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Type | Label | Description |
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Statement | ||
Syntax | cdjh 35701 | Extend class notation with subspace join for DVecH vector space. |
class joinH | ||
Definition | df-djh 35702* | 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 35703* | 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 35704* | 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 35705 | 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 35706 | 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 35707 | 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 35708 | 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 35709 | 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 35710 | 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 35711 | 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))) | ||
Theorem | djhcom 35712 | Subspace join commutes. (Contributed by NM, 8-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋)) | ||
Theorem | djhspss 35713 | Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘(𝑋 ∪ 𝑌)) ⊆ (𝑋 ∨ 𝑌)) | ||
Theorem | djhsumss 35714 | Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ⊕ 𝑌) ⊆ (𝑋 ∨ 𝑌)) | ||
Theorem | dihsumssj 35715 | The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) ⇒ ⊢ (𝜑 → ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ⊆ (𝐼‘(𝑋 ∨ 𝑌))) | ||
Theorem | djhunssN 35716 | Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∪ 𝑌) ⊆ (𝑋 ∨ 𝑌)) | ||
Theorem | dochdmm1 35717 | De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌))) | ||
Theorem | djhexmid 35718 | Excluded middle property of DVecH vector space closed subspace join. (Contributed by NM, 22-Jul-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) ⇒ ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 𝑉) | ||
Theorem | djh01 35719 | Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ∨ { 0 }) = 𝑋) | ||
Theorem | djh02 35720 | Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) ⇒ ⊢ (𝜑 → ({ 0 } ∨ 𝑋) = 𝑋) | ||
Theorem | djhlsmcl 35721 | A closed subspace sum equals subspace join. (shjshseli 27736 analog.) (Contributed by NM, 13-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑌 ∈ 𝑆) ⇒ ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 ↔ (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌))) | ||
Theorem | djhcvat42 35722* | A covering property. (cvrat42 33748 analog.) (Contributed by NM, 17-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑆 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ((𝑆 ≠ { 0 } ∧ (𝑁‘{𝑋}) ⊆ (𝑆 ∨ (𝑁‘{𝑌}))) → ∃𝑧 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑧}) ⊆ 𝑆 ∧ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑧}) ∨ (𝑁‘{𝑌}))))) | ||
Theorem | dihjatb 35723 | Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) | ||
Theorem | dihjatc 35724 | Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃))) | ||
Theorem | dihjatcclem1 35725 | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-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) & ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉))) | ||
Theorem | dihjatcclem2 35726 | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-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) & ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) & ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) ⇒ ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) | ||
Theorem | dihjatcclem3 35727* | Lemma for dihjatcc 35729. (Contributed by NM, 28-Sep-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) & ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) & ⊢ 𝐶 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃) & ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄) ⇒ ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉) | ||
Theorem | dihjatcclem4 35728* | Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-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) & ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) & ⊢ 𝐶 = ((oc‘𝐾)‘𝑊) & ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) & ⊢ 𝑅 = ((trL‘𝐾)‘𝑊) & ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊) & ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃) & ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄) & ⊢ 𝑁 = (𝑎 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ◡(𝑎‘𝑑))) & ⊢ 0 = (𝑑 ∈ 𝑇 ↦ ( I ↾ 𝐵)) & ⊢ 𝐽 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ((𝑎‘𝑑) ∘ (𝑏‘𝑑)))) ⇒ ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) | ||
Theorem | dihjatcc 35729 | Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.) |
⊢ ≤ = (le‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) & ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊)) ⇒ ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) | ||
Theorem | dihjat 35730 | Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄))) | ||
Theorem | dihprrnlem1N 35731 | Lemma for dihprrn 35733, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ ≤ = (le‘𝐾) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑌 ≠ 0 ) & ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊) & ⊢ (𝜑 → ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) | ||
Theorem | dihprrnlem2 35732 | Lemma for dihprrn 35733. (Contributed by NM, 29-Sep-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) | ||
Theorem | dihprrn 35733 | The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼) | ||
Theorem | djhlsmat 35734 | The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 35733; should we directly use dihjat 35730? (Contributed by NM, 13-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑋}) ∨ (𝑁‘{𝑌}))) | ||
Theorem | dihjat1lem 35735 | Subspace sum of a closed subspace and an atom. (pmapjat1 34157 analog.) TODO: merge into dihjat1 35736? (Contributed by NM, 18-Aug-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) | ||
Theorem | dihjat1 35736 | Subspace sum of a closed subspace and an atom. (pmapjat1 34157 analog.) (Contributed by NM, 1-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇}))) | ||
Theorem | dihsmsprn 35737 | Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ ran 𝐼) | ||
Theorem | dihjat2 35738 | The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ∨ = ((joinH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∨ 𝑄) = (𝑋 ⊕ 𝑄)) | ||
Theorem | dihjat3 35739 | Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃))) | ||
Theorem | dihjat4 35740 | Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ⊕ 𝑄) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄)))) | ||
Theorem | dihjat6 35741 | Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.) |
⊢ ∨ = (join‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (◡𝐼‘(𝑋 ⊕ 𝑄)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄))) | ||
Theorem | dihsmsnrn 35742 | The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran 𝐼) | ||
Theorem | dihsmatrn 35743 | The subspace sum of a closed subspace and an atom is closed. TODO: see if proof at http://math.stackexchange.com/a/1233211/50776 and Mon, 13 Apr 2015 20:44:07 -0400 email could be used instead of this and dihjat2 35738. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ⊕ 𝑄) ∈ ran 𝐼) | ||
Theorem | dihjat5N 35744 | Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ∨ = (join‘𝐾) & ⊢ 𝐴 = (Atoms‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) ⇒ ⊢ (𝜑 → (𝑋 ∨ 𝑃) = (◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))) | ||
Theorem | dvh4dimat 35745* | There is an atom that is outside the subspace sum of 3 others. (Contributed by NM, 25-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) & ⊢ (𝜑 → 𝑅 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ ((𝑃 ⊕ 𝑄) ⊕ 𝑅)) | ||
Theorem | dvh3dimatN 35746* | There is an atom that is outside the subspace sum of 2 others. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 ¬ 𝑠 ⊆ (𝑃 ⊕ 𝑄)) | ||
Theorem | dvh2dimatN 35747* | Given an atom, there exists another. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑃 ∈ 𝐴) ⇒ ⊢ (𝜑 → ∃𝑠 ∈ 𝐴 𝑠 ≠ 𝑃) | ||
Theorem | dvh1dimat 35748* | There exists an atom. (Contributed by NM, 25-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ∃𝑠 𝑠 ∈ 𝐴) | ||
Theorem | dvh1dim 35749* | There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 𝑧 ≠ 0 ) | ||
Theorem | dvh4dimlem 35750* | Lemma for dvh4dimN 35754. (Contributed by NM, 22-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑌 ≠ 0 ) & ⊢ (𝜑 → 𝑍 ≠ 0 ) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍})) | ||
Theorem | dvhdimlem 35751* | Lemma for dvh2dim 35752 and dvh3dim 35753. TODO: make this obsolete and use dvh4dimlem 35750 directly? (Contributed by NM, 24-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ≠ 0 ) & ⊢ (𝜑 → 𝑌 ≠ 0 ) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | dvh2dim 35752* | There is a vector that is outside the span of another. (Contributed by NM, 25-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋})) | ||
Theorem | dvh3dim 35753* | There is a vector that is outside the span of 2 others. (Contributed by NM, 24-Apr-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | dvh4dimN 35754* | There is a vector that is outside the span of 3 others. (Contributed by NM, 22-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍})) | ||
Theorem | dvh3dim2 35755* | There is a vector that is outside of 2 spans with a common vector. (Contributed by NM, 13-May-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑍}))) | ||
Theorem | dvh3dim3N 35756* | There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 35755 everywhere. If this one is needed, make dvh3dim2 35755 into a lemma. (Contributed by NM, 21-May-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) & ⊢ (𝜑 → 𝑇 ∈ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑧 ∈ 𝑉 (¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇}))) | ||
Theorem | dochsnnz 35757 | The orthocomplement of a singleton is nonzero. (Contributed by NM, 13-Jun-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 }) | ||
Theorem | dochsatshp 35758 | The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 27-Jul-2014.) (Revised by Mario Carneiro, 1-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝑌) | ||
Theorem | dochsatshpb 35759 | The orthocomplement of a subspace atom is a hyperplane. (Contributed by NM, 29-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑄 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ( ⊥ ‘𝑄) ∈ 𝑌)) | ||
Theorem | dochsnshp 35760 | The orthocomplement of a nonzero singleton is a hyperplane. (Contributed by NM, 3-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ 𝑌) | ||
Theorem | dochshpsat 35761 | A hyperplane is closed iff its orthocomplement is an atom. (Contributed by NM, 29-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝑌 = (LSHyp‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑌) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋 ↔ ( ⊥ ‘𝑋) ∈ 𝐴)) | ||
Theorem | dochkrsat 35762 | The orthocomplement of a kernel is an atom iff it is nonzero. (Contributed by NM, 1-Nov-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ≠ { 0 } ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) | ||
Theorem | dochkrsat2 35763 | The orthocomplement of a kernel is an atom iff the double orthocomplement is not the vector space. (Contributed by NM, 1-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴)) | ||
Theorem | dochsat0 35764 | The orthocomplement of a kernel is either an atom or zero. (Contributed by NM, 29-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴 ∨ ( ⊥ ‘(𝐿‘𝐺)) = { 0 })) | ||
Theorem | dochkrsm 35765 | The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 35721 can be used to convert sum to join.) (Contributed by NM, 29-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘(𝐿‘𝐺))) ∈ ran 𝐼) | ||
Theorem | dochexmidat 35766 | Special case of excluded middle for the singleton of a vector. (Contributed by NM, 27-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (( ⊥ ‘{𝑋}) ⊕ (𝑁‘{𝑋})) = 𝑉) | ||
Theorem | dochexmidlem1 35767 | Lemma for dochexmid 35775. Holland's proof implicitly requires 𝑞 ≠ 𝑟, which we prove here. (Contributed by NM, 14-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ∈ 𝐴) & ⊢ (𝜑 → 𝑟 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) & ⊢ (𝜑 → 𝑟 ⊆ 𝑋) ⇒ ⊢ (𝜑 → 𝑞 ≠ 𝑟) | ||
Theorem | dochexmidlem2 35768 | Lemma for dochexmid 35775. (Contributed by NM, 14-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ∈ 𝐴) & ⊢ (𝜑 → 𝑟 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) & ⊢ (𝜑 → 𝑟 ⊆ 𝑋) & ⊢ (𝜑 → 𝑝 ⊆ (𝑟 ⊕ 𝑞)) ⇒ ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) | ||
Theorem | dochexmidlem3 35769 | Lemma for dochexmid 35775. Use atom exchange lsatexch1 33351 to swap 𝑝 and 𝑞. (Contributed by NM, 14-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ∈ 𝐴) & ⊢ (𝜑 → 𝑟 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ⊆ ( ⊥ ‘𝑋)) & ⊢ (𝜑 → 𝑟 ⊆ 𝑋) & ⊢ (𝜑 → 𝑞 ⊆ (𝑟 ⊕ 𝑝)) ⇒ ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) | ||
Theorem | dochexmidlem4 35770 | Lemma for dochexmid 35775. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ (𝜑 → 𝑞 ∈ 𝐴) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑀 = (𝑋 ⊕ 𝑝) & ⊢ (𝜑 → 𝑋 ≠ { 0 }) & ⊢ (𝜑 → 𝑞 ⊆ (( ⊥ ‘𝑋) ∩ 𝑀)) ⇒ ⊢ (𝜑 → 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) | ||
Theorem | dochexmidlem5 35771 | Lemma for dochexmid 35775. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑀 = (𝑋 ⊕ 𝑝) & ⊢ (𝜑 → 𝑋 ≠ { 0 }) & ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) ⇒ ⊢ (𝜑 → (( ⊥ ‘𝑋) ∩ 𝑀) = { 0 }) | ||
Theorem | dochexmidlem6 35772 | Lemma for dochexmid 35775. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑀 = (𝑋 ⊕ 𝑝) & ⊢ (𝜑 → 𝑋 ≠ { 0 }) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) & ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) ⇒ ⊢ (𝜑 → 𝑀 = 𝑋) | ||
Theorem | dochexmidlem7 35773 | Lemma for dochexmid 35775. Contradict dochexmidlem6 35772. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → 𝑝 ∈ 𝐴) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝑀 = (𝑋 ⊕ 𝑝) & ⊢ (𝜑 → 𝑋 ≠ { 0 }) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) & ⊢ (𝜑 → ¬ 𝑝 ⊆ (𝑋 ⊕ ( ⊥ ‘𝑋))) ⇒ ⊢ (𝜑 → 𝑀 ≠ 𝑋) | ||
Theorem | dochexmidlem8 35774 | Lemma for dochexmid 35775. The contradiction of dochexmidlem6 35772 and dochexmidlem7 35773 shows that there can be no atom 𝑝 that is not in 𝑋 + ( ⊥ ‘𝑋), which is therefore the whole atom space. (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ 𝑁 = (LSpan‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ 𝐴 = (LSAtoms‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ 0 = (0g‘𝑈) & ⊢ (𝜑 → 𝑋 ≠ { 0 }) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ⇒ ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉) | ||
Theorem | dochexmid 35775 | Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 35684. Lemma 3.3(2) in [Holland95] p. 215. In our proof, we use the variables 𝑋, 𝑀, 𝑝, 𝑞, 𝑟 in place of Hollands' l, m, P, Q, L respectively. (pexmidALTN 34282 analog.) (Contributed by NM, 15-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑆 = (LSubSp‘𝑈) & ⊢ ⊕ = (LSSum‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ 𝑆) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) ⇒ ⊢ (𝜑 → (𝑋 ⊕ ( ⊥ ‘𝑋)) = 𝑉) | ||
Theorem | dochsnkrlem1 35776 | Lemma for dochsnkr 35779. (Contributed by NM, 1-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) | ||
Theorem | dochsnkrlem2 35777 | Lemma for dochsnkr 35779. (Contributed by NM, 1-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) & ⊢ 𝐴 = (LSAtoms‘𝑈) ⇒ ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ∈ 𝐴) | ||
Theorem | dochsnkrlem3 35778 | Lemma for dochsnkr 35779. (Contributed by NM, 2-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)) | ||
Theorem | dochsnkr 35779 | A (closed) kernel expressed in terms of a nonzero vector in its orthocomplement. TODO: consolidate lemmas unless they're needed for something else (in which case break out as theorems). (Contributed by NM, 2-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) | ||
Theorem | dochsnkr2 35780* | Kernel of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkr 33422. (Contributed by NM, 27-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝐷) & ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋})) | ||
Theorem | dochsnkr2cl 35781* | The 𝑋 determining functional 𝐺 belongs to the atom formed by the orthocomplement of the kernel. (Contributed by NM, 4-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝐷) & ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) | ||
Theorem | dochflcl 35782* | Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 33421. (Contributed by NM, 27-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝐷) & ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) ⇒ ⊢ (𝜑 → 𝐺 ∈ 𝐹) | ||
Theorem | dochfl1 35783* | The value of the explicit functional 𝐺 is 1 at the 𝑋 that determines it. (Contributed by NM, 27-Oct-2014.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ + = (+g‘𝑈) & ⊢ · = ( ·𝑠 ‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐷 = (Scalar‘𝑈) & ⊢ 𝑅 = (Base‘𝐷) & ⊢ 1 = (1r‘𝐷) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) & ⊢ 𝐺 = (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋)))) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) = 1 ) | ||
Theorem | dochfln0 35784 | The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 𝑁 = (0g‘𝑅) & ⊢ 0 = (0g‘𝑈) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })) ⇒ ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁) | ||
Theorem | dochkr1 35785* | A nonzero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 33375. (Contributed by NM, 2-Jan-2015.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑈) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 })(𝐺‘𝑥) = 1 ) | ||
Theorem | dochkr1OLDN 35786* | A nonzero functional has a value of 1 at some argument belonging to the orthocomplement of its kernel (when its kernel is a closed hyperplane). Tighter version of lfl1 33375. (Contributed by NM, 2-Jan-2015.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑉 = (Base‘𝑈) & ⊢ 𝑅 = (Scalar‘𝑈) & ⊢ 0 = (0g‘𝑅) & ⊢ 1 = (1r‘𝑅) & ⊢ 𝐹 = (LFnl‘𝑈) & ⊢ 𝐿 = (LKer‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉) ⇒ ⊢ (𝜑 → ∃𝑥 ∈ ( ⊥ ‘(𝐿‘𝐺))(𝐺‘𝑥) = 1 ) | ||
Syntax | clpoN 35787 | Extend class notation with all polarities of a left module or left vector space. |
class LPol | ||
Definition | df-lpolN 35788* | Define the set of all polarities of a left module or left vector space. A polarity is a kind of complementation operation on a subspace. The double polarity of a subspace is a closure operation. Based on Definition 3.2 of [Holland95] p. 214 for projective geometry polarities. For convenience, we open up the domain to include all vector subsets and not just subspaces, but any more restricted polarity can be converted to this one by taking the span of its argument. (Contributed by NM, 24-Nov-2014.) |
⊢ LPol = (𝑤 ∈ V ↦ {𝑜 ∈ ((LSubSp‘𝑤) ↑𝑚 𝒫 (Base‘𝑤)) ∣ ((𝑜‘(Base‘𝑤)) = {(0g‘𝑤)} ∧ ∀𝑥∀𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜‘𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}) | ||
Theorem | lpolsetN 35789* | The set of polarities of a left module or left vector space. (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝑃 = {𝑜 ∈ (𝑆 ↑𝑚 𝒫 𝑉) ∣ ((𝑜‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → (𝑜‘𝑦) ⊆ (𝑜‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 ((𝑜‘𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜‘𝑥)) = 𝑥))}) | ||
Theorem | islpolN 35790* | The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ :𝒫 𝑉⟶𝑆 ∧ (( ⊥ ‘𝑉) = { 0 } ∧ ∀𝑥∀𝑦((𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) ∧ ∀𝑥 ∈ 𝐴 (( ⊥ ‘𝑥) ∈ 𝐻 ∧ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥))))) | ||
Theorem | islpoldN 35791* | Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) & ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) & ⊢ ((𝜑 ∧ (𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦)) → ( ⊥ ‘𝑦) ⊆ ( ⊥ ‘𝑥)) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘𝑥) ∈ 𝐻) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥) ⇒ ⊢ (𝜑 → ⊥ ∈ 𝑃) | ||
Theorem | lpolfN 35792 | Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ ∈ 𝑃) ⇒ ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶𝑆) | ||
Theorem | lpolvN 35793 | The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ ∈ 𝑃) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑉) = { 0 }) | ||
Theorem | lpolconN 35794 | Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ ∈ 𝑃) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) & ⊢ (𝜑 → 𝑌 ⊆ 𝑉) & ⊢ (𝜑 → 𝑋 ⊆ 𝑌) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) | ||
Theorem | lpolsatN 35795 | The polarity of an atomic subspace is a hyperplane. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝐻 = (LSHyp‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ ∈ 𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → ( ⊥ ‘𝑄) ∈ 𝐻) | ||
Theorem | lpolpolsatN 35796 | Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.) |
⊢ 𝐴 = (LSAtoms‘𝑊) & ⊢ 𝑃 = (LPol‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ 𝑋) & ⊢ (𝜑 → ⊥ ∈ 𝑃) & ⊢ (𝜑 → 𝑄 ∈ 𝐴) ⇒ ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑄)) = 𝑄) | ||
Theorem | dochpolN 35797 | The subspace orthocomplement for the DVecH vector space is a polarity. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.) |
⊢ 𝐻 = (LHyp‘𝐾) & ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) & ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) & ⊢ 𝑃 = (LPol‘𝑈) & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) ⇒ ⊢ (𝜑 → ⊥ ∈ 𝑃) | ||
Theorem | lcfl1lem 35798* | Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.) |
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} ⇒ ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) | ||
Theorem | lcfl1 35799* | Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.) |
⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))) | ||
Theorem | lcfl2 35800* | Property of a functional with a closed kernel. (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) & ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)} & ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) & ⊢ (𝜑 → 𝐺 ∈ 𝐹) ⇒ ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ∨ (𝐿‘𝐺) = 𝑉))) |
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