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Type | Label | Description |
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Statement | ||
Theorem | dihmeetlem3N 38601 | 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 38602* | 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 38603 | 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 38604 | 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 38605 | 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 38606 | 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 38607 | 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 38608 | 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 38609 | 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 38610 | 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 38611 | 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 38612 | 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 38613 | 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 38614 | 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 38615* | 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 38616 | 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 38617 | 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 38618 | 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 38619 | 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 38620 | 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 38621 | 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 38622 | 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 38623 | 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 38624* | Lemma for dih1dimat 38626. (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 38625* | Lemma for dih1dimat 38626. (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 38626 | 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 38627 | 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 38628 | 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 38629 | 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 38630 | 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 38631 | 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 38632* | The value of isomorphism H at the fiducial atom 𝑃 is determined by the vector 〈0, 𝑆〉 (the zero translation ltrnid 37431 and a nonzero member of the endomorphism ring). In particular, 𝑆 can be replaced with the ring unit ( 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 38633 | 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 38634* | 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 38635* | 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 38636* | 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 38637* | 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 38638* | 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 38639 | 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 38640 | 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 38641 | 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 38642 | 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 38643 | Extend class notation with subspace orthocomplement for DVecH vector space. |
class ocH | ||
Definition | df-doch 38644* | 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 38645* | 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 38646* | 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 38647* | 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 38648* | 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 38649 | 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 38650 | 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 38651 | 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 38652 | 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 38653 | 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 38654 | 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 38655 | 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 38656 | 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 38657 | 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 38658 | 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 38659 | 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 38660* | 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 38661 | 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 38662 | 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 38663 | 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 38664 | 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 38665 | 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 38666 | 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 38667 | 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 38668 | 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 38669 | 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 38670 | 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 38671 | 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 38672 | Version of dihoml4 38673 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 38673 | Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 37249 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 38674 | 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 38675 | 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 38676 | 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 38677 | 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 38678 | 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 38679 | 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 38680 | 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 38681 | 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 38682 | 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 38683 | 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 38684 | 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 38685 | 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 38686 | 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 38687 | 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 38688 | 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 38689 | 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 38690 | Extend class notation with subspace join for DVecH vector space. |
class joinH | ||
Definition | df-djh 38691* | 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 38692* | 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 38693* | 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 38694 | 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)))) | ||
Theorem | djhval2 38695 | Value of subspace join for DVecH vector space. (Contributed by NM, 6-Aug-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)))) | ||
Theorem | djhcl 38696 | Closure of subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) ∈ ran 𝐼) | ||
Theorem | djhlj 38697 | Transfer lattice join to DVecH vector space closed subspace join. (Contributed by NM, 19-Jul-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌))) | ||
Theorem | djhljjN 38698 | Lattice join in terms of DVecH vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌)))) | ||
Theorem | djhjlj 38699 | DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))) | ||
Theorem | djhj 38700 | DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))) |
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