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Theorem List for Metamath Proof Explorer - 37701-37800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdibval 37701* 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 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
 
TheoremdibopelvalN 37702* 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 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋 ∈ dom 𝐽) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
 
Theoremdibval2 37703* 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 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ((𝐽𝑋) × { 0 }))
 
Theoremdibopelval2 37704* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ (𝐹 ∈ (𝐽𝑋) ∧ 𝑆 = 0 )))
 
Theoremdibval3N 37705* Value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = ({𝑓𝑇 ∣ (𝑅𝑓) 𝑋} × { 0 }))
 
Theoremdibelval3 37706* Member of the partial isomorphism B. (Contributed by NM, 26-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝑌 ∈ (𝐼𝑋) ↔ ∃𝑓𝑇 (𝑌 = ⟨𝑓, 0 ⟩ ∧ (𝑅𝑓) 𝑋)))
 
Theoremdibopelval3 37707* Member of the partial isomorphism B. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &    0 = (𝑔𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 0 )))
 
Theoremdibelval1st 37708 Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ (𝐽𝑋))
 
Theoremdibelval1st1 37709 Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (1st𝑌) ∈ 𝑇)
 
Theoremdibelval1st2N 37710 Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (𝑅‘(1st𝑌)) 𝑋)
 
Theoremdibelval2nd 37711* Membership in value of the partial isomorphism B for a lattice 𝐾. (Contributed by NM, 13-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &    0 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ 𝑌 ∈ (𝐼𝑋)) → (2nd𝑌) = 0 )
 
Theoremdibn0 37712 The value of the partial isomorphism B is not empty. (Contributed by NM, 18-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ≠ ∅)
 
Theoremdibfna 37713 Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 Fn dom 𝐽)
 
Theoremdibdiadm 37714 Domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → dom 𝐼 = dom 𝐽)
 
TheoremdibfnN 37715* Functionality and domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
 
TheoremdibdmN 37716* Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
 
TheoremdibeldmN 37717 Member of domain of the partial isomorphism B. (Contributed by NM, 17-Jan-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → (𝑋 ∈ dom 𝐼 ↔ (𝑋𝐵𝑋 𝑊)))
 
Theoremdibord 37718 The isomorphism B for a lattice 𝐾 is order-preserving in the region under co-atom 𝑊. (Contributed by NM, 24-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → ((𝐼𝑋) ⊆ (𝐼𝑌) ↔ 𝑋 𝑌))
 
Theoremdib11N 37719 The isomorphism B for a lattice 𝐾 is one-to-one in the region under co-atom 𝑊. (Contributed by NM, 24-Feb-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊) ∧ (𝑌𝐵𝑌 𝑊)) → ((𝐼𝑋) = (𝐼𝑌) ↔ 𝑋 = 𝑌))
 
Theoremdibf11N 37720 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 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
 
TheoremdibclN 37721 Closure of partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼𝑋) ∈ ran 𝐼)
 
Theoremdibvalrel 37722 The value of partial isomorphism B is a relation. (Contributed by NM, 8-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
 
Theoremdib0 37723 The value of partial isomorphism B at the lattice zero is the singleton of the zero vector i.e. the zero subspace. (Contributed by NM, 27-Mar-2014.)
0 = (0.‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑂 = (0g𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝐼0 ) = {𝑂})
 
Theoremdib1dim 37724* Two expressions for the 1-dimensional subspaces of vector space H. (Contributed by NM, 24-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = {𝑔 ∈ (𝑇 × 𝐸) ∣ ∃𝑠𝐸 𝑔 = ⟨(𝑠𝐹), 𝑂⟩})
 
TheoremdibglbN 37725* Partial isomorphism B of a lattice glb. (Contributed by NM, 9-Mar-2014.) (New usage is discouraged.)
𝐺 = (glb‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆 ⊆ dom 𝐼𝑆 ≠ ∅)) → (𝐼‘(𝐺𝑆)) = 𝑥𝑆 (𝐼𝑥))
 
TheoremdibintclN 37726 The intersection of partial isomorphism B closed subspaces is a closed subspace. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑆 ⊆ ran 𝐼𝑆 ≠ ∅)) → 𝑆 ∈ ran 𝐼)
 
Theoremdib1dim2 37727* Two expressions for a 1-dimensional subspace of vector space H (when 𝐹 is a nonzero vector i.e. non-identity translation). (Contributed by NM, 24-Feb-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))
 
Theoremdibss 37728 The partial isomorphism B maps to a set of vectors in full vector space H. (Contributed by NM, 1-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ⊆ 𝑉)
 
Theoremdiblss 37729 The value of partial isomorphism B is a subspace of partial vector space H. TODO: use dib* specific theorems instead of dia* ones to shorten proof? (Contributed by NM, 11-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
 
Theoremdiblsmopel 37730* Membership in subspace sum for partial isomorphism B. (Contributed by NM, 21-Sep-2014.) (Revised by Mario Carneiro, 6-May-2015.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝑉 = ((DVecA‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑉)    &    = (LSSum‘𝑈)    &   𝐽 = ((DIsoA‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))    &   (𝜑 → (𝑋𝐵𝑋 𝑊))    &   (𝜑 → (𝑌𝐵𝑌 𝑊))       (𝜑 → (⟨𝐹, 𝑆⟩ ∈ ((𝐼𝑋) (𝐼𝑌)) ↔ (𝐹 ∈ ((𝐽𝑋) (𝐽𝑌)) ∧ 𝑆 = 𝑂)))
 
Syntaxcdic 37731 Extend class notation with isomorphism C.
class DIsoC
 
Definitiondf-dic 37732* Isomorphism C has domain of lattice atoms that are not less than or equal to the fiducial co-atom 𝑤. The value is a one-dimensional subspace generated by the pair consisting of the vector below and the endomorphism ring unit. Definition of phi(q) in [Crawley] p. 121. Note that we use the fixed atom ((oc k ) 𝑤) to represent the p in their "Choose an atom p..." on line 21. (Contributed by NM, 15-Dec-2013.)
DIsoC = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑞 ∈ {𝑟 ∈ (Atoms‘𝑘) ∣ ¬ 𝑟(le‘𝑘)𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝑘)‘𝑤)(𝑔‘((oc‘𝑘)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝑘)‘𝑤))})))
 
Theoremdicffval 37733* The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoC‘𝐾) = (𝑤𝐻 ↦ (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑤} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔 ∈ ((LTrn‘𝐾)‘𝑤)(𝑔‘((oc‘𝐾)‘𝑤)) = 𝑞)) ∧ 𝑠 ∈ ((TEndo‘𝐾)‘𝑤))})))
 
Theoremdicfval 37734* The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑞 ∈ {𝑟𝐴 ∣ ¬ 𝑟 𝑊} ↦ {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑞)) ∧ 𝑠𝐸)}))
 
Theoremdicval 37735* The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 22-Sep-2015.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑠𝐸)})
 
Theoremdicopelval 37736* Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 15-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝐹 ∈ V    &   𝑆 ∈ V       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ 𝑆𝐸)))
 
TheoremdicelvalN 37737* Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)) ∧ (2nd𝑌) ∈ 𝐸))))
 
Theoremdicval2 37738* The partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = {⟨𝑓, 𝑠⟩ ∣ (𝑓 = (𝑠𝐺) ∧ 𝑠𝐸)})
 
Theoremdicelval3 37739* Member of the partial isomorphism C. (Contributed by NM, 26-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ ∃𝑠𝐸 𝑌 = ⟨(𝑠𝐺), 𝑠⟩))
 
Theoremdicopelval2 37740* Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 20-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)    &   𝐹 ∈ V    &   𝑆 ∈ V       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑄) ↔ (𝐹 = (𝑆𝐺) ∧ 𝑆𝐸)))
 
Theoremdicelval2N 37741* Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 25-Feb-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝐺 = (𝑔𝑇 (𝑔𝑃) = 𝑄)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝑌 ∈ (𝐼𝑄) ↔ (𝑌 ∈ (V × V) ∧ ((1st𝑌) = ((2nd𝑌)‘𝐺) ∧ (2nd𝑌) ∈ 𝐸))))
 
TheoremdicfnN 37742* Functionality and domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
 
TheoremdicdmN 37743* Domain of the partial isomorphism C. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑝𝐴 ∣ ¬ 𝑝 𝑊})
 
TheoremdicvalrelN 37744 The value of partial isomorphism C is a relation. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       ((𝐾𝑉𝑊𝐻) → Rel (𝐼𝑋))
 
Theoremdicssdvh 37745 The partial isomorphism C maps to a set of vectors in full vector space H. (Contributed by NM, 19-Jan-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ⊆ 𝑉)
 
Theoremdicelval1sta 37746* Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) = ((2nd𝑌)‘(𝑔𝑇 (𝑔𝑃) = 𝑄)))
 
Theoremdicelval1stN 37747 Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.) (New usage is discouraged.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (1st𝑌) ∈ 𝑇)
 
Theoremdicelval2nd 37748 Membership in value of the partial isomorphism C for a lattice 𝐾. (Contributed by NM, 16-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ 𝑌 ∈ (𝐼𝑄)) → (2nd𝑌) ∈ 𝐸)
 
Theoremdicvaddcl 37749 Membership in value of the partial isomorphism C is closed under vector sum. (Contributed by NM, 16-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &    + = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋 ∈ (𝐼𝑄) ∧ 𝑌 ∈ (𝐼𝑄))) → (𝑋 + 𝑌) ∈ (𝐼𝑄))
 
Theoremdicvscacl 37750 Membership in value of the partial isomorphism C is closed under scalar product. (Contributed by NM, 16-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &    · = ( ·𝑠𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑋𝐸𝑌 ∈ (𝐼𝑄))) → (𝑋 · 𝑌) ∈ (𝐼𝑄))
 
Theoremdicn0 37751 The value of the partial isomorphism C is not empty. (Contributed by NM, 15-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ≠ ∅)
 
Theoremdiclss 37752 The value of partial isomorphism C is a subspace of partial vector space H. (Contributed by NM, 16-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) ∈ 𝑆)
 
Theoremdiclspsn 37753* The value of isomorphism C is spanned by vector 𝐹. Part of proof of Lemma N of [Crawley] p. 121 line 29. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐼 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑄)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}))
 
Theoremcdlemn2 37754* Part of proof of Lemma N of [Crawley] p. 121 line 30. (Contributed by NM, 21-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐹 = (𝑇 (𝑄) = 𝑆)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑅𝐹) 𝑋)
 
Theoremcdlemn2a 37755* Part of proof of Lemma N of [Crawley] p. 121. (Contributed by NM, 24-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (𝑇 (𝑄) = 𝑆)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑆 (𝑄 𝑋)) → (𝑁‘{⟨𝐹, 𝑂⟩}) ⊆ (𝐼𝑋))
 
Theoremcdlemn3 37756* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)    &   𝐽 = (𝑇 (𝑄) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝐽𝐹) = 𝐺)
 
Theoremcdlemn4 37757* Part of proof of Lemma N of [Crawley] p. 121 line 31. (Contributed by NM, 21-Feb-2014.) (Revised by Mario Carneiro, 24-Jun-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)    &   𝐽 = (𝑇 (𝑄) = 𝑅)    &    + = (+g𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨𝐹, ( I ↾ 𝑇)⟩ +𝐽, 𝑂⟩))
 
Theoremcdlemn4a 37758* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 24-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)    &   𝐽 = (𝑇 (𝑄) = 𝑅)    &   𝑁 = (LSpan‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) → (𝑁‘{⟨𝐺, ( I ↾ 𝑇)⟩}) ⊆ ((𝑁‘{⟨𝐹, ( I ↾ 𝑇)⟩}) (𝑁‘{⟨𝐽, 𝑂⟩})))
 
Theoremcdlemn5pre 37759* Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)    &   𝑀 = (𝑇 (𝑄) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑅 (𝑄 𝑋)) → (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋)))
 
Theoremcdlemn5 37760 Part of proof of Lemma N of [Crawley] p. 121 line 32. (Contributed by NM, 25-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ 𝑅 (𝑄 𝑋)) → (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋)))
 
Theoremcdlemn6 37761* Part of proof of Lemma N of [Crawley] p. 121 line 35. (Contributed by NM, 26-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇)) → (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩) = ⟨((𝑠𝐹) ∘ 𝑔), 𝑠⟩)
 
Theoremcdlemn7 37762* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝐺 = ((𝑠𝐹) ∘ 𝑔) ∧ ( I ↾ 𝑇) = 𝑠))
 
Theoremcdlemn8 37763* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 26-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → 𝑔 = (𝐺𝐹))
 
Theoremcdlemn9 37764* Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝐺, ( I ↾ 𝑇)⟩ = (⟨(𝑠𝐹), 𝑠+𝑔, 𝑂⟩))) → (𝑔𝑄) = 𝑅)
 
Theoremcdlemn10 37765 Part of proof of Lemma N of [Crawley] p. 121 line 36. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑆𝐴 ∧ ¬ 𝑆 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝑔𝑇 ∧ (𝑔𝑄) = 𝑆 ∧ (𝑅𝑔) 𝑋)) → 𝑆 (𝑄 𝑋))
 
Theoremcdlemn11a 37766* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ (𝐽𝑁))
 
Theoremcdlemn11b 37767* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ⟨𝐺, ( I ↾ 𝑇)⟩ ∈ ((𝐽𝑄) (𝐼𝑋)))
 
Theoremcdlemn11c 37768* Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → ∃𝑦 ∈ (𝐽𝑄)∃𝑧 ∈ (𝐼𝑋)⟨𝐺, ( I ↾ 𝑇)⟩ = (𝑦 + 𝑧))
 
Theoremcdlemn11pre 37769* Part of proof of Lemma N of [Crawley] p. 121 line 37. TODO: combine cdlemn11a 37766, cdlemn11b 37767, cdlemn11c 37768, cdlemn11pre into one? (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &    = (LSSum‘𝑈)    &   𝐹 = (𝑇 (𝑃) = 𝑄)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑁) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑁 (𝑄 𝑋))
 
Theoremcdlemn11 37770 Part of proof of Lemma N of [Crawley] p. 121 line 37. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊)) ∧ (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋))) → 𝑅 (𝑄 𝑋))
 
Theoremcdlemn 37771 Lemma N of [Crawley] p. 121 line 27. (Contributed by NM, 27-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ (𝑋𝐵𝑋 𝑊))) → (𝑅 (𝑄 𝑋) ↔ (𝐽𝑅) ⊆ ((𝐽𝑄) (𝐼𝑋))))
 
Theoremdihordlem6 37772* Part of proof of Lemma N of [Crawley] p. 122 line 35. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇)) → (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩) = ⟨((𝑠𝐺) ∘ 𝑔), 𝑠⟩)
 
Theoremdihordlem7 37773* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑓 = ((𝑠𝐺) ∘ 𝑔) ∧ 𝑂 = 𝑠))
 
Theoremdihordlem7b 37774* Part of proof of Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑅)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑠𝐸𝑔𝑇 ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑓 = 𝑔𝑂 = 𝑠))
 
Theoremdihjustlem 37775 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑋𝐵) ∧ (𝑄 (𝑋 𝑊)) = (𝑅 (𝑋 𝑊))) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑋 𝑊))))
 
Theoremdihjust 37776 Part of proof after Lemma N of [Crawley] p. 122 line 4, "the definition of phi(x) is independent of the atom q." (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊) ∧ 𝑋𝐵) ∧ (𝑄 (𝑋 𝑊)) = (𝑅 (𝑋 𝑊))) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) = ((𝐽𝑅) (𝐼‘(𝑋 𝑊))))
 
Theoremdihord1 37777 Part of proof after Lemma N of [Crawley] p. 122. Forward ordering property. TODO: change (𝑄 (𝑋 𝑊)) = 𝑋 to 𝑄 𝑋 using lhpmcvr3 36584, here and all theorems below. (Contributed by NM, 2-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌𝑋 𝑌)) → ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
 
Theoremdihord2a 37778 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑅 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))) → 𝑄 (𝑅 (𝑌 𝑊)))
 
Theoremdihord2b 37779 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑅𝐴 ∧ ¬ 𝑅 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊)))) → (𝐼‘(𝑋 𝑊)) ⊆ ((𝐽𝑅) (𝐼‘(𝑌 𝑊))))
 
Theoremdihord2cN 37780* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: needed? shorten other proof with it? (Contributed by NM, 3-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝐵 ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ (𝐼‘(𝑋 𝑊)))
 
Theoremdihord11b 37781* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       (((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊)))) ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ⟨𝑓, 𝑂⟩ ∈ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))
 
Theoremdihord10 37782* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊)) ∧ ((𝑠𝐸𝑔𝑇) ∧ (𝑅𝑔) (𝑌 𝑊) ∧ ⟨𝑓, 𝑂⟩ = (⟨(𝑠𝐺), 𝑠+𝑔, 𝑂⟩))) → (𝑅𝑓) (𝑌 𝑊))
 
Theoremdihord11c 37783* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ (((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))) ∧ 𝑓𝑇 ∧ (𝑅𝑓) (𝑋 𝑊))) → ∃𝑦 ∈ (𝐽𝑁)∃𝑧 ∈ (𝐼‘(𝑌 𝑊))⟨𝑓, 𝑂⟩ = (𝑦 + 𝑧))
 
Theoremdihord2pre 37784* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 3-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊)))) → (𝑋 𝑊) (𝑌 𝑊))
 
Theoremdihord2pre2 37785* Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑃 = ((oc‘𝐾)‘𝑊)    &   𝐸 = ((TEndo‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝐺 = (𝑇 (𝑃) = 𝑁)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑁 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))) → (𝑄 (𝑋 𝑊)) (𝑁 (𝑌 𝑊)))
 
Theoremdihord2 37786 Part of proof after Lemma N of [Crawley] p. 122. Reverse ordering property. TODO: do we need ¬ 𝑋 𝑊 and ¬ 𝑌 𝑊? (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoB‘𝐾)‘𝑊)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       ((((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑁𝐴 ∧ ¬ 𝑁 𝑊)) ∧ (𝑋𝐵𝑌𝐵) ∧ ((𝑄 (𝑋 𝑊)) = 𝑋 ∧ (𝑁 (𝑌 𝑊)) = 𝑌 ∧ ((𝐽𝑄) (𝐼‘(𝑋 𝑊))) ⊆ ((𝐽𝑁) (𝐼‘(𝑌 𝑊))))) → 𝑋 𝑌)
 
Syntaxcdih 37787 Extend class notation with isomorphism H.
class DIsoH
 
Definitiondf-dih 37788* Define isomorphism H. (Contributed by NM, 28-Jan-2014.)
DIsoH = (𝑘 ∈ V ↦ (𝑤 ∈ (LHyp‘𝑘) ↦ (𝑥 ∈ (Base‘𝑘) ↦ if(𝑥(le‘𝑘)𝑤, (((DIsoB‘𝑘)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝑘)‘𝑤))∀𝑞 ∈ (Atoms‘𝑘)((¬ 𝑞(le‘𝑘)𝑤 ∧ (𝑞(join‘𝑘)(𝑥(meet‘𝑘)𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝑘)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝑘)‘𝑤))(((DIsoB‘𝑘)‘𝑤)‘(𝑥(meet‘𝑘)𝑤)))))))))
 
Theoremdihffval 37789* The isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoH‘𝐾) = (𝑤𝐻 ↦ (𝑥𝐵 ↦ if(𝑥 𝑤, (((DIsoB‘𝐾)‘𝑤)‘𝑥), (𝑢 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑤))∀𝑞𝐴 ((¬ 𝑞 𝑤 ∧ (𝑞 (𝑥 𝑤)) = 𝑥) → 𝑢 = ((((DIsoC‘𝐾)‘𝑤)‘𝑞)(LSSum‘((DVecH‘𝐾)‘𝑤))(((DIsoB‘𝐾)‘𝑤)‘(𝑥 𝑤)))))))))
 
Theoremdihfval 37790* Isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 28-Jan-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       ((𝐾𝑉𝑊𝐻) → 𝐼 = (𝑥𝐵 ↦ if(𝑥 𝑊, (𝐷𝑥), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑥 𝑊)) = 𝑥) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑥 𝑊))))))))
 
Theoremdihval 37791* Value of isomorphism H for a lattice 𝐾. Definition of isomorphism map in [Crawley] p. 122 line 3. (Contributed by NM, 3-Feb-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾𝑉𝑊𝐻) ∧ 𝑋𝐵) → (𝐼𝑋) = if(𝑋 𝑊, (𝐷𝑋), (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊)))))))
 
Theoremdihvalc 37792* Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊. (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝐼𝑋) = (𝑢𝑆𝑞𝐴 ((¬ 𝑞 𝑊 ∧ (𝑞 (𝑋 𝑊)) = 𝑋) → 𝑢 = ((𝐶𝑞) (𝐷‘(𝑋 𝑊))))))
 
Theoremdihlsscpre 37793 Closure of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊. (Contributed by NM, 6-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊)) → (𝐼𝑋) ∈ 𝑆)
 
Theoremdihvalcqpre 37794 Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊, given auxiliary atom 𝑄. (Contributed by NM, 6-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝐷‘(𝑋 𝑊))))
 
Theoremdihvalcq 37795 Value of isomorphism H for a lattice 𝐾 when ¬ 𝑋 𝑊, given auxiliary atom 𝑄. TODO: Use dihvalcq2 37806 (with lhpmcvr3 36584 for (𝑄 (𝑋 𝑊)) = 𝑋 simplification) that changes 𝐶 and 𝐷 to 𝐼 and make this obsolete. Do to other theorems as well. (Contributed by NM, 6-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &    = (join‘𝐾)    &    = (meet‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)    &   𝐶 = ((DIsoC‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑋𝐵 ∧ ¬ 𝑋 𝑊) ∧ ((𝑄𝐴 ∧ ¬ 𝑄 𝑊) ∧ (𝑄 (𝑋 𝑊)) = 𝑋)) → (𝐼𝑋) = ((𝐶𝑄) (𝐷‘(𝑋 𝑊))))
 
Theoremdihvalb 37796 Value of isomorphism H for a lattice 𝐾 when 𝑋 𝑊. (Contributed by NM, 4-Mar-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝐷 = ((DIsoB‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (𝐼𝑋) = (𝐷𝑋))
 
TheoremdihopelvalbN 37797* Ordered pair member of the partial isomorphism H for argument under 𝑊. (Contributed by NM, 21-Mar-2014.) (New usage is discouraged.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑔𝑇 ↦ ( I ↾ 𝐵))    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾𝑉𝑊𝐻) ∧ (𝑋𝐵𝑋 𝑊)) → (⟨𝐹, 𝑆⟩ ∈ (𝐼𝑋) ↔ ((𝐹𝑇 ∧ (𝑅𝐹) 𝑋) ∧ 𝑆 = 𝑂)))
 
Theoremdihvalcqat 37798 Value of isomorphism H for a lattice 𝐾 at an atom not under 𝑊. (Contributed by NM, 27-Mar-2014.)
= (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝐽 = ((DIsoC‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴 ∧ ¬ 𝑄 𝑊)) → (𝐼𝑄) = (𝐽𝑄))
 
Theoremdih1dimb 37799* Two expressions for a 1-dimensional subspace of vector space H (when 𝐹 is a nonzero vector i.e. non-identity translation). (Contributed by NM, 27-Apr-2014.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑅 = ((trL‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝐹𝑇) → (𝐼‘(𝑅𝐹)) = (𝑁‘{⟨𝐹, 𝑂⟩}))
 
Theoremdih1dimb2 37800* Isomorphism H at an atom under 𝑊. (Contributed by NM, 27-Apr-2014.)
𝐵 = (Base‘𝐾)    &    = (le‘𝐾)    &   𝐴 = (Atoms‘𝐾)    &   𝐻 = (LHyp‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑂 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐼 = ((DIsoH‘𝐾)‘𝑊)    &   𝑁 = (LSpan‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑄𝐴𝑄 𝑊)) → ∃𝑓𝑇 (𝑓 ≠ ( I ↾ 𝐵) ∧ (𝐼𝑄) = (𝑁‘{⟨𝑓, 𝑂⟩})))
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