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Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-djh 37001* 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))))))

Theoremdjhffval 37002* Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑋 → (joinH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))

Theoremdjhfval 37003* 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)       ((𝐾𝑋𝑊𝐻) → = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))

Theoremdjhval 37004 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 ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))

Theoremdjhval2 37005 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 ∧ 𝑊𝐻) ∧ 𝑋𝑉𝑌𝑉) → (𝑋 𝑌) = ( ‘( ‘(𝑋𝑌))))

Theoremdjhcl 37006 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 ∧ 𝑊𝐻) ∧ (𝑋𝑉𝑌𝑉)) → (𝑋 𝑌) ∈ ran 𝐼)

Theoremdjhlj 37007 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 ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑌𝐵)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋)𝐽(𝐼𝑌)))

TheoremdjhljjN 37008 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → (𝑋 𝑌) = (𝐼‘((𝐼𝑋)𝐽(𝐼𝑌))))

Theoremdjhjlj 37009 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝐽𝑌) = (𝐼‘((𝐼𝑋) (𝐼𝑌))))

Theoremdjhj 37010 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝐼‘(𝑋𝐽𝑌)) = ((𝐼𝑋) (𝐼𝑌)))

Theoremdjhcom 37011 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 𝑌) = (𝑌 𝑋))

Theoremdjhspss 37012 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘(𝑋𝑌)) ⊆ (𝑋 𝑌))

Theoremdjhsumss 37013 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 𝑌) ⊆ (𝑋 𝑌))

Theoremdihsumssj 37014 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑌𝐵)       (𝜑 → ((𝐼𝑋) (𝐼𝑌)) ⊆ (𝐼‘(𝑋 𝑌)))

TheoremdjhunssN 37015 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋𝑌) ⊆ (𝑋 𝑌))

Theoremdochdmm1 37016 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → ( ‘(𝑋𝑌)) = (( 𝑋) ( 𝑌)))

Theoremdjhexmid 37017 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 ∧ 𝑊𝐻) ∧ 𝑋𝑉) → (𝑋 ( 𝑋)) = 𝑉)

Theoremdjh01 37018 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((joinH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)       (𝜑 → (𝑋 { 0 }) = 𝑋)

Theoremdjh02 37019 Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((joinH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)       (𝜑 → ({ 0 } 𝑋) = 𝑋)

Theoremdjhlsmcl 37020 A closed subspace sum equals subspace join. (shjshseli 28480 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)       (𝜑 → ((𝑋 𝑌) ∈ ran 𝐼 ↔ (𝑋 𝑌) = (𝑋 𝑌)))

Theoremdjhcvat42 37021* A covering property. (cvrat42 35048 analog.) (Contributed by NM, 17-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((joinH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑆 ∈ ran 𝐼)    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ((𝑆 ≠ { 0 } ∧ (𝑁‘{𝑋}) ⊆ (𝑆 (𝑁‘{𝑌}))) → ∃𝑧 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑧}) ⊆ 𝑆 ∧ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑧}) (𝑁‘{𝑌})))))

Theoremdihjatb 37022 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴𝑃 𝑊))    &   (𝜑 → (𝑄𝐴𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) = ((𝐼𝑃) (𝐼𝑄)))

Theoremdihjatc 37023 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑋𝐵𝑋 𝑊))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))       (𝜑 → (𝐼‘(𝑋 𝑃)) = ((𝐼𝑋) (𝐼𝑃)))

Theoremdihjatcclem1 37024 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) = (((𝐼𝑃) (𝐼𝑄)) (𝐼𝑉)))

Theoremdihjatcclem2 37025 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))    &   (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))       (𝜑 → (𝐼‘(𝑃 𝑄)) = ((𝐼𝑃) (𝐼𝑄)))

Theoremdihjatcclem3 37026* Lemma for dihjatcc 37028. (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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))    &   𝐶 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)    &   𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)       (𝜑 → (𝑅‘(𝐺𝐷)) = 𝑉)

Theoremdihjatcclem4 37027* 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))    &   𝐶 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐺 = (𝑑𝑇 (𝑑𝐶) = 𝑃)    &   𝐷 = (𝑑𝑇 (𝑑𝐶) = 𝑄)    &   𝑁 = (𝑎𝐸 ↦ (𝑑𝑇(𝑎𝑑)))    &    0 = (𝑑𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑑𝑇 ↦ ((𝑎𝑑) ∘ (𝑏𝑑))))       (𝜑 → (𝐼𝑉) ⊆ ((𝐼𝑃) (𝐼𝑄)))

Theoremdihjatcc 37028 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 ∧ 𝑊𝐻))    &   (𝜑 → (𝑃𝐴 ∧ ¬ 𝑃 𝑊))    &   (𝜑 → (𝑄𝐴 ∧ ¬ 𝑄 𝑊))       (𝜑 → (𝐼‘(𝑃 𝑄)) = ((𝐼𝑃) (𝐼𝑄)))

Theoremdihjat 37029 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 ∧ 𝑊𝐻))    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)       (𝜑 → (𝐼‘(𝑃 𝑄)) = ((𝐼𝑃) (𝐼𝑄)))

Theoremdihprrnlem1N 37030 Lemma for dihprrn 37032, 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    = (le‘𝐾)    &    0 = (0g𝑈)    &   (𝜑𝑌0 )    &   (𝜑 → (𝐼‘(𝑁‘{𝑋})) 𝑊)    &   (𝜑 → ¬ (𝐼‘(𝑁‘{𝑌})) 𝑊)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Theoremdihprrnlem2 37031 Lemma for dihprrn 37032. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )       (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Theoremdihprrn 37032 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)

Theoremdjhlsmat 37033 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 37032; should we directly use dihjat 37029? (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) (𝑁‘{𝑌})) = ((𝑁‘{𝑋}) (𝑁‘{𝑌})))

Theoremdihjat1lem 37034 Subspace sum of a closed subspace and an atom. (pmapjat1 35457 analog.) TODO: merge into dihjat1 37035? (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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &    0 = (0g𝑈)    &   (𝜑𝑇 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝑋 (𝑁‘{𝑇})) = (𝑋 (𝑁‘{𝑇})))

Theoremdihjat1 37035 Subspace sum of a closed subspace and an atom. (pmapjat1 35457 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑇𝑉)       (𝜑 → (𝑋 (𝑁‘{𝑇})) = (𝑋 (𝑁‘{𝑇})))

Theoremdihsmsprn 37036 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑇𝑉)       (𝜑 → (𝑋 (𝑁‘{𝑇})) ∈ ran 𝐼)

Theoremdihjat2 37037 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑋 𝑄) = (𝑋 𝑄))

Theoremdihjat3 37038 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑃𝐴)       (𝜑 → (𝐼‘(𝑋 𝑃)) = ((𝐼𝑋) (𝐼𝑃)))

Theoremdihjat4 37039 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑋 𝑄) = (𝐼‘((𝐼𝑋) (𝐼𝑄))))

Theoremdihjat6 37040 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑄𝐴)       (𝜑 → (𝐼‘(𝑋 𝑄)) = ((𝐼𝑋) (𝐼𝑄)))

Theoremdihsmsnrn 37041 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) (𝑁‘{𝑌})) ∈ ran 𝐼)

Theoremdihsmatrn 37042 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 37037. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑄𝐴)       (𝜑 → (𝑋 𝑄) ∈ ran 𝐼)

Theoremdihjat5N 37043 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝐵)    &   (𝜑𝑃𝐴)       (𝜑 → (𝑋 𝑃) = (𝐼‘((𝐼𝑋) (𝐼𝑃))))

Theoremdvh4dimat 37044* 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 ∧ 𝑊𝐻))    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)    &   (𝜑𝑅𝐴)       (𝜑 → ∃𝑠𝐴 ¬ 𝑠 ⊆ ((𝑃 𝑄) 𝑅))

Theoremdvh3dimatN 37045* 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 ∧ 𝑊𝐻))    &   (𝜑𝑃𝐴)    &   (𝜑𝑄𝐴)       (𝜑 → ∃𝑠𝐴 ¬ 𝑠 ⊆ (𝑃 𝑄))

Theoremdvh2dimatN 37046* 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 ∧ 𝑊𝐻))    &   (𝜑𝑃𝐴)       (𝜑 → ∃𝑠𝐴 𝑠𝑃)

Theoremdvh1dimat 37047* There exists an atom. (Contributed by NM, 25-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ∃𝑠 𝑠𝐴)

Theoremdvh1dim 37048* There exists a nonzero vector. (Contributed by NM, 26-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ∃𝑧𝑉 𝑧0 )

Theoremdvh4dimlem 37049* Lemma for dvh4dimN 37053. (Contributed by NM, 22-May-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )    &   (𝜑𝑍0 )       (𝜑 → ∃𝑧𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍}))

Theoremdvhdimlem 37050* Lemma for dvh2dim 37051 and dvh3dim 37052. TODO: make this obsolete and use dvh4dimlem 37049 directly? (Contributed by NM, 24-Apr-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &    0 = (0g𝑈)    &   (𝜑𝑋0 )    &   (𝜑𝑌0 )       (𝜑 → ∃𝑧𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremdvh2dim 37051* 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑧𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋}))

Theoremdvh3dim 37052* 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ∃𝑧𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremdvh4dimN 37053* 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → ∃𝑧𝑉 ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑌, 𝑍}))

Theoremdvh3dim2 37054* 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → ∃𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑋, 𝑍})))

Theoremdvh3dim3N 37055* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 37054 everywhere. If this one is needed, make dvh3dim2 37054 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)    &   (𝜑𝑇𝑉)       (𝜑 → ∃𝑧𝑉𝑧 ∈ (𝑁‘{𝑋, 𝑌}) ∧ ¬ 𝑧 ∈ (𝑁‘{𝑍, 𝑇})))

Theoremdochsnnz 37056 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 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘{𝑋}) ≠ { 0 })

Theoremdochsatshp 37057 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 ∧ 𝑊𝐻))    &   (𝜑𝑄𝐴)       (𝜑 → ( 𝑄) ∈ 𝑌)

Theoremdochsatshpb 37058 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 ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)       (𝜑 → (𝑄𝐴 ↔ ( 𝑄) ∈ 𝑌))

Theoremdochsnshp 37059 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 = (0g𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ( ‘{𝑋}) ∈ 𝑌)

Theoremdochshpsat 37060 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘( 𝑋)) = 𝑋 ↔ ( 𝑋) ∈ 𝐴))

Theoremdochkrsat 37061 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 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘(𝐿𝐺)) ≠ { 0 } ↔ ( ‘(𝐿𝐺)) ∈ 𝐴))

Theoremdochkrsat2 37062 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 ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ ( ‘(𝐿𝐺)) ∈ 𝐴))

Theoremdochsat0 37063 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 = (0g𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘(𝐿𝐺)) ∈ 𝐴 ∨ ( ‘(𝐿𝐺)) = { 0 }))

Theoremdochkrsm 37064 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 37020 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 ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝐺𝐹)       (𝜑 → (𝑋 ( ‘(𝐿𝐺))) ∈ ran 𝐼)

Theoremdochexmidat 37065 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 = (0g𝑈)    &   𝑁 = (LSpan‘𝑈)    &    = (LSSum‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (( ‘{𝑋}) (𝑁‘{𝑋})) = 𝑉)

Theoremdochexmidlem1 37066 Lemma for dochexmid 37074. Holland's proof implicitly requires 𝑞𝑟, 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &   (𝜑𝑞𝐴)    &   (𝜑𝑟𝐴)    &   (𝜑𝑞 ⊆ ( 𝑋))    &   (𝜑𝑟𝑋)       (𝜑𝑞𝑟)

Theoremdochexmidlem2 37067 Lemma for dochexmid 37074. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &   (𝜑𝑞𝐴)    &   (𝜑𝑟𝐴)    &   (𝜑𝑞 ⊆ ( 𝑋))    &   (𝜑𝑟𝑋)    &   (𝜑𝑝 ⊆ (𝑟 𝑞))       (𝜑𝑝 ⊆ (𝑋 ( 𝑋)))

Theoremdochexmidlem3 37068 Lemma for dochexmid 37074. Use atom exchange lsatexch1 34651 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &   (𝜑𝑞𝐴)    &   (𝜑𝑟𝐴)    &   (𝜑𝑞 ⊆ ( 𝑋))    &   (𝜑𝑟𝑋)    &   (𝜑𝑞 ⊆ (𝑟 𝑝))       (𝜑𝑝 ⊆ (𝑋 ( 𝑋)))

Theoremdochexmidlem4 37069 Lemma for dochexmid 37074. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &   (𝜑𝑞𝐴)    &    0 = (0g𝑈)    &   𝑀 = (𝑋 𝑝)    &   (𝜑𝑋 ≠ { 0 })    &   (𝜑𝑞 ⊆ (( 𝑋) ∩ 𝑀))       (𝜑𝑝 ⊆ (𝑋 ( 𝑋)))

Theoremdochexmidlem5 37070 Lemma for dochexmid 37074. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &    0 = (0g𝑈)    &   𝑀 = (𝑋 𝑝)    &   (𝜑𝑋 ≠ { 0 })    &   (𝜑 → ¬ 𝑝 ⊆ (𝑋 ( 𝑋)))       (𝜑 → (( 𝑋) ∩ 𝑀) = { 0 })

Theoremdochexmidlem6 37071 Lemma for dochexmid 37074. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &    0 = (0g𝑈)    &   𝑀 = (𝑋 𝑝)    &   (𝜑𝑋 ≠ { 0 })    &   (𝜑 → ( ‘( 𝑋)) = 𝑋)    &   (𝜑 → ¬ 𝑝 ⊆ (𝑋 ( 𝑋)))       (𝜑𝑀 = 𝑋)

Theoremdochexmidlem7 37072 Lemma for dochexmid 37074. Contradict dochexmidlem6 37071. (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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑝𝐴)    &    0 = (0g𝑈)    &   𝑀 = (𝑋 𝑝)    &   (𝜑𝑋 ≠ { 0 })    &   (𝜑 → ( ‘( 𝑋)) = 𝑋)    &   (𝜑 → ¬ 𝑝 ⊆ (𝑋 ( 𝑋)))       (𝜑𝑀𝑋)

Theoremdochexmidlem8 37073 Lemma for dochexmid 37074. The contradiction of dochexmidlem6 37071 and dochexmidlem7 37072 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &    0 = (0g𝑈)    &   (𝜑𝑋 ≠ { 0 })    &   (𝜑 → ( ‘( 𝑋)) = 𝑋)       (𝜑 → (𝑋 ( 𝑋)) = 𝑉)

Theoremdochexmid 37074 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 36983. 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 35582 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 ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑 → ( ‘( 𝑋)) = 𝑋)       (𝜑 → (𝑋 ( 𝑋)) = 𝑉)

Theoremdochsnkrlem1 37075 Lemma for dochsnkr 37078. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))       (𝜑 → ( ‘( ‘(𝐿𝐺))) ≠ 𝑉)

Theoremdochsnkrlem2 37076 Lemma for dochsnkr 37078. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))    &   𝐴 = (LSAtoms‘𝑈)       (𝜑 → ( ‘(𝐿𝐺)) ∈ 𝐴)

Theoremdochsnkrlem3 37077 Lemma for dochsnkr 37078. (Contributed by NM, 2-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))       (𝜑 → ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺))

Theoremdochsnkr 37078 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 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))       (𝜑 → (𝐿𝐺) = ( ‘{𝑋}))

Theoremdochsnkr2 37079* Kernel of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkr 34722. (Contributed by NM, 27-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐷)    &   𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → (𝐿𝐺) = ( ‘{𝑋}))

Theoremdochsnkr2cl 37080* 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 = (0g𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐷 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐷)    &   𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))

Theoremdochflcl 37081* Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 34721. (Contributed by NM, 27-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐷 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐷)    &   𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑𝐺𝐹)

Theoremdochfl1 37082* 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)    &    + = (+g𝑈)    &    · = ( ·𝑠𝑈)    &    0 = (0g𝑈)    &   𝐷 = (Scalar‘𝑈)    &   𝑅 = (Base‘𝐷)    &    1 = (1r𝐷)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))    &   𝐺 = (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑋})𝑣 = (𝑤 + (𝑘 · 𝑋))))       (𝜑 → (𝐺𝑋) = 1 )

Theoremdochfln0 37083 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 = (0g𝑅)    &    0 = (0g𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑𝑋 ∈ (( ‘(𝐿𝐺)) ∖ { 0 }))       (𝜑 → (𝐺𝑋) ≠ 𝑁)

Theoremdochkr1 37084* 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 34675. (Contributed by NM, 2-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑅 = (Scalar‘𝑈)    &    0 = (0g𝑈)    &    1 = (1r𝑅)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑 → ( ‘( ‘(𝐿𝐺))) ≠ 𝑉)       (𝜑 → ∃𝑥 ∈ (( ‘(𝐿𝐺)) ∖ { 0 })(𝐺𝑥) = 1 )

Theoremdochkr1OLDN 37085* 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 34675. (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 = (0g𝑅)    &    1 = (1r𝑅)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)    &   (𝜑 → ( ‘( ‘(𝐿𝐺))) ≠ 𝑉)       (𝜑 → ∃𝑥 ∈ ( ‘(𝐿𝐺))(𝐺𝑥) = 1 )

20.23.14  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 37086 Extend class notation with all polarities of a left module or left vector space.
class LPol

Definitiondf-lpolN 37087* 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)) = {(0g𝑤)} ∧ ∀𝑥𝑦((𝑥 ⊆ (Base‘𝑤) ∧ 𝑦 ⊆ (Base‘𝑤) ∧ 𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑤)((𝑜𝑥) ∈ (LSHyp‘𝑤) ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})

TheoremlpolsetN 37088* 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 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝑃 = (LPol‘𝑊)       (𝑊𝑋𝑃 = {𝑜 ∈ (𝑆𝑚 𝒫 𝑉) ∣ ((𝑜𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → (𝑜𝑦) ⊆ (𝑜𝑥)) ∧ ∀𝑥𝐴 ((𝑜𝑥) ∈ 𝐻 ∧ (𝑜‘(𝑜𝑥)) = 𝑥))})

TheoremislpolN 37089* The predicate "is a polarity". (Contributed by NM, 24-Nov-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝑃 = (LPol‘𝑊)       (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))

TheoremislpoldN 37090* Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &    0 = (0g𝑊)    &   𝐴 = (LSAtoms‘𝑊)    &   𝐻 = (LSHyp‘𝑊)    &   𝑃 = (LPol‘𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑 :𝒫 𝑉𝑆)    &   (𝜑 → ( 𝑉) = { 0 })    &   ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑥𝑦)) → ( 𝑦) ⊆ ( 𝑥))    &   ((𝜑𝑥𝐴) → ( 𝑥) ∈ 𝐻)    &   ((𝜑𝑥𝐴) → ( ‘( 𝑥)) = 𝑥)       (𝜑𝑃)

TheoremlpolfN 37091 Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑃 = (LPol‘𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑃)       (𝜑 :𝒫 𝑉𝑆)

TheoremlpolvN 37092 The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑃 = (LPol‘𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑃)       (𝜑 → ( 𝑉) = { 0 })

TheoremlpolconN 37093 Contraposition property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
𝑉 = (Base‘𝑊)    &   𝑃 = (LPol‘𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑃)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑋𝑌)       (𝜑 → ( 𝑌) ⊆ ( 𝑋))

TheoremlpolsatN 37094 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)    &   (𝜑𝑊𝑋)    &   (𝜑𝑃)    &   (𝜑𝑄𝐴)       (𝜑 → ( 𝑄) ∈ 𝐻)

TheoremlpolpolsatN 37095 Property of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
𝐴 = (LSAtoms‘𝑊)    &   𝑃 = (LPol‘𝑊)    &   (𝜑𝑊𝑋)    &   (𝜑𝑃)    &   (𝜑𝑄𝐴)       (𝜑 → ( ‘( 𝑄)) = 𝑄)

TheoremdochpolN 37096 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 ∧ 𝑊𝐻))       (𝜑𝑃)

Theoremlcfl1lem 37097* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}       (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))

Theoremlcfl1 37098* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))

Theoremlcfl2 37099* 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 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ∨ (𝐿𝐺) = 𝑉)))

Theoremlcfl3 37100* 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)    &   𝐴 = (LSAtoms‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ (( ‘(𝐿𝐺)) ∈ 𝐴 ∨ (𝐿𝐺) = 𝑉)))

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