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Theorem List for Metamath Proof Explorer - 35301-35400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremtendoinvcl 35301* 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 35179. (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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → ((𝑁𝑆) ∈ 𝐸 ∧ (𝑁𝑆) ≠ 𝑂))

Theoremtendolinv 35302* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → ((𝑁𝑆) ∘ 𝑆) = ( I ↾ 𝑇))

Theoremtendorinv 35303* 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 = (𝑇 ↦ ( I ↾ 𝐵))    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝐹 = (Scalar‘𝑈)    &   𝑁 = (invr𝐹)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑆𝐸𝑆𝑂) → (𝑆 ∘ (𝑁𝑆)) = ( I ↾ 𝑇))

Theoremdvhgrp 35304 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)    &    = (+g𝐷)    &    + = (+g𝑈)    &    0 = (0g𝐷)    &   𝐼 = (invg𝐷)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ Grp)

Theoremdvhlveclem 35305 Lemma for dvhlvec 35306. 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)    &    = (+g𝐷)    &    + = (+g𝑈)    &    0 = (0g𝐷)    &   𝐼 = (invg𝐷)    &    × = (.r𝐷)    &    · = ( ·𝑠𝑈)       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝑈 ∈ LVec)

Theoremdvhlvec 35306 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LVec)

Theoremdvhlmod 35307 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 ∧ 𝑊𝐻))       (𝜑𝑈 ∈ LMod)

Theoremdvh0g 35308* 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 = (0g𝑈)    &   𝑂 = (𝑓𝑇 ↦ ( I ↾ 𝐵))       ((𝐾 ∈ HL ∧ 𝑊𝐻) → 0 = ⟨( I ↾ 𝐵), 𝑂⟩)

Theoremdvheveccl 35309 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 35313 and dihpN 35533. (Contributed by NM, 27-Mar-2015.)
𝐻 = (LHyp‘𝐾)    &   𝐵 = (Base‘𝐾)    &   𝑇 = ((LTrn‘𝐾)‘𝑊)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &   𝑉 = (Base‘𝑈)    &    0 = (0g𝑈)    &   𝐸 = ⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩    &   (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))       (𝜑𝐸 ∈ (𝑉 ∖ { 0 }))

TheoremdvhopclN 35310 Closure of a DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
((𝐹𝑇𝑈𝐸) → ⟨𝐹, 𝑈⟩ ∈ (𝑇 × 𝐸))

TheoremdvhopaddN 35311* 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) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)       (((𝐹𝑇𝑈𝐸) ∧ (𝐺𝑇𝑉𝐸)) → (⟨𝐹, 𝑈𝐴𝐺, 𝑉⟩) = ⟨(𝐹𝐺), (𝑈𝑃𝑉)⟩)

TheoremdvhopspN 35312* Scalar product of DVecH vector expressed as ordered pair. (Contributed by NM, 20-Nov-2013.) (New usage is discouraged.)
𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)       ((𝑅𝐸 ∧ (𝐹𝑇𝑈𝐸)) → (𝑅𝑆𝐹, 𝑈⟩) = ⟨(𝑅𝐹), (𝑅𝑈)⟩)

TheoremdvhopN 35313* 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 = (𝑎𝐸, 𝑏𝐸 ↦ (𝑐𝑇 ↦ ((𝑎𝑐) ∘ (𝑏𝑐))))    &   𝐴 = (𝑓 ∈ (𝑇 × 𝐸), 𝑔 ∈ (𝑇 × 𝐸) ↦ ⟨((1st𝑓) ∘ (1st𝑔)), ((2nd𝑓)𝑃(2nd𝑔))⟩)    &   𝑆 = (𝑠𝐸, 𝑓 ∈ (𝑇 × 𝐸) ↦ ⟨(𝑠‘(1st𝑓)), (𝑠 ∘ (2nd𝑓))⟩)    &   𝑂 = (𝑐𝑇 ↦ ( I ↾ 𝐵))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝐹𝑇𝑈𝐸)) → ⟨𝐹, 𝑈⟩ = (⟨𝐹, 𝑂𝐴(𝑈𝑆⟨( I ↾ 𝐵), ( I ↾ 𝑇)⟩)))

Theoremdvhopellsm 35314* Ordered pair membership in a subspace sum. (Contributed by NM, 12-Mar-2014.)
𝐻 = (LHyp‘𝐾)    &   𝑈 = ((DVecH‘𝐾)‘𝑊)    &    + = (+g𝑈)    &   𝑆 = (LSubSp‘𝑈)    &    = (LSSum‘𝑈)       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ 𝑋𝑆𝑌𝑆) → (⟨𝐹, 𝑇⟩ ∈ (𝑋 𝑌) ↔ ∃𝑥𝑦𝑧𝑤((⟨𝑥, 𝑦⟩ ∈ 𝑋 ∧ ⟨𝑧, 𝑤⟩ ∈ 𝑌) ∧ ⟨𝐹, 𝑇⟩ = (⟨𝑥, 𝑦+𝑧, 𝑤⟩))))

Theoremcdlemm10N 35315* The image of the map 𝐺 is the entire one-dimensional subspace (𝐼𝑉). 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 = {𝑟𝐴 ∣ (𝑟 (𝑃 𝑉) ∧ ¬ 𝑟 𝑊)}    &   𝐹 = (𝑓𝑇 (𝑓𝑃) = 𝑠)    &   𝐺 = (𝑞𝐶 ↦ (𝑓𝑇 (𝑓𝑃) = 𝑞))       (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑃𝐴 ∧ ¬ 𝑃 𝑊) ∧ (𝑉𝐴𝑉 𝑊)) → ran 𝐺 = (𝐼𝑉))

SyntaxcocaN 35316 Extend class notation with subspace orthocomplement for DVecA partial vector space.
class ocA

Definitiondf-docaN 35317* 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) ∣ 𝑥𝑧}))(join‘𝑘)((oc‘𝑘)‘𝑤))(meet‘𝑘)𝑤)))))

TheoremdocaffvalN 35318* 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)       (𝐾𝑉 → (ocA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((DIsoA‘𝐾)‘𝑤)‘((( ‘(((DIsoA‘𝐾)‘𝑤)‘ {𝑧 ∈ ran ((DIsoA‘𝐾)‘𝑤) ∣ 𝑥𝑧})) ( 𝑤)) 𝑤)))))

TheoremdocafvalN 35319* 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)       ((𝐾𝑉𝑊𝐻) → 𝑁 = (𝑥 ∈ 𝒫 𝑇 ↦ (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑥𝑧})) ( 𝑊)) 𝑊))))

TheoremdocavalN 35320* 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 ∧ 𝑊𝐻) ∧ 𝑋𝑇) → (𝑁𝑋) = (𝐼‘((( ‘(𝐼 {𝑧 ∈ ran 𝐼𝑋𝑧})) ( 𝑊)) 𝑊)))

TheoremdocaclN 35321 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 ∧ 𝑊𝐻) ∧ 𝑋𝑇) → ( 𝑋) ∈ ran 𝐼)

TheoremdiaocN 35322 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 ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → (𝐼‘((( 𝑋) ( 𝑊)) 𝑊)) = (𝑁‘(𝐼𝑋)))

Theoremdoca2N 35323 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 ∧ 𝑊𝐻) ∧ 𝑋 ∈ dom 𝐼) → ( ‘( ‘(𝐼𝑋))) = (𝐼𝑋))

Theoremdoca3N 35324 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 ∧ 𝑊𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ‘( 𝑋)) = 𝑋)

TheoremdvadiaN 35325 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 ∧ 𝑊𝐻) ∧ (𝑋𝑆 ∧ ( ‘( 𝑋)) = 𝑋)) → 𝑋 ∈ ran 𝐼)

TheoremdiarnN 35326* 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 ∧ 𝑊𝐻) → ran 𝐼 = {𝑥𝑆 ∣ ( ‘( 𝑥)) = 𝑥})

Theoremdiaf1oN 35327* 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 35232 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 ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→{𝑥𝑆 ∣ ( ‘( 𝑥)) = 𝑥})

SyntaxcdjaN 35328 Extend class notation with subspace join for DVecA partial vector space.
class vA

Definitiondf-djaN 35329* 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))))))

TheoremdjaffvalN 35330* Subspace join for DVecA partial vector space. (Contributed by NM, 6-Dec-2013.) (New usage is discouraged.)
𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (vA‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤), 𝑦 ∈ 𝒫 ((LTrn‘𝐾)‘𝑤) ↦ (((ocA‘𝐾)‘𝑤)‘((((ocA‘𝐾)‘𝑤)‘𝑥) ∩ (((ocA‘𝐾)‘𝑤)‘𝑦))))))

TheoremdjafvalN 35331* 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)       ((𝐾𝑉𝑊𝐻) → 𝐽 = (𝑥 ∈ 𝒫 𝑇, 𝑦 ∈ 𝒫 𝑇 ↦ ( ‘(( 𝑥) ∩ ( 𝑦)))))

TheoremdjavalN 35332 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 ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) = ( ‘(( 𝑋) ∩ ( 𝑌))))

TheoremdjaclN 35333 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 ∧ 𝑊𝐻) ∧ (𝑋𝑇𝑌𝑇)) → (𝑋𝐽𝑌) ∈ ran 𝐼)

TheoremdjajN 35334 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 ∧ 𝑊𝐻) ∧ (𝑋 ∈ dom 𝐼𝑌 ∈ dom 𝐼)) → (𝐼‘(𝑋 𝑌)) = ((𝐼𝑋)𝐽(𝐼𝑌)))

Syntaxcdib 35335 Extend class notation with isomorphism B.
class DIsoB

Definitiondf-dib 35336* 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)))}))))

Theoremdibffval 35337* The partial isomorphism B for a lattice 𝐾. (Contributed by NM, 8-Dec-2013.)
𝐵 = (Base‘𝐾)    &   𝐻 = (LHyp‘𝐾)       (𝐾𝑉 → (DIsoB‘𝐾) = (𝑤𝐻 ↦ (𝑥 ∈ dom ((DIsoA‘𝐾)‘𝑤) ↦ ((((DIsoA‘𝐾)‘𝑤)‘𝑥) × {(𝑓 ∈ ((LTrn‘𝐾)‘𝑤) ↦ ( I ↾ 𝐵))}))))

Theoremdibfval 35338* 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) → 𝐼 = (𝑥 ∈ dom 𝐽 ↦ ((𝐽𝑥) × { 0 })))

Theoremdibval 35339* 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 35340* 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 35341* 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 35342* 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 35343* 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 35344* 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 35345* 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 35346 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 35347 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 35348 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 35349* 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 35350 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 35351 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 35352 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 35353* 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 35354* 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 35355 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 35356 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 35357 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 35358 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 35359 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 35360 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 35361 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 35362* 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 35363* 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 35364 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 35365* 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 35366 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 35367 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 35368* 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 35369 Extend class notation with isomorphism C.
class DIsoC

Definitiondf-dic 35370* 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 35371* 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 35372* 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 35373* 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 35374* 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 35375* 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 35376* 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 35377* 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 35378* 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 35379* 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 35380* 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 35381* 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 35382 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 35383 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 35384* 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 35385 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 35386 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 35387 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 35388 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 35389 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 35390 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 35391* 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 35392* 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 35393* 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 35394* 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 35395* 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 35396* 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 35397* 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 35398 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 35399* 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 35400* 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))

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