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Theorem List for Metamath Proof Explorer - 36501-36600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdochnel 36501 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 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))       (𝜑 → ¬ 𝑋 ∈ ( ‘{𝑋}))

Syntaxcdjh 36502 Extend class notation with subspace join for DVecH vector space.
class joinH

Definitiondf-djh 36503* 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 36504* 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 36505* 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 36506 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 36507 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 36508 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 36509 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 36510 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 36511 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 36512 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 36513 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 36514 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 36515 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 36516 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 36517 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 36518 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 36519 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 36520 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 36521 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 36522 A closed subspace sum equals subspace join. (shjshseli 28322 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 36523* A covering property. (cvrat42 34549 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 36524 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 36525 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 36526 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 36527 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 36528* Lemma for dihjatcc 36530. (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 36529* 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 36530 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 36531 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 36532 Lemma for dihprrn 36534, 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 36533 Lemma for dihprrn 36534. (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 36534 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 36535 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 36534; should we directly use dihjat 36531? (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 36536 Subspace sum of a closed subspace and an atom. (pmapjat1 34958 analog.) TODO: merge into dihjat1 36537? (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 36537 Subspace sum of a closed subspace and an atom. (pmapjat1 34958 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 36538 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 36539 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 36540 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 36541 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 36542 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 36543 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 36544 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 36539. (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 36545 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 36546* 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 36547* 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 36548* 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 36549* 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 36550* 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 36551* Lemma for dvh4dimN 36555. (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 36552* Lemma for dvh2dim 36553 and dvh3dim 36554. TODO: make this obsolete and use dvh4dimlem 36551 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 36553* 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 36554* 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 36555* 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 36556* 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 36557* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 36556 everywhere. If this one is needed, make dvh3dim2 36556 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 36558 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 36559 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 36560 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 36561 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 36562 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 36563 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 36564 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 36565 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 36566 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 36522 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 36567 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 36568 Lemma for dochexmid 36576. 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 36569 Lemma for dochexmid 36576. (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 36570 Lemma for dochexmid 36576. Use atom exchange lsatexch1 34152 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 36571 Lemma for dochexmid 36576. (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 36572 Lemma for dochexmid 36576. (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 36573 Lemma for dochexmid 36576. (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 36574 Lemma for dochexmid 36576. Contradict dochexmidlem6 36573. (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 36575 Lemma for dochexmid 36576. The contradiction of dochexmidlem6 36573 and dochexmidlem7 36574 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 36576 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 36485. 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 35083 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 36577 Lemma for dochsnkr 36580. (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 36578 Lemma for dochsnkr 36580. (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 36579 Lemma for dochsnkr 36580. (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 36580 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 36581* Kernel of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkr 34223. (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 36582* 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 36583* Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 34222. (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 36584* 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 36585 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 36586* 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 34176. (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 36587* 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 34176. (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 )

20.22.14  Construction of involution and inner product from a Hilbert lattice

SyntaxclpoN 36588 Extend class notation with all polarities of a left module or left vector space.
class LPol

Definitiondf-lpolN 36589* 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 36590* 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 36591* 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 36592* 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 36593 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 36594 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 36595 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 36596 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 36597 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 36598 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 36599* Property of a functional with a closed kernel. (Contributed by NM, 28-Dec-2014.)
𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}       (𝐺𝐶 ↔ (𝐺𝐹 ∧ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))

Theoremlcfl1 36600* Property of a functional with a closed kernel. (Contributed by NM, 31-Dec-2014.)
𝐶 = {𝑓𝐹 ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}    &   (𝜑𝐺𝐹)       (𝜑 → (𝐺𝐶 ↔ ( ‘( ‘(𝐿𝐺))) = (𝐿𝐺)))

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