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Theorem List for Metamath Proof Explorer - 19001-19100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlssssr 19001* Conclude subspace ordering from nonzero vector membership. (ssrdv 3642 analog.) (Contributed by NM, 17-Aug-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑆)    &   ((𝜑𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥𝑇𝑥𝑈))       (𝜑𝑇𝑈)

Theoremlssvacl 19002 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
+ = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝑈𝑌𝑈)) → (𝑋 + 𝑌) ∈ 𝑈)

Theoremlssvscl 19003 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝐵𝑌𝑈)) → (𝑋 · 𝑌) ∈ 𝑈)

Theoremlssvnegcl 19004 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (invg𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁𝑋) ∈ 𝑈)

Theoremlsssubg 19005 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))

Theoremlsssssubg 19006 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))

Theoremislss3 19007 A linear subspace of a module is a subset which is a module in its own right. (Contributed by Stefan O'Rear, 6-Dec-2014.) (Revised by Mario Carneiro, 30-Apr-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → (𝑈𝑆 ↔ (𝑈𝑉𝑋 ∈ LMod)))

Theoremlsslmod 19008 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑋 ∈ LMod)

Theoremlsslss 19009 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &   𝑇 = (LSubSp‘𝑋)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑉𝑇 ↔ (𝑉𝑆𝑉𝑈)))

Theoremislss4 19010* A linear subspace is a subgroup which respects scalar multiplication. (Contributed by Stefan O'Rear, 11-Dec-2014.) (Revised by Mario Carneiro, 19-Apr-2016.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → (𝑈𝑆 ↔ (𝑈 ∈ (SubGrp‘𝑊) ∧ ∀𝑎𝐵𝑏𝑈 (𝑎 · 𝑏) ∈ 𝑈)))

Theoremlss1d 19011* One-dimensional subspace (or zero-dimensional if 𝑋 is the zero vector). (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → {𝑣 ∣ ∃𝑘𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆)

Theoremlssintcl 19012 The intersection of a nonempty set of subspaces is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐴𝑆𝐴 ≠ ∅) → 𝐴𝑆)

Theoremlssincl 19013 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑆𝑈𝑆) → (𝑇𝑈) ∈ 𝑆)

Theoremlssmre 19014 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐵 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ∈ (Moore‘𝐵))

Theoremlssacs 19015 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐵 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ∈ (ACS‘𝐵))

Theoremprdsvscacl 19016* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)    &   𝐾 = (Base‘𝑆)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝑅:𝐼⟶LMod)    &   (𝜑𝐹𝐾)    &   (𝜑𝐺𝐵)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)       (𝜑 → (𝐹 · 𝐺) ∈ 𝐵)

Theoremprdslmodd 19017* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑦𝐼) → (Scalar‘(𝑅𝑦)) = 𝑆)       (𝜑𝑌 ∈ LMod)

Theorempwslmod 19018 The product of a family of left modules is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ LMod ∧ 𝐼𝑉) → 𝑌 ∈ LMod)

Syntaxclspn 19019 Extend class notation with span of a set of vectors.
class LSpan

Definitiondf-lsp 19020* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSpan = (𝑤 ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘𝑤) ↦ {𝑡 ∈ (LSubSp‘𝑤) ∣ 𝑠𝑡}))

Theoremlspfval 19021* The span function for a left vector space (or a left module). (df-span 28296 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊𝑋𝑁 = (𝑠 ∈ 𝒫 𝑉 {𝑡𝑆𝑠𝑡}))

Theoremlspf 19022 The span operator on a left module maps subsets to subsets. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → 𝑁:𝒫 𝑉𝑆)

Theoremlspval 19023* The span of a set of vectors (in a left module). (spanval 28320 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) = {𝑡𝑆𝑈𝑡})

Theoremlspcl 19024 The span of a set of vectors is a subspace. (spancl 28323 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) ∈ 𝑆)

Theoremlspsncl 19025 The span of a singleton is a subspace (frequently used special case of lspcl 19024). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)

Theoremlspprcl 19026 The span of a pair is a subspace (frequently used special case of lspcl 19024). (Contributed by NM, 11-Apr-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆)

Theoremlsptpcl 19027 The span of an unordered triple is a subspace (frequently used special case of lspcl 19024). (Contributed by NM, 22-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝑍𝑉)       (𝜑 → (𝑁‘{𝑋, 𝑌, 𝑍}) ∈ 𝑆)

Theoremlspsnsubg 19028 The span of a singleton is an additive subgroup (frequently used special case of lspcl 19024). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))

Theorem00lsp 19029 fvco4i 6315 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
∅ = (LSpan‘∅)

Theoremlspid 19030 The span of a subspace is itself. (spanid 28334 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → (𝑁𝑈) = 𝑈)

Theoremlspssv 19031 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁𝑈) ⊆ 𝑉)

Theoremlspss 19032 Span preserves subset ordering. (spanss 28335 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉𝑇𝑈) → (𝑁𝑇) ⊆ (𝑁𝑈))

Theoremlspssid 19033 A set of vectors is a subset of its span. (spanss2 28332 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → 𝑈 ⊆ (𝑁𝑈))

Theoremlspidm 19034 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑉) → (𝑁‘(𝑁𝑈)) = (𝑁𝑈))

Theoremlspun 19035 The span of union is the span of the union of spans. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑇𝑉𝑈𝑉) → (𝑁‘(𝑇𝑈)) = (𝑁‘((𝑁𝑇) ∪ (𝑁𝑈))))

Theoremlspssp 19036 If a set of vectors is a subset of a subspace, then the span of those vectors is also contained in the subspace. (Contributed by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑇𝑈) → (𝑁𝑇) ⊆ 𝑈)

Theoremmrclsp 19037 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝑈 = (LSubSp‘𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝐹 = (mrCls‘𝑈)       (𝑊 ∈ LMod → 𝐾 = 𝐹)

Theoremlspsnss 19038 The span of the singleton of a subspace member is included in the subspace. (spansnss 28558 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆𝑋𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈)

Theoremlspsnel3 19039 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 28559 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))       (𝜑𝑌𝑈)

Theoremlspprss 19040 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑈)       (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈)

Theoremlspsnid 19041 A vector belongs to the span of its singleton. (spansnid 28550 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → 𝑋 ∈ (𝑁‘{𝑋}))

Theoremlspsnel6 19042 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)       (𝜑 → (𝑋𝑈 ↔ (𝑋𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)))

Theoremlspsnel5 19043 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑋𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈))

Theoremlspsnel5a 19044 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)       (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈)

Theoremlspprid1 19045 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑋 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlspprid2 19046 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑𝑌 ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlspprvacl 19047 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌}))

Theoremlssats2 19048* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)       (𝜑𝑈 = 𝑥𝑈 (𝑁‘{𝑥}))

Theoremlspsneli 19049 A scalar product with a vector belongs to the span of its singleton. (spansnmul 28551 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋}))

Theoremlspsn 19050* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) = {𝑣 ∣ ∃𝑘𝐾 𝑣 = (𝑘 · 𝑋)})

Theoremlspsnel 19051* Member of span of the singleton of a vector. (elspansn 28553 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘𝐾 𝑈 = (𝑘 · 𝑋)))

Theoremlspsnvsi 19052 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝑉) → (𝑁‘{(𝑅 · 𝑋)}) ⊆ (𝑁‘{𝑋}))

Theoremlspsnss2 19053* Comparable spans of singletons must have proportional vectors. See lspsneq 19170 for equal span version. (Contributed by NM, 7-Jun-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)    &    · = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ ∃𝑘𝐾 𝑋 = (𝑘 · 𝑌)))

Theoremlspsnneg 19054 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑀 = (invg𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{(𝑀𝑋)}) = (𝑁‘{𝑋}))

Theoremlspsnsub 19055 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (𝑁‘{(𝑋 𝑌)}) = (𝑁‘{(𝑌 𝑋)}))

Theoremlspsn0 19056 Span of the singleton of the zero vector. (spansn0 28528 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 })

Theoremlsp0 19057 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })

Theoremlspuni0 19058 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = 0 )

Theoremlspun0 19059 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)       (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁𝑋))

Theoremlspsneq0 19060 Span of the singleton is the zero subspace iff the vector is zero. (Contributed by NM, 27-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 ))

Theoremlspsneq0b 19061 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌}))       (𝜑 → (𝑋 = 0𝑌 = 0 ))

Theoremlmodindp1 19062 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))       (𝜑 → (𝑋 + 𝑌) ≠ 0 )

Theoremlsslsp 19063 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap 𝑀𝐺 and 𝑁𝐺 since we are computing a property of 𝑁𝐺? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
𝑋 = (𝑊s 𝑈)    &   𝑀 = (LSpan‘𝑊)    &   𝑁 = (LSpan‘𝑋)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝐿𝐺𝑈) → (𝑀𝐺) = (𝑁𝐺))

Theoremlss0v 19064 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
𝑋 = (𝑊s 𝑈)    &    0 = (0g𝑊)    &   𝑍 = (0g𝑋)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝐿) → 𝑍 = 0 )

Theoremlsspropd 19065* If two structures have the same components (properties), they have the same subspace structure. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝑃 = (Base‘(Scalar‘𝐾)))    &   (𝜑𝑃 = (Base‘(Scalar‘𝐿)))       (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿))

Theoremlsppropd 19066* If two structures have the same components (properties), they have the same span function. (Contributed by Mario Carneiro, 9-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐵𝑊)    &   ((𝜑 ∧ (𝑥𝑊𝑦𝑊)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) ∈ 𝑊)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   (𝜑𝑃 = (Base‘(Scalar‘𝐾)))    &   (𝜑𝑃 = (Base‘(Scalar‘𝐿)))    &   (𝜑𝐾 ∈ V)    &   (𝜑𝐿 ∈ V)       (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))

10.6.3  Homomorphisms and isomorphisms of left modules

Syntaxclmhm 19067 Extend class notation with the generator of left module hom-sets.
class LMHom

Syntaxclmim 19068 The class of left module isomorphism sets.
class LMIso

Syntaxclmic 19069 The class of the left module isomorphism relation.
class 𝑚

Definitiondf-lmhm 19070* A homomorphism of left modules is a group homomorphism which additionally preserves the scalar product. This requires both structures to be left modules over the same ring. (Contributed by Stefan O'Rear, 31-Dec-2014.)
LMHom = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑡) ∣ [(Scalar‘𝑠) / 𝑤]((Scalar‘𝑡) = 𝑤 ∧ ∀𝑥 ∈ (Base‘𝑤)∀𝑦 ∈ (Base‘𝑠)(𝑓‘(𝑥( ·𝑠𝑠)𝑦)) = (𝑥( ·𝑠𝑡)(𝑓𝑦)))})

Definitiondf-lmim 19071* An isomorphism of modules is a homomorphism which is also a bijection, i.e. it preserves equality as well as the group and scalar operations. (Contributed by Stefan O'Rear, 21-Jan-2015.)
LMIso = (𝑠 ∈ LMod, 𝑡 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑡) ∣ 𝑔:(Base‘𝑠)–1-1-onto→(Base‘𝑡)})

Definitiondf-lmic 19072 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑚 = ( LMIso “ (V ∖ 1𝑜))

Theoremreldmlmhm 19073 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom LMHom

Theoremlmimfn 19074 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso Fn (LMod × LMod)

Theoremislmhm 19075* Property of being a homomorphism of left modules. (Contributed by Stefan O'Rear, 1-Jan-2015.) (Proof shortened by Mario Carneiro, 30-Apr-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))

Theoremislmhm3 19076* Property of a module homomorphism, similar to ismhm 17384. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥𝐵𝑦𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))))

Theoremlmhmlem 19077 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))

Theoremlmhmsca 19078 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐿 = 𝐾)

Theoremlmghm 19079 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))

Theoremlmhmlmod2 19080 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑇 ∈ LMod)

Theoremlmhmlmod1 19081 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → 𝑆 ∈ LMod)

Theoremlmhmf 19082 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐹:𝐵𝐶)

Theoremlmhmlin 19083 A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalar‘𝑆)    &   𝐵 = (Base‘𝐾)    &   𝐸 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝐵𝑌𝐸) → (𝐹‘(𝑋 · 𝑌)) = (𝑋 × (𝐹𝑌)))

Theoremlmodvsinv 19084 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝑀 = (invg𝐹)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝐵) → ((𝑀𝑅) · 𝑋) = (𝑁‘(𝑅 · 𝑋)))

Theoremlmodvsinv2 19085 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ LMod ∧ 𝑅𝐾𝑋𝐵) → (𝑅 · (𝑁𝑋)) = (𝑁‘(𝑅 · 𝑋)))

Theoremislmhm2 19086* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 18981. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)    &   𝐾 = (Scalar‘𝑆)    &   𝐿 = (Scalar‘𝑇)    &   𝐸 = (Base‘𝐾)    &    + = (+g𝑆)    &    = (+g𝑇)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)       ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐵𝐶𝐿 = 𝐾 ∧ ∀𝑥𝐸𝑦𝐵𝑧𝐵 (𝐹‘((𝑥 · 𝑦) + 𝑧)) = ((𝑥 × (𝐹𝑦)) (𝐹𝑧)))))

Theoremislmhmd 19087* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑋 = (Base‘𝑆)    &    · = ( ·𝑠𝑆)    &    × = ( ·𝑠𝑇)    &   𝐾 = (Scalar‘𝑆)    &   𝐽 = (Scalar‘𝑇)    &   𝑁 = (Base‘𝐾)    &   (𝜑𝑆 ∈ LMod)    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐽 = 𝐾)    &   (𝜑𝐹 ∈ (𝑆 GrpHom 𝑇))    &   ((𝜑 ∧ (𝑥𝑁𝑦𝑋)) → (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹𝑦)))       (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))

Theorem0lmhm 19088 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0g𝑁)    &   𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑇 = (Scalar‘𝑁)       ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) → (𝐵 × { 0 }) ∈ (𝑀 LMHom 𝑁))

Theoremidlmhm 19089 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐵 = (Base‘𝑀)       (𝑀 ∈ LMod → ( I ↾ 𝐵) ∈ (𝑀 LMHom 𝑀))

Theoreminvlmhm 19090 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐼 = (invg𝑀)       (𝑀 ∈ LMod → 𝐼 ∈ (𝑀 LMHom 𝑀))

Theoremlmhmco 19091 The composition of two module-linear functions is module-linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
((𝐹 ∈ (𝑁 LMHom 𝑂) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝐺) ∈ (𝑀 LMHom 𝑂))

Theoremlmhmplusg 19092 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+g𝑁)       ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) → (𝐹𝑓 + 𝐺) ∈ (𝑀 LMHom 𝑁))

Theoremlmhmvsca 19093 The pointwise scalar product of a linear function and a constant is linear, over a commutative ring. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝑉 = (Base‘𝑀)    &    · = ( ·𝑠𝑁)    &   𝐽 = (Scalar‘𝑁)    &   𝐾 = (Base‘𝐽)       ((𝐽 ∈ CRing ∧ 𝐴𝐾𝐹 ∈ (𝑀 LMHom 𝑁)) → ((𝑉 × {𝐴}) ∘𝑓 · 𝐹) ∈ (𝑀 LMHom 𝑁))

Theoremlmhmf1o 19094 A bijective module homomorphism is also converse homomorphic. (Contributed by Stefan O'Rear, 25-Jan-2015.)
𝑋 = (Base‘𝑆)    &   𝑌 = (Base‘𝑇)       (𝐹 ∈ (𝑆 LMHom 𝑇) → (𝐹:𝑋1-1-onto𝑌𝐹 ∈ (𝑇 LMHom 𝑆)))

Theoremlmhmima 19095 The image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (LSubSp‘𝑆)    &   𝑌 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑋) → (𝐹𝑈) ∈ 𝑌)

Theoremlmhmpreima 19096 The inverse image of a subspace under a homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑋 = (LSubSp‘𝑆)    &   𝑌 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑌) → (𝐹𝑈) ∈ 𝑋)

Theoremlmhmlsp 19097 Homomorphisms preserve spans. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑉 = (Base‘𝑆)    &   𝐾 = (LSpan‘𝑆)    &   𝐿 = (LSpan‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑈𝑉) → (𝐹 “ (𝐾𝑈)) = (𝐿‘(𝐹𝑈)))

Theoremlmhmrnlss 19098 The range of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → ran 𝐹 ∈ (LSubSp‘𝑇))

Theoremlmhmkerlss 19099 The kernel of a homomorphism is a submodule. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (𝐹 “ { 0 })    &    0 = (0g𝑇)    &   𝑈 = (LSubSp‘𝑆)       (𝐹 ∈ (𝑆 LMHom 𝑇) → 𝐾𝑈)

Theoremreslmhm 19100 Restriction of a homomorphism to a subspace. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝑈 = (LSubSp‘𝑆)    &   𝑅 = (𝑆s 𝑋)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋𝑈) → (𝐹𝑋) ∈ (𝑅 LMHom 𝑇))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42879
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