P. 399 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 39801-39900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	dihatexv 39801*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 16-Aug-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })(𝐼‘𝑄) = (𝑁‘{𝑥})))
Theorem	dihatexv2 39802*	There is a nonzero vector that maps to every lattice atom. (Contributed by NM, 17-Aug-2014.)
⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ↔ ∃𝑥 ∈ (𝑉 ∖ { 0 })𝑄 = (◡𝐼‘(𝑁‘{𝑥}))))
Theorem	dihglblem6 39803*	Isomorphism H of a lattice glb. (Contributed by NM, 9-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑃 = (LSubSp‘𝑈)    &   ⊢ 𝐷 = (LSAtoms‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	dihglb 39804*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ 𝐵 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	dihglb2 39805*	Isomorphism H of a lattice glb. (Contributed by NM, 11-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ⊆ 𝑉) → (𝐼‘(𝐺‘{𝑥 ∈ 𝐵 ∣ 𝑆 ⊆ (𝐼‘𝑥)})) = ∩ {𝑦 ∈ ran 𝐼 ∣ 𝑆 ⊆ 𝑦})
Theorem	dihmeet 39806	Isomorphism H of a lattice meet. (Contributed by NM, 13-Apr-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	dihintcl 39807	The intersection of closed subspaces (the range of isomorphism H) is a closed subspace. (Contributed by NM, 14-Apr-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)
Theorem	dihmeetcl 39808	Closure of closed subspace meet for DVecH vector space. (Contributed by NM, 5-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (𝑋 ∩ 𝑌) ∈ ran 𝐼)
Theorem	dihmeet2 39809	Reverse isomorphism H of a closed subspace intersection. (Contributed by NM, 15-Jan-2015.)
⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (◡𝐼‘(𝑋 ∩ 𝑌)) = ((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌)))
Syntax	coch 39810	Extend class notation with subspace orthocomplement for DVecH vector space.
class ocH
Definition	df-doch 39811*	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)}))))))
Theorem	dochffval 39812*	Subspace orthocomplement for DVecH vector space. (Contributed by NM, 14-Mar-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐺 = (glb‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (ocH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((DIsoH‘𝐾)‘𝑤)‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (((DIsoH‘𝐾)‘𝑤)‘𝑦)}))))))
Theorem	dochfval 39813*	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)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑉 ↦ (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑥 ⊆ (𝐼‘𝑦)})))))
Theorem	dochval 39814*	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)    ⇒   ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(𝐺‘{𝑦 ∈ 𝐵 ∣ 𝑋 ⊆ (𝐼‘𝑦)}))))
Theorem	dochval2 39815*	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧}))))
Theorem	dochcl 39816	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ ran 𝐼)
Theorem	dochlss 39817	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ∈ 𝑆)
Theorem	dochssv 39818	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → ( ⊥ ‘𝑋) ⊆ 𝑉)
Theorem	dochfN 39819	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ⊥ :𝒫 𝑉⟶ran 𝐼)
Theorem	dochvalr 39820	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋))))
Theorem	doch0 39821	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 = (0g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{ 0 }) = 𝑉)
Theorem	doch1 39822	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 = (0g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘𝑉) = { 0 })
Theorem	dochoc0 39823	The zero subspace is closed. (Contributed by NM, 16-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{ 0 })) = { 0 })
Theorem	dochoc1 39824	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 ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
Theorem	dochvalr2 39825	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝐼‘𝑋)) = (𝐼‘( ⊥ ‘𝑋)))
Theorem	dochvalr3 39826	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋)))
Theorem	doch2val2 39827*	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = ∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})
Theorem	dochss 39828	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ 𝑉 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))
Theorem	dochocss 39829	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → 𝑋 ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	dochoc 39830	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	dochsscl 39831	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ 𝑌))
Theorem	dochoccl 39832	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ ran 𝐼 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋))
Theorem	dochord 39833	Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)))
Theorem	dochord2N 39834	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (( ⊥ ‘𝑋) ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ 𝑋))
Theorem	dochord3 39835	Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋)))
Theorem	doch11 39836	Orthocomplement is one-to-one. (Contributed by NM, 12-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (( ⊥ ‘𝑋) = ( ⊥ ‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	dochsordN 39837	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ⊊ 𝑌 ↔ ( ⊥ ‘𝑌) ⊊ ( ⊥ ‘𝑋)))
Theorem	dochn0nv 39838	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 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (( ⊥ ‘𝑋) ≠ { 0 } ↔ ( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑉))
Theorem	dihoml4c 39839	Version of dihoml4 39840 with closed subspaces. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = 𝑋)
Theorem	dihoml4 39840	Orthomodular law for constructed vector space H. Lemma 3.3(1) in [Holland95] p. 215. (poml4N 38416 analog.) (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    &   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌)    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑌)    ⇒   ⊢ (𝜑 → (( ⊥ ‘(( ⊥ ‘𝑋) ∩ 𝑌)) ∩ 𝑌) = ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	dochspss 39841	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘𝑋) ⊆ ( ⊥ ‘( ⊥ ‘𝑋)))
Theorem	dochocsp 39842	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘(𝑁‘𝑋)) = ( ⊥ ‘𝑋))
Theorem	dochspocN 39843	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘( ⊥ ‘𝑋)) = ( ⊥ ‘(𝑁‘𝑋)))
Theorem	dochocsn 39844	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘{𝑋})) = (𝑁‘{𝑋}))
Theorem	dochsncom 39845	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑌}) ↔ 𝑌 ∈ ( ⊥ ‘{𝑋})))
Theorem	dochsat 39846	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑄 ∈ 𝑆)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑄)) ∈ 𝐴 ↔ 𝑄 ∈ 𝐴))
Theorem	dochshpncl 39847	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑌)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑋)) ≠ 𝑋 ↔ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑉))
Theorem	dochlkr 39848	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
Theorem	dochkrshp 39849	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ∈ 𝑌))
Theorem	dochkrshp2 39850	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ∈ 𝑌)))
Theorem	dochkrshp3 39851	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ∧ (𝐿‘𝐺) ≠ 𝑉)))
Theorem	dochkrshp4 39852	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 ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝐺 ∈ 𝐹)    ⇒   ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺) ↔ (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ∨ (𝐿‘𝐺) = 𝑉)))
Theorem	dochdmj1 39853	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 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → ( ⊥ ‘(𝑋 ∪ 𝑌)) = (( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌)))
Theorem	dochnoncon 39854	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 = (0g‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆) → (𝑋 ∩ ( ⊥ ‘𝑋)) = { 0 })
Theorem	dochnel2 39855	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 = (0g‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑇 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑇 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘𝑇))
Theorem	dochnel 39856	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 = (0g‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋}))
Syntax	cdjh 39857	Extend class notation with subspace join for DVecH vector space.
class joinH
Definition	df-djh 39858*	Define (closed) subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
⊢ joinH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝑘)‘𝑤)) ↦ (((ocH‘𝑘)‘𝑤)‘((((ocH‘𝑘)‘𝑤)‘𝑥) ∩ (((ocH‘𝑘)‘𝑤)‘𝑦))))))
Theorem	djhffval 39859*	Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑋 → (joinH‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)), 𝑦 ∈ 𝒫 (Base‘((DVecH‘𝐾)‘𝑤)) ↦ (((ocH‘𝐾)‘𝑤)‘((((ocH‘𝐾)‘𝑤)‘𝑥) ∩ (((ocH‘𝐾)‘𝑤)‘𝑦))))))
Theorem	djhfval 39860*	Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → ∨ = (𝑥 ∈ 𝒫 𝑉, 𝑦 ∈ 𝒫 𝑉 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
Theorem	djhval 39861	Subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
Theorem	djhval2 39862	Value of subspace join for DVecH vector space. (Contributed by NM, 6-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉) → (𝑋 ∨ 𝑌) = ( ⊥ ‘( ⊥ ‘(𝑋 ∪ 𝑌))))
Theorem	djhcl 39863	Closure of subspace join for DVecH vector space. (Contributed by NM, 19-Jul-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑉 ∧ 𝑌 ⊆ 𝑉)) → (𝑋 ∨ 𝑌) ∈ ran 𝐼)
Theorem	djhlj 39864	Transfer lattice join to DVecH vector space closed subspace join. (Contributed by NM, 19-Jul-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
Theorem	djhljjN 39865	Lattice join in terms of DVecH vector space closed subspace join. (Contributed by NM, 17-Aug-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (◡𝐼‘((𝐼‘𝑋)𝐽(𝐼‘𝑌))))
Theorem	djhjlj 39866	DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 9-Aug-2014.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋𝐽𝑌) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌))))
Theorem	djhj 39867	DVecH vector space closed subspace join in terms of lattice join. (Contributed by NM, 17-Aug-2014.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (◡𝐼‘(𝑋𝐽𝑌)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑌)))
Theorem	djhcom 39868	Subspace join commutes. (Contributed by NM, 8-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑌) = (𝑌 ∨ 𝑋))
Theorem	djhspss 39869	Subspace span of union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘(𝑋 ∪ 𝑌)) ⊆ (𝑋 ∨ 𝑌))
Theorem	djhsumss 39870	Subspace sum is a subset of subspace join. (Contributed by NM, 6-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ 𝑌) ⊆ (𝑋 ∨ 𝑌))
Theorem	dihsumssj 39871	The subspace sum of two isomorphisms of lattice elements is less than the isomorphism of their lattice join. (Contributed by NM, 23-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    ⇒   ⊢ (𝜑 → ((𝐼‘𝑋) ⊕ (𝐼‘𝑌)) ⊆ (𝐼‘(𝑋 ∨ 𝑌)))
Theorem	djhunssN 39872	Subspace union is a subset of subspace join. (Contributed by NM, 6-Aug-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ⊆ 𝑉)    &   ⊢ (𝜑 → 𝑌 ⊆ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∪ 𝑌) ⊆ (𝑋 ∨ 𝑌))
Theorem	dochdmm1 39873	De Morgan-like law for closed subspace orthocomplement. (Contributed by NM, 13-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑌 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → ( ⊥ ‘(𝑋 ∩ 𝑌)) = (( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑌)))
Theorem	djhexmid 39874	Excluded middle property of DVecH vector space closed subspace join. (Contributed by NM, 22-Jul-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑉) → (𝑋 ∨ ( ⊥ ‘𝑋)) = 𝑉)
Theorem	djh01 39875	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ∨ { 0 }) = 𝑋)
Theorem	djh02 39876	Closed subspace join with zero. (Contributed by NM, 9-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    ⇒   ⊢ (𝜑 → ({ 0 } ∨ 𝑋) = 𝑋)
Theorem	djhlsmcl 39877	A closed subspace sum equals subspace join. (shjshseli 30435 analog.) (Contributed by NM, 13-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑆)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑆)    ⇒   ⊢ (𝜑 → ((𝑋 ⊕ 𝑌) ∈ ran 𝐼 ↔ (𝑋 ⊕ 𝑌) = (𝑋 ∨ 𝑌)))
Theorem	djhcvat42 39878*	A covering property. (cvrat42 37907 analog.) (Contributed by NM, 17-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑆 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))    &   ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → ((𝑆 ≠ { 0 } ∧ (𝑁‘{𝑋}) ⊆ (𝑆 ∨ (𝑁‘{𝑌}))) → ∃𝑧 ∈ (𝑉 ∖ { 0 })((𝑁‘{𝑧}) ⊆ 𝑆 ∧ (𝑁‘{𝑋}) ⊆ ((𝑁‘{𝑧}) ∨ (𝑁‘{𝑌})))))
Theorem	dihjatb 39879	Isomorphism H of lattice join of two atoms under the fiducial hyperplane. (Contributed by NM, 23-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ 𝑄 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjatc 39880	Isomorphism H of lattice join of an element under the fiducial hyperplane with atom not under it. (Contributed by NM, 26-Aug-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
Theorem	dihjatcclem1 39881	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = (((𝐼‘𝑃) ⊕ (𝐼‘𝑄)) ⊕ (𝐼‘𝑉)))
Theorem	dihjatcclem2 39882	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjatcclem3 39883*	Lemma for dihjatcc 39885. (Contributed by NM, 28-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    &   ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)    &   ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)    ⇒   ⊢ (𝜑 → (𝑅‘(𝐺 ∘ ◡𝐷)) = 𝑉)
Theorem	dihjatcclem4 39884*	Lemma for isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = ((𝑃 ∨ 𝑄) ∧ 𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    &   ⊢ 𝐶 = ((oc‘𝐾)‘𝑊)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐺 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑃)    &   ⊢ 𝐷 = (℩𝑑 ∈ 𝑇 (𝑑‘𝐶) = 𝑄)    &   ⊢ 𝑁 = (𝑎 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ◡(𝑎‘𝑑)))    &   ⊢ 0 = (𝑑 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑑 ∈ 𝑇 ↦ ((𝑎‘𝑑) ∘ (𝑏‘𝑑))))    ⇒   ⊢ (𝜑 → (𝐼‘𝑉) ⊆ ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjatcc 39885	Isomorphism H of lattice join of two atoms not under the fiducial hyperplane. (Contributed by NM, 29-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihjat 39886	Isomorphism H of lattice join of two atoms. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑃 ∨ 𝑄)) = ((𝐼‘𝑃) ⊕ (𝐼‘𝑄)))
Theorem	dihprrnlem1N 39887	Lemma for dihprrn 39889, showing one of 4 cases. (Contributed by NM, 30-Aug-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    &   ⊢ (𝜑 → (◡𝐼‘(𝑁‘{𝑋})) ≤ 𝑊)    &   ⊢ (𝜑 → ¬ (◡𝐼‘(𝑁‘{𝑌})) ≤ 𝑊)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	dihprrnlem2 39888	Lemma for dihprrn 39889. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑋 ≠ 0 )    &   ⊢ (𝜑 → 𝑌 ≠ 0 )    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	dihprrn 39889	The span of a vector pair belongs to the range of isomorphism H i.e. is a closed subspace. (Contributed by NM, 29-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ ran 𝐼)
Theorem	djhlsmat 39890	The sum of two subspace atoms equals their join. TODO: seems convoluted to go via dihprrn 39889; should we directly use dihjat 39886? (Contributed by NM, 13-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) = ((𝑁‘{𝑋}) ∨ (𝑁‘{𝑌})))
Theorem	dihjat1lem 39891	Subspace sum of a closed subspace and an atom. (pmapjat1 38316 analog.) TODO: merge into dihjat1 39892? (Contributed by NM, 18-Aug-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ (𝜑 → 𝑇 ∈ (𝑉 ∖ { 0 }))    ⇒   ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇})))
Theorem	dihjat1 39892	Subspace sum of a closed subspace and an atom. (pmapjat1 38316 analog.) (Contributed by NM, 1-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ∨ (𝑁‘{𝑇})) = (𝑋 ⊕ (𝑁‘{𝑇})))
Theorem	dihsmsprn 39893	Subspace sum of a closed subspace and the span of a singleton. (Contributed by NM, 17-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑇 ∈ 𝑉)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ (𝑁‘{𝑇})) ∈ ran 𝐼)
Theorem	dihjat2 39894	The subspace sum of a closed subspace and an atom is the same as their subspace join. (Contributed by NM, 1-Oct-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ ∨ = ((joinH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑄) = (𝑋 ⊕ 𝑄))
Theorem	dihjat3 39895	Isomorphism H of lattice join with an atom. (Contributed by NM, 25-Apr-2015.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑋 ∨ 𝑃)) = ((𝐼‘𝑋) ⊕ (𝐼‘𝑃)))
Theorem	dihjat4 39896	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ 𝑄) = (𝐼‘((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄))))
Theorem	dihjat6 39897	Transfer the subspace sum of a closed subspace and an atom back to lattice join. (Contributed by NM, 25-Apr-2015.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (◡𝐼‘(𝑋 ⊕ 𝑄)) = ((◡𝐼‘𝑋) ∨ (◡𝐼‘𝑄)))
Theorem	dihsmsnrn 39898	The subspace sum of two singleton spans is closed. (Contributed by NM, 27-Feb-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝑁 = (LSpan‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊕ (𝑁‘{𝑌})) ∈ ran 𝐼)
Theorem	dihsmatrn 39899	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 39894. (Contributed by NM, 15-Jan-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐴 = (LSAtoms‘𝑈)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ ran 𝐼)    &   ⊢ (𝜑 → 𝑄 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ⊕ 𝑄) ∈ ran 𝐼)
Theorem	dihjat5N 39900	Transfer lattice join with atom to subspace sum. (Contributed by NM, 25-Apr-2015.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑈)    &   ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑃 ∈ 𝐴)    ⇒   ⊢ (𝜑 → (𝑋 ∨ 𝑃) = (◡𝐼‘((𝐼‘𝑋) ⊕ (𝐼‘𝑃))))