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Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdochsat 38401 The double orthocomplement of an atom is an atom. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &   𝐴 = (LSAtoms‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑄𝑆)       (𝜑 → (( ‘( 𝑄)) ∈ 𝐴𝑄𝐴))
 
Theoremdochshpncl 38402 If a hyperplane is not closed, its closure equals the vector space. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘( 𝑋)) ≠ 𝑋 ↔ ( ‘( 𝑋)) = 𝑉))
 
Theoremdochlkr 38403 Equivalent conditions for the closure of a kernel to be a hyperplane. (Contributed by NM, 29-Oct-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (LFnl‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ∈ 𝑌 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ∈ 𝑌)))
 
Theoremdochkrshp 38404 The closure of a kernel is a hyperplane iff it doesn't contain all vectors. (Contributed by NM, 1-Nov-2014.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ ( ‘( ‘(𝐿𝐺))) ∈ 𝑌))
 
Theoremdochkrshp2 38405 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑌 = (LSHyp‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ∈ 𝑌)))
 
Theoremdochkrshp3 38406 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ↔ (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ∧ (𝐿𝐺) ≠ 𝑉)))
 
Theoremdochkrshp4 38407 Properties of the closure of the kernel of a functional. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐹 = (LFnl‘𝑈)    &   𝐿 = (LKer‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝐺𝐹)       (𝜑 → (( ‘( ‘(𝐿𝐺))) = (𝐿𝐺) ↔ (( ‘( ‘(𝐿𝐺))) ≠ 𝑉 ∨ (𝐿𝐺) = 𝑉)))
 
Theoremdochdmj1 38408 De Morgan-like law for subspace orthocomplement. (Contributed by NM, 5-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉𝑌𝑉) → ( ‘(𝑋𝑌)) = (( 𝑋) ∩ ( 𝑌)))
 
Theoremdochnoncon 38409 Law of noncontradiction. The intersection of a subspace and its orthocomplement is the zero subspace. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    0 = (0g𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑆) → (𝑋 ∩ ( 𝑋)) = { 0 })
 
Theoremdochnel2 38410 A nonzero member of a subspace doesn't belong to the orthocomplement of the subspace. (Contributed by NM, 28-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    0 = (0g𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑇𝑆)    &   (𝜑𝑋 ∈ (𝑇 ∖ { 0 }))       (𝜑 → ¬ 𝑋 ∈ ( 𝑇))
 
Theoremdochnel 38411 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 38412 Extend class notation with subspace join for DVecH vector space.
class joinH
 
Definitiondf-djh 38413* 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 38414* 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 38415* 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 38416 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 38417 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 38418 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 38419 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 38420 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 38421 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 38422 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 38423 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 38424 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 38425 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 38426 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 38427 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 38428 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 38429 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 38430 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 38431 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 38432 A closed subspace sum equals subspace join. (shjshseli 29198 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 38433* A covering property. (cvrat42 36462 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 38434 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 38435 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 38436 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 38437 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 38438* Lemma for dihjatcc 38440. (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 38439* 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 38440 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 38441 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 38442 Lemma for dihprrn 38444, 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 38443 Lemma for dihprrn 38444. (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 38444 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 38445 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 38444; should we directly use dihjat 38441? (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 38446 Subspace sum of a closed subspace and an atom. (pmapjat1 36871 analog.) TODO: merge into dihjat1 38447? (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 38447 Subspace sum of a closed subspace and an atom. (pmapjat1 36871 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 38448 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 38449 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 38450 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 38451 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 38452 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 38453 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 38454 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 38449. (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 38455 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 38456* 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 38457* 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 38458* 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 38459* 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 38460* 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 38461* Lemma for dvh4dimN 38465. (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 38462* Lemma for dvh2dim 38463 and dvh3dim 38464. TODO: make this obsolete and use dvh4dimlem 38461 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 38463* 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 38464* 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 38465* 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 38466* 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 38467* There is a vector that is outside of 2 spans. TODO: decide to use either this or dvh3dim2 38466 everywhere. If this one is needed, make dvh3dim2 38466 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 38468 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 38469 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 38470 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 38471 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 38472 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 38473 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 38474 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 38475 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 38476 The subspace sum of a closed subspace and a kernel orthocomplement is closed. (djhlsmcl 38432 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 38477 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 38478 Lemma for dochexmid 38486. 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 38479 Lemma for dochexmid 38486. (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 38480 Lemma for dochexmid 38486. Use atom exchange lsatexch1 36064 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 38481 Lemma for dochexmid 38486. (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 38482 Lemma for dochexmid 38486. (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 38483 Lemma for dochexmid 38486. (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 38484 Lemma for dochexmid 38486. Contradict dochexmidlem6 38483. (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 38485 Lemma for dochexmid 38486. The contradiction of dochexmidlem6 38483 and dochexmidlem7 38484 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 38486 Excluded middle law for closed subspaces, which is equivalent to (and derived from) the orthomodular law dihoml4 38395. 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 36996 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 38487 Lemma for dochsnkr 38490. (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 38488 Lemma for dochsnkr 38490. (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 38489 Lemma for dochsnkr 38490. (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 38490 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 38491* Kernel of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkr 36135. (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 38492* 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 38493* Closure of the explicit functional 𝐺 determined by a nonzero vector 𝑋. Compare the more general lshpkrcl 36134. (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 38494* 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 38495 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 38496* 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 36088. (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 38497* 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 36088. (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 )
 
20.24.14  Construction of involution and inner product from a Hilbert lattice
 
SyntaxclpoN 38498 Extend class notation with all polarities of a left module or left vector space.
class LPol
 
Definitiondf-lpolN 38499* 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 38500* 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))})
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