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Theorem List for Metamath Proof Explorer - 38601-38700   *Has distinct variable group(s)
TypeLabelDescription
Statement

Syntaxcoch 38601 Extend class notation with subspace orthocomplement for DVecH vector space.
class ocH

Definitiondf-doch 38602* Define subspace orthocomplement for DVecH vector space. Temporarily, we are using the range of the isomorphism instead of the set of closed subspaces. Later, when inner product is introduced, we will show that these are the same. (Contributed by NM, 14-Mar-2014.)
ocH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((DIsoH‘𝑘)‘𝑤)‘((oc‘𝑘)‘((glb‘𝑘)‘{𝑦 ∈ (Base‘𝑘) ∣ 𝑥 ⊆ (((DIsoH‘𝑘)‘𝑤)‘𝑦)}))))))

Theoremdochffval 38603* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (ocH‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))

Theoremdochfval 38604* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       ((𝐾𝑋𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑥 ⊆ (𝐼𝑦)})))))

Theoremdochval 38605* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
𝐵 = (Base‘𝐾)    &   𝐺 = (glb‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾𝑌𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐺‘{𝑦𝐵𝑋 ⊆ (𝐼𝑦)}))))

Theoremdochval2 38606* Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Apr-2014.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → (𝑁𝑋) = (𝐼‘( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧}))))

Theoremdochcl 38607 Closure of subspace orthocomplement for DVecH vector space. (Contributed by NM, 9-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ∈ ran 𝐼)

Theoremdochlss 38608 A subspace orthocomplement is a subspace of the DVecH vector space. (Contributed by NM, 22-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑆 = (LSubSp‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ∈ 𝑆)

Theoremdochssv 38609 A subspace orthocomplement belongs to the DVecH vector space. (Contributed by NM, 22-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → ( 𝑋) ⊆ 𝑉)

TheoremdochfN 38610 Domain and codomain of the subspace orthocomplement for the DVecH vector space. (Contributed by NM, 27-Dec-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 :𝒫 𝑉⟶ran 𝐼)

Theoremdochvalr 38611 Orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁𝑋) = (𝐼‘( ‘(𝐼𝑋))))

Theoremdoch0 38612 Orthocomplement of the zero subspace. (Contributed by NM, 19-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( ‘{ 0 }) = 𝑉)

Theoremdoch1 38613 Orthocomplement of the unit subspace (all vectors). (Contributed by NM, 19-Jun-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → ( 𝑉) = { 0 })

Theoremdochoc0 38614 The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ( ‘( ‘{ 0 })) = { 0 })

Theoremdochoc1 38615 The unit subspace (all vectors) is closed. (Contributed by NM, 16-Feb-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑 → ( ‘( 𝑉)) = 𝑉)

Theoremdochvalr2 38616 Orthocomplement of a closed subspace. (Contributed by NM, 21-Jul-2014.)
𝐵 = (Base‘𝐾)    &    = (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵) → (𝑁‘(𝐼𝑋)) = (𝐼‘( 𝑋)))

Theoremdochvalr3 38617 Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.)
= (oc‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)       (𝜑 → ( ‘(𝐼𝑋)) = (𝐼‘(𝑁𝑋)))

Theoremdoch2val2 38618* Double orthocomplement for DVecH vector space. (Contributed by NM, 26-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘( 𝑋)) = {𝑧 ∈ ran 𝐼𝑋𝑧})

Theoremdochss 38619 Subset law for orthocomplement. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑌𝑉𝑋𝑌) → ( 𝑌) ⊆ ( 𝑋))

Theoremdochocss 38620 Double negative law for orthocomplement of an arbitrary set of vectors. (Contributed by NM, 16-Apr-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑉) → 𝑋 ⊆ ( ‘( 𝑋)))

Theoremdochoc 38621 Double negative law for orthocomplement of a closed subspace. (Contributed by NM, 14-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ‘( 𝑋)) = 𝑋)

Theoremdochsscl 38622 If a set of vectors is included in a closed set, so is its closure. (Contributed by NM, 17-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( ‘( 𝑋)) ⊆ 𝑌))

Theoremdochoccl 38623 A set of vectors is closed iff it equals its double orthocomplent. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋 ∈ ran 𝐼 ↔ ( ‘( 𝑋)) = 𝑋))

Theoremdochord 38624 Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( 𝑌) ⊆ ( 𝑋)))

Theoremdochord2N 38625 Ordering law for orthocomplement. (Contributed by NM, 29-Oct-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (( 𝑋) ⊆ 𝑌 ↔ ( 𝑌) ⊆ 𝑋))

Theoremdochord3 38626 Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋 ⊆ ( 𝑌) ↔ 𝑌 ⊆ ( 𝑋)))

Theoremdoch11 38627 Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (( 𝑋) = ( 𝑌) ↔ 𝑋 = 𝑌))

TheoremdochsordN 38628 Strict ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)       (𝜑 → (𝑋𝑌 ↔ ( 𝑌) ⊊ ( 𝑋)))

Theoremdochn0nv 38629 An orthocomplement is nonzero iff the double orthocomplement is not the whole vector space. (Contributed by NM, 1-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (( 𝑋) ≠ { 0 } ↔ ( ‘( 𝑋)) ≠ 𝑉))

Theoremdihoml4c 38630 Version of dihoml4 38631 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋 ∈ ran 𝐼)    &   (𝜑𝑌 ∈ ran 𝐼)    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)

Theoremdihoml4 38631 Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 37207 analog.) (Contributed by NM, 15-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = ((ocH‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑆)    &   (𝜑𝑌𝑆)    &   (𝜑 → ( ‘( 𝑌)) = 𝑌)    &   (𝜑𝑋𝑌)       (𝜑 → (( ‘(( 𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ‘( 𝑋)))

Theoremdochspss 38632 The span of a set of vectors is included in their double orthocomplement. (Contributed by NM, 26-Jul-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁𝑋) ⊆ ( ‘( 𝑋)))

Theoremdochocsp 38633 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘(𝑁𝑋)) = ( 𝑋))

TheoremdochspocN 38634 The span of an orthocomplement equals the orthocomplement of the span. (Contributed by NM, 7-Aug-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁‘( 𝑋)) = ( ‘(𝑁𝑋)))

Theoremdochocsn 38635 The double orthocomplement of a singleton is its span. (Contributed by NM, 13-Jan-2015.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   𝑁 = (LSpan‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)       (𝜑 → ( ‘( ‘{𝑋})) = (𝑁‘{𝑋}))

Theoremdochsncom 38636 Swap vectors in an orthocomplement of a singleton. (Contributed by NM, 17-Jun-2015.)
𝐻 = (LHyp‘𝐾)    &    = ((ocH‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 ∈ ( ‘{𝑌}) ↔ 𝑌 ∈ ( ‘{𝑋})))

Theoremdochsat 38637 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 38638 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 38639 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 38640 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 38641 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 38642 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 38643 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 38644 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 38645 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 38646 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 38647 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 38648 Extend class notation with subspace join for DVecH vector space.
class joinH

Definitiondf-djh 38649* 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 38650* 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 38651* 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 38652 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 38653 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 38654 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 38655 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 38656 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 38657 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 38658 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 38659 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 38660 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 38661 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 38662 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 38663 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 38664 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 38665 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 38666 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 38667 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 38668 A closed subspace sum equals subspace join. (shjshseli 29274 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 38669* A covering property. (cvrat42 36698 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 38670 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 38671 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 38672 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 38673 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 38674* Lemma for dihjatcc 38676. (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 38675* 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 38676 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 38677 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 38678 Lemma for dihprrn 38680, 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 38679 Lemma for dihprrn 38680. (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 38680 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 38681 The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 38680; should we directly use dihjat 38677? (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 38682 Subspace sum of a closed subspace and an atom. (pmapjat1 37107 analog.) TODO: merge into dihjat1 38683? (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 38683 Subspace sum of a closed subspace and an atom. (pmapjat1 37107 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 38684 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 38685 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 38686 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 38687 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 38688 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 38689 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 38690 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 38685. (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 38691 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 38692* 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 38693* 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 38694* 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 38695* 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 38696* 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 38697* Lemma for dvh4dimN 38701. (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 38698* Lemma for dvh2dim 38699 and dvh3dim 38700. TODO: make this obsolete and use dvh4dimlem 38697 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 38699* 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 38700* 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}))

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