P. 373 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	diaelrnN 37201	Any value of the partial isomorphism A is a set of translations i.e. a set of vectors. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ ran 𝐼) → 𝑆 ⊆ 𝑇)
Theorem	dialss 37202	The value of partial isomorphism A is a subspace of partial vector space A. Part of Lemma M of [Crawley] p. 120 line 26. (Contributed by NM, 17-Jan-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ∈ 𝑆)
Theorem	diaord 37203	The partial isomorphism A for a lattice 𝐾 is order-preserving in the region under co-atom 𝑊. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 26-Nov-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) ⊆ (𝐼‘𝑌) ↔ 𝑋 ≤ 𝑌))
Theorem	dia11N 37204	The partial isomorphism A for a lattice 𝐾 is one-to-one in the region under co-atom 𝑊. Part of Lemma M of [Crawley] p. 120 line 28. (Contributed by NM, 25-Nov-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊) ∧ (𝑌 ∈ 𝐵 ∧ 𝑌 ≤ 𝑊)) → ((𝐼‘𝑋) = (𝐼‘𝑌) ↔ 𝑋 = 𝑌))
Theorem	diaf11N 37205	The partial isomorphism A for a lattice 𝐾 is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)
Theorem	diaclN 37206	Closure of partial isomorphism A for a lattice 𝐾. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘𝑋) ∈ ran 𝐼)
Theorem	diacnvclN 37207	Closure of partial isomorphism A converse. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ dom 𝐼)
Theorem	dia0 37208	The value of the partial isomorphism A at the lattice zero is the singleton of the identity translation i.e. the zero subspace. (Contributed by NM, 26-Nov-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 0 = (0.‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘ 0 ) = {( I ↾ 𝐵)})
Theorem	dia1N 37209	The value of the partial isomorphism A at the fiducial co-atom is the set of all translations i.e. the entire vector space. (Contributed by NM, 26-Nov-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝐼‘𝑊) = 𝑇)
Theorem	dia1elN 37210	The largest subspace in the range of partial isomorphism A. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑇 ∈ ran 𝐼)
Theorem	diaglbN 37211*	Partial isomorphism A of a lattice glb. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
⊢ 𝐺 = (glb‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ dom 𝐼 ∧ 𝑆 ≠ ∅)) → (𝐼‘(𝐺‘𝑆)) = ∩ 𝑥 ∈ 𝑆 (𝐼‘𝑥))
Theorem	diameetN 37212	Partial isomorphism A of a lattice meet. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∧ 𝑌)) = ((𝐼‘𝑋) ∩ (𝐼‘𝑌)))
Theorem	diainN 37213	Inverse partial isomorphism A of an intersection. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ ran 𝐼 ∧ 𝑌 ∈ ran 𝐼)) → (𝑋 ∩ 𝑌) = (𝐼‘((◡𝐼‘𝑋) ∧ (◡𝐼‘𝑌))))
Theorem	diaintclN 37214	The intersection of partial isomorphism A closed subspaces is a closed subspace. (Contributed by NM, 3-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑆 ⊆ ran 𝐼 ∧ 𝑆 ≠ ∅)) → ∩ 𝑆 ∈ ran 𝐼)
Theorem	diasslssN 37215	The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 ⊆ 𝑆)
Theorem	diassdvaN 37216	The partial isomorphism A maps to a set of vectors in partial vector space A. (Contributed by NM, 1-Jan-2014.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ (((𝐾 ∈ 𝑌 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) ⊆ 𝑉)
Theorem	dia1dim 37217*	Two expressions for the 1-dimensional subspaces of partial vector space A (when 𝐹 is a nonzero vector i.e. non-identity translation). Remark after Lemma L in [Crawley] p. 120 line 21. (Contributed by NM, 15-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = {𝑔 ∣ ∃𝑠 ∈ 𝐸 𝑔 = (𝑠‘𝐹)})
Theorem	dia1dim2 37218	Two expressions for a 1-dimensional subspace of partial vector space A (when 𝐹 is a nonzero vector i.e. non-identity translation). (Contributed by NM, 15-Jan-2014.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → (𝐼‘(𝑅‘𝐹)) = (𝑁‘{𝐹}))
Theorem	dia1dimid 37219	A vector (translation) belongs to the 1-dim subspace it generates. (Contributed by NM, 8-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇) → 𝐹 ∈ (𝐼‘(𝑅‘𝐹)))
Theorem	dia2dimlem1 37220	Lemma for dia2dim 37233. Show properties of the auxiliary atom 𝑄. Part of proof of Lemma M in [Crawley] p. 121 line 3. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    ⇒   ⊢ (𝜑 → (𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊))
Theorem	dia2dimlem2 37221	Lemma for dia2dim 37233. Define a translation 𝐺 whose trace is atom 𝑈. Part of proof of Lemma M in [Crawley] p. 121 line 4. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄)    ⇒   ⊢ (𝜑 → (𝑅‘𝐺) = 𝑈)
Theorem	dia2dimlem3 37222	Lemma for dia2dim 37233. Define a translation 𝐷 whose trace is atom 𝑉. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))    ⇒   ⊢ (𝜑 → (𝑅‘𝐷) = 𝑉)
Theorem	dia2dimlem4 37223	Lemma for dia2dim 37233. Show that the composition (sum) of translations (vectors) 𝐺 and 𝐷 equals 𝐹. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))    ⇒   ⊢ (𝜑 → (𝐷 ∘ 𝐺) = 𝐹)
Theorem	dia2dimlem5 37224	Lemma for dia2dim 37233. The sum of vectors 𝐺 and 𝐷 belongs to the sum of the subspaces generated by them. Thus, 𝐹 = (𝐺 ∘ 𝐷) belongs to the subspace sum. Part of proof of Lemma M in [Crawley] p. 121 line 5. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    &   ⊢ (𝜑 → 𝐺 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐺‘𝑃) = 𝑄)    &   ⊢ (𝜑 → 𝐷 ∈ 𝑇)    &   ⊢ (𝜑 → (𝐷‘𝑄) = (𝐹‘𝑃))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem6 37225	Lemma for dia2dim 37233. Eliminate auxiliary translations 𝐺 and 𝐷. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → (𝐹 ∈ 𝑇 ∧ (𝐹‘𝑃) ≠ 𝑃))    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem7 37226	Lemma for dia2dim 37233. Eliminate (𝐹‘𝑃) ≠ 𝑃 condition. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑄 = ((𝑃 ∨ 𝑈) ∧ ((𝐹‘𝑃) ∨ 𝑉))    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem8 37227	Lemma for dia2dim 37233. Eliminate no-longer used auxiliary atoms 𝑃 and 𝑄. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑈)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≠ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem9 37228	Lemma for dia2dim 37233. Eliminate (𝑅‘𝐹) ≠ 𝑈, 𝑉 conditions. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem10 37229	Lemma for dia2dim 37233. Convert membership in closed subspace (𝐼‘(𝑈 ∨ 𝑉)) to a lattice ordering. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)))    ⇒   ⊢ (𝜑 → (𝑅‘𝐹) ≤ (𝑈 ∨ 𝑉))
Theorem	dia2dimlem11 37230	Lemma for dia2dim 37233. Convert ordering hypothesis on 𝑅‘𝐹 to subspace membership 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)). (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → 𝐹 ∈ 𝑇)    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    &   ⊢ (𝜑 → 𝐹 ∈ (𝐼‘(𝑈 ∨ 𝑉)))    ⇒   ⊢ (𝜑 → 𝐹 ∈ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem12 37231	Lemma for dia2dim 37233. Obtain subset relation. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    &   ⊢ (𝜑 → 𝑈 ≠ 𝑉)    ⇒   ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dimlem13 37232	Lemma for dia2dim 37233. Eliminate 𝑈 ≠ 𝑉 condition. (Contributed by NM, 8-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑌)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝑁 = (LSpan‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Theorem	dia2dim 37233	A two-dimensional subspace of partial vector space A is closed, or equivalently, the isomorphism of a join of two atoms is a subset of the subspace sum of the isomorphisms of each atom (and thus they are equal, as shown later for the full vector space H). (Contributed by NM, 9-Sep-2014.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑌 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑌)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    &   ⊢ (𝜑 → (𝑈 ∈ 𝐴 ∧ 𝑈 ≤ 𝑊))    &   ⊢ (𝜑 → (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊))    ⇒   ⊢ (𝜑 → (𝐼‘(𝑈 ∨ 𝑉)) ⊆ ((𝐼‘𝑈) ⊕ (𝐼‘𝑉)))
Syntax	cdvh 37234	Extend class notation with constructed full vector space H.
class DVecH
Definition	df-dvech 37235*	Define constructed full vector space H. (Contributed by NM, 17-Oct-2013.)
⊢ DVecH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ ({⟨(Base‘ndx), (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)), 𝑔 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝑘)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝑘)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝑘)‘𝑤), 𝑓 ∈ (((LTrn‘𝑘)‘𝑤) × ((TEndo‘𝑘)‘𝑤)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
Theorem	dvhfset 37236*	The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (DVecH‘𝐾) = (𝑤 ∈ 𝐻 ↦ ({⟨(Base‘ndx), (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤))⟩, ⟨(+g‘ndx), (𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)), 𝑔 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ ((LTrn‘𝐾)‘𝑤) ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), ((EDRing‘𝐾)‘𝑤)⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ ((TEndo‘𝐾)‘𝑤), 𝑓 ∈ (((LTrn‘𝐾)‘𝑤) × ((TEndo‘𝐾)‘𝑤)) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩})))
Theorem	dvhset 37237*	The constructed full vector space H for a lattice 𝐾. (Contributed by NM, 17-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑈 = ({⟨(Base‘ndx), (𝑇 × 𝐸)⟩, ⟨(+g‘ndx), (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), (ℎ ∈ 𝑇 ↦ (((2nd ‘𝑓)‘ℎ) ∘ ((2nd ‘𝑔)‘ℎ)))⟩)⟩, ⟨(Scalar‘ndx), 𝐷⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)⟩}))
Theorem	dvhsca 37238	The ring of scalars of the constructed full vector space H. (Contributed by NM, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐷 = ((EDRing‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐹 = 𝐷)
Theorem	dvhbase 37239	The ring base set of the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝐶 = (Base‘𝐹)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝐶 = 𝐸)
Theorem	dvhfplusr 37240*	Ring addition operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))    &   ⊢ ✚ = (+g‘𝐹)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → ✚ = + )
Theorem	dvhfmulr 37241*	Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ · = (.r‘𝐹)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑠 ∘ 𝑡)))
Theorem	dvhmulr 37242	Ring multiplication operation for the constructed full vector space H. (Contributed by NM, 29-Oct-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ · = (.r‘𝐹)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝑆 ∈ 𝐸)) → (𝑅 · 𝑆) = (𝑅 ∘ 𝑆))
Theorem	dvhvbase 37243	The vectors (vector base set) of the constructed full vector space H are all translations (for a fiducial co-atom 𝑊). (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ ((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) → 𝑉 = (𝑇 × 𝐸))
Theorem	dvhelvbasei 37244	Vector membership in the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    ⇒   ⊢ (((𝐾 ∈ 𝑋 ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑆 ∈ 𝐸)) → ⟨𝐹, 𝑆⟩ ∈ 𝑉)
Theorem	dvhvaddcbv 37245*	Change bound variables to isolate them later. (Contributed by NM, 3-Nov-2013.)
⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩)    ⇒   ⊢ + = (ℎ ∈ (𝑇 × 𝐸), 𝑖 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘ℎ) ∘ (1st ‘𝑖)), ((2nd ‘ℎ) ⨣ (2nd ‘𝑖))⟩)
Theorem	dvhvaddval 37246*	The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.)
⊢ + = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩)    ⇒   ⊢ ((𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸)) → (𝐹 + 𝐺) = ⟨((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))⟩)
Theorem	dvhfvadd 37247*	The vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ ✚ = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓) ⨣ (2nd ‘𝑔))⟩)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → + = ✚ )
Theorem	dvhvadd 37248	The vector sum operation for the constructed full vector space H. (Contributed by NM, 11-Feb-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = ⟨((1st ‘𝐹) ∘ (1st ‘𝐺)), ((2nd ‘𝐹) ⨣ (2nd ‘𝐺))⟩)
Theorem	dvhopvadd 37249	The vector sum operation for the constructed full vector space H. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 6-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ + = (+g‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ + ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄 ⨣ 𝑅)⟩)
Theorem	dvhopvadd2 37250*	The vector sum operation for the constructed full vector space H. TODO: check if this will shorten proofs that use dvhopvadd 37249 and/or dvhfplusr 37240. (Contributed by NM, 26-Sep-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ + = (𝑠 ∈ 𝐸, 𝑡 ∈ 𝐸 ↦ (𝑓 ∈ 𝑇 ↦ ((𝑠‘𝑓) ∘ (𝑡‘𝑓))))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ ✚ = (+g‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑅 ∈ 𝐸)) → (⟨𝐹, 𝑄⟩ ✚ ⟨𝐺, 𝑅⟩) = ⟨(𝐹 ∘ 𝐺), (𝑄 + 𝑅)⟩)
Theorem	dvhvaddcl 37251	Closure of the vector sum operation for the constructed full vector space H. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) ∈ (𝑇 × 𝐸))
Theorem	dvhvaddcomN 37252	Commutativity of vector sum. (Contributed by NM, 26-Oct-2013.) (Revised by Mario Carneiro, 23-Jun-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸))) → (𝐹 + 𝐺) = (𝐺 + 𝐹))
Theorem	dvhvaddass 37253	Associativity of vector sum. (Contributed by NM, 31-Oct-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ + = (+g‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ (𝑇 × 𝐸) ∧ 𝐺 ∈ (𝑇 × 𝐸) ∧ 𝐼 ∈ (𝑇 × 𝐸))) → ((𝐹 + 𝐺) + 𝐼) = (𝐹 + (𝐺 + 𝐼)))
Theorem	dvhvscacbv 37254*	Change bound variables to isolate them later. (Contributed by NM, 20-Nov-2013.)
⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)    ⇒   ⊢ · = (𝑡 ∈ 𝐸, 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑡‘(1st ‘𝑔)), (𝑡 ∘ (2nd ‘𝑔))⟩)
Theorem	dvhvscaval 37255*	The scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Nov-2013.)
⊢ · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)    ⇒   ⊢ ((𝑈 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸)) → (𝑈 · 𝐹) = ⟨(𝑈‘(1st ‘𝐹)), (𝑈 ∘ (2nd ‘𝐹))⟩)
Theorem	dvhfvsca 37256*	Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑈)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → · = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩))
Theorem	dvhvsca 37257	Scalar product operation for the constructed full vector space H. (Contributed by NM, 2-Nov-2013.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑈)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) = ⟨(𝑅‘(1st ‘𝐹)), (𝑅 ∘ (2nd ‘𝐹))⟩)
Theorem	dvhopvsca 37258	Scalar product operation for the constructed full vector space H. (Contributed by NM, 20-Feb-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑈)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐸)) → (𝑅 · ⟨𝐹, 𝑋⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑋)⟩)
Theorem	dvhvscacl 37259	Closure of the scalar product operation for the constructed full vector space H. (Contributed by NM, 12-Feb-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑅 ∈ 𝐸 ∧ 𝐹 ∈ (𝑇 × 𝐸))) → (𝑅 · 𝐹) ∈ (𝑇 × 𝐸))
Theorem	tendoinvcl 37260*	Closure of multiplicative inverse for endomorphism. We use the scalar inverse of the vector space since it is much simpler than the direct inverse of cdleml8 37139. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝑁 = (invr‘𝐹)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∈ 𝐸 ∧ (𝑁‘𝑆) ≠ 𝑂))
Theorem	tendolinv 37261*	Left multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝑁 = (invr‘𝐹)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → ((𝑁‘𝑆) ∘ 𝑆) = ( I ↾ 𝑇))
Theorem	tendorinv 37262*	Right multiplicative inverse for endomorphism. (Contributed by NM, 10-Apr-2014.) (Revised by Mario Carneiro, 23-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑂 = (ℎ ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑈)    &   ⊢ 𝑁 = (invr‘𝐹)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑆 ∈ 𝐸 ∧ 𝑆 ≠ 𝑂) → (𝑆 ∘ (𝑁‘𝑆)) = ( I ↾ 𝑇))
Theorem	dvhgrp 37263	The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a group. (Contributed by NM, 19-Oct-2013.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐼 = (invg‘𝐷)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ Grp)
Theorem	dvhlveclem 37264	Lemma for dvhlvec 37265. TODO: proof substituting inner part first shorter/longer than substituting outer part first? TODO: break up into smaller lemmas? TODO: does 𝜑 → method shorten proof? (Contributed by NM, 22-Oct-2013.) (Proof shortened by Mario Carneiro, 24-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝐷 = (Scalar‘𝑈)    &   ⊢ ⨣ = (+g‘𝐷)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 0 = (0g‘𝐷)    &   ⊢ 𝐼 = (invg‘𝐷)    &   ⊢ × = (.r‘𝐷)    &   ⊢ · = ( ·𝑠 ‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝑈 ∈ LVec)
Theorem	dvhlvec 37265	The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a left module. (Contributed by NM, 23-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑈 ∈ LVec)
Theorem	dvhlmod 37266	The full vector space 𝑈 constructed from a Hilbert lattice 𝐾 (given a fiducial hyperplane 𝑊) is a left module. (Contributed by NM, 23-May-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝑈 ∈ LMod)
Theorem	dvh0g 37267*	The zero vector of vector space H has the zero translation as its first member and the zero trace-preserving endomorphism as the second. (Contributed by NM, 9-Mar-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝑂 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)
Theorem	dvheveccl 37268	Properties of a unit vector that we will use later as a convenient reference vector. This vector is called "e" in the remark after Lemma M of [Crawley] p. 121. line 17. See also dvhopN 37272 and dihpN 37492. (Contributed by NM, 27-Mar-2015.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑈)    &   ⊢ 0 = (0g‘𝑈)    &   ⊢ 𝐸 = ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩    &   ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))    ⇒   ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ { 0 }))
Theorem	dvhopclN 37269	Closure of a DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
⊢ ((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))
Theorem	dvhopaddN 37270*	Sum of DVecH vectors expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)    ⇒   ⊢ (((𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸) ∧ (𝐺 ∈ 𝑇 ∧ 𝑉 ∈ 𝐸)) → (⟨𝐹, 𝑈⟩𝐴⟨𝐺, 𝑉⟩) = ⟨(𝐹 ∘ 𝐺), (𝑈𝑃𝑉)⟩)
Theorem	dvhopspN 37271*	Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)    ⇒   ⊢ ((𝑅 ∈ 𝐸 ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → (𝑅𝑆⟨𝐹, 𝑈⟩) = ⟨(𝑅‘𝐹), (𝑅 ∘ 𝑈)⟩)
Theorem	dvhopN 37272*	Decompose a DVecH vector expressed as an ordered pair into the sum of two components, the first from the translation group vector base of DVecA and the other from the one-dimensional vector subspace 𝐸. Part of Lemma M of [Crawley] p. 121, line 18. We represent their e, sigma, f by ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩, 𝑈, ⟨𝐹, 𝑂⟩. We swapped the order of vector sum (their juxtaposition i.e. composition) to show ⟨𝐹, 𝑂⟩ first. Note that 𝑂 and ( I ↾ 𝑇) are the zero and one of the division ring 𝐸, and ( I ↾ 𝐵) is the zero of the translation group. 𝑆 is the scalar product. (Contributed by NM, 21-Nov-2013.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐸 = ((TEndo‘𝐾)‘𝑊)    &   ⊢ 𝑃 = (𝑎 ∈ 𝐸, 𝑏 ∈ 𝐸 ↦ (𝑐 ∈ 𝑇 ↦ ((𝑎‘𝑐) ∘ (𝑏‘𝑐))))    &   ⊢ 𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st ‘𝑓) ∘ (1st ‘𝑔)), ((2nd ‘𝑓)𝑃(2nd ‘𝑔))⟩)    &   ⊢ 𝑆 = (𝑠 ∈ 𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st ‘𝑓)), (𝑠 ∘ (2nd ‘𝑓))⟩)    &   ⊢ 𝑂 = (𝑐 ∈ 𝑇 ↦ ( I ↾ 𝐵))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐹 ∈ 𝑇 ∧ 𝑈 ∈ 𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂⟩𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))
Theorem	dvhopellsm 37273*	Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)    &   ⊢ + = (+g‘𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    &   ⊢ ⊕ = (LSSum‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (⟨𝐹, 𝑇⟩ ∈ (𝑋 ⊕ 𝑌) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((⟨𝑥, 𝑦⟩ ∈ 𝑋 ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑌) ∧ ⟨𝐹, 𝑇⟩ = (⟨𝑥, 𝑦⟩ + ⟨𝑧, 𝑤⟩))))
Theorem	cdlemm10N 37274*	The image of the map 𝐺 is the entire one-dimensional subspace (𝐼‘𝑉). Remark after Lemma M of [Crawley] p. 121 line 23. (Contributed by NM, 24-Nov-2013.) (New usage is discouraged.)
⊢ ≤ = (le‘𝐾)    &   ⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐴 = (Atoms‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝑅 = ((trL‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐶 = {𝑟 ∈ 𝐴 ∣ (𝑟 ≤ (𝑃 ∨ 𝑉) ∧ ¬ 𝑟 ≤ 𝑊)}    &   ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑠)    &   ⊢ 𝐺 = (𝑞 ∈ 𝐶 ↦ (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑞))    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊)) → ran 𝐺 = (𝐼‘𝑉))
Syntax	cocaN 37275	Extend class notation with subspace orthocomplement for DVecA partial vector space.
class ocA
Definition	df-docaN 37276*	Define subspace orthocomplement for DVecA partial 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, 6-Dec-2013.)
⊢ ocA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((DIsoA‘𝑘)‘𝑤)‘((((oc‘𝑘)‘(◡((DIsoA‘𝑘)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝑘)‘𝑤) ∣ 𝑥 ⊆ 𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))
Theorem	docaffvalN 37277*	Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (ocA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ⊥ ‘(◡((DIsoA‘𝐾)‘𝑤)‘∩ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑤)) ∧ 𝑤)))))
Theorem	docafvalN 37278*	Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑥 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊))))
Theorem	docavalN 37279*	Subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → (𝑁‘𝑋) = (𝐼‘((( ⊥ ‘(◡𝐼‘∩ {𝑧 ∈ ran 𝐼 ∣ 𝑋 ⊆ 𝑧})) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)))
Theorem	docaclN 37280	Closure of subspace orthocomplement for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ 𝑇) → ( ⊥ ‘𝑋) ∈ ran 𝐼)
Theorem	diaocN 37281	Value of partial isomorphism A at lattice orthocomplement (using a Sasaki projection to get orthocomplement relative to the fiducial co-atom 𝑊). (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ ∧ = (meet‘𝐾)    &   ⊢ ⊥ = (oc‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝑁 = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((( ⊥ ‘𝑋) ∨ ( ⊥ ‘𝑊)) ∧ 𝑊)) = (𝑁‘(𝐼‘𝑋)))
Theorem	doca2N 37282	Double orthocomplement of partial isomorphism A. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ⊥ ‘( ⊥ ‘(𝐼‘𝑋))) = (𝐼‘𝑋))
Theorem	doca3N 37283	Double orthocomplement of partial isomorphism A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)
Theorem	dvadiaN 37284	Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝑆 ∧ ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋)) → 𝑋 ∈ ran 𝐼)
Theorem	diarnN 37285*	Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = {𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥})
Theorem	diaf1oN 37286*	The partial isomorphism A for a lattice 𝐾 is a one-to-one, onto function. Part of Lemma M of [Crawley] p. 121 line 12, with closed subspaces rather than subspaces. See diadm 37191 for the domain. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑈 = ((DVecA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    &   ⊢ 𝑆 = (LSubSp‘𝑈)    ⇒   ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→{𝑥 ∈ 𝑆 ∣ ( ⊥ ‘( ⊥ ‘𝑥)) = 𝑥})
Syntax	cdjaN 37287	Extend class notation with subspace join for DVecA partial vector space.
class vA
Definition	df-djaN 37288*	Define (closed) subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.)
⊢ vA = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝑘)‘𝑤) ↦ (((ocA‘𝑘)‘𝑤)‘((((ocA‘𝑘)‘𝑤)‘𝑥) ∩ (((ocA‘𝑘)‘𝑤)‘𝑦))))))
Theorem	djaffvalN 37289*	Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (vA‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))
Theorem	djafvalN 37290*	Subspace join for DVecA partial vector space. TODO: take out hypothesis .i, no longer used. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ⊥ ‘(( ⊥ ‘𝑥) ∩ ( ⊥ ‘𝑦)))))
Theorem	djavalN 37291	Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ ⊥ = ((ocA‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) = ( ⊥ ‘(( ⊥ ‘𝑋) ∩ ( ⊥ ‘𝑌))))
Theorem	djaclN 37292	Closure of subspace join for DVecA partial vector space. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ⊆ 𝑇 ∧ 𝑌 ⊆ 𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼)
Theorem	djajN 37293	Transfer lattice join to DVecA partial vector space closed subspace join. Part of Lemma M of [Crawley] p. 120 line 29, with closed subspace join rather than subspace sum. (Contributed by NM, 5-Dec-2013.) (New usage is discouraged.)
⊢ ∨ = (join‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐽 = ((vA‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ dom 𝐼 ∧ 𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 ∨ 𝑌)) = ((𝐼‘𝑋)𝐽(𝐼‘𝑌)))
Syntax	cdib 37294	Extend class notation with isomorphism B.
class DIsoB
Definition	df-dib 37295*	Isomorphism B is isomorphism A extended with an extra dimension set to the zero vector component i.e. the zero endormorphism. Its domain is lattice elements less than or equal to the fiducial co-atom 𝑤. (Contributed by NM, 8-Dec-2013.)
⊢ DIsoB = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ dom ((DIsoA‘𝑘)‘𝑤) ↦ ((((DIsoA‘𝑘)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝑘)‘𝑤) ↦ ( I ↾ (Base‘𝑘)))}))))
Theorem	dibffval 37296*	The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    ⇒   ⊢ (𝐾 ∈ 𝑉 → (DIsoB‘𝐾) = (𝑤 ∈ 𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))
Theorem	dibfval 37297*	The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)    ⇒   ⊢ ((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽‘𝑥) × { 0 })))
Theorem	dibval 37298*	The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))
Theorem	dibopelvalN 37299*	Member of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼‘𝑋) ↔ (𝐹 ∈ (𝐽‘𝑋) ∧ 𝑆 = 0 )))
Theorem	dibval2 37300*	Value of the partial isomorphism B. (Contributed by NM, 18-Jan-2014.)
⊢ 𝐵 = (Base‘𝐾)    &   ⊢ ≤ = (le‘𝐾)    &   ⊢ 𝐻 = (LHyp‘𝐾)    &   ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊)    &   ⊢ 0 = (𝑓 ∈ 𝑇 ↦ ( I ↾ 𝐵))    &   ⊢ 𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   ⊢ 𝐼 = ((DIsoB‘𝐾)‘𝑊)    ⇒   ⊢ (((𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻) ∧ (𝑋 ∈ 𝐵 ∧ 𝑋 ≤ 𝑊)) → (𝐼‘𝑋) = ((𝐽‘𝑋) × { 0 }))