
| Syntax | cdjh 41396 | Extend class notation with subspace join for DVecH vector space. |
| class joinH | ||
| Definition | df-djh 41397* | Define (closed) subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.) |
| ⊢ joinH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦)))))) | ||
| Theorem | djhffval 41398* | Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) ⇒ ⊢ (𝐾 ∈ 𝑋 → (joinH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦)))))) | ||
| Theorem | djhfval 41399* | Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) ⇒ ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦))))) | ||
| Theorem | djhval 41400 | Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.) |
| ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))) | ||
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