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Theorem List for Metamath Proof Explorer - 18801-18900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlspsnsubg 18801 The span of a singleton is an additive subgroup (frequently used special case of lspcl 18797). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))

Theorem00lsp 18802 fvco4i 6186 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
∅ = (LSpan‘∅)

Theoremlspid 18803 The span of a subspace is itself. (spanid 27590 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 18804 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 18805 Span preserves subset ordering. (spanss 27591 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 18806 A set of vectors is a subset of its span. (spanss2 27588 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 18807 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 18808 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 18809 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 18810 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 18811 The span of the singleton of a subspace member is included in the subspace. (spansnss 27814 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 18812 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 27815 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))       (𝜑𝑌𝑈)

Theoremlspprss 18813 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 18814 A vector belongs to the span of its singleton. (spansnid 27806 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 18815 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 18816 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 18817 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 18818 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 18819 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 18820 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 18821* 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 18822 A scalar product with a vector belongs to the span of its singleton. (spansnmul 27807 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 18823* 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 18824* Member of span of the singleton of a vector. (elspansn 27809 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 18825 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 18826* Comparable spans of singletons must have proportional vectors. See lspsneq 18943 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 18827 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 18828 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 18829 Span of the singleton of the zero vector. (spansn0 27784 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 18830 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })

Theoremlspuni0 18831 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = 0 )

Theoremlspun0 18832 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 18833 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 18834 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 18835 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 18836 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 18837 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 18838* 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 18839* 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 18840 Extend class notation with the generator of left module hom-sets.
class LMHom

Syntaxclmim 18841 The class of left module isomorphism sets.
class LMIso

Syntaxclmic 18842 The class of the left module isomorphism relation.
class 𝑚

Definitiondf-lmhm 18843* 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 18844* 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 18845 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 18846 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom LMHom

Theoremlmimfn 18847 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso Fn (LMod × LMod)

Theoremislmhm 18848* 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 18849* Property of a module homomorphism, similar to ismhm 17160. (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 18850 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 18851 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 18852 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 18853 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 18854 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 18855 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 18856 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 18857 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 18858 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 18859* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 18754. (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 18860* 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 18861 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 18862 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 18863 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 18864 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 18865 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 18866 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 18867 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 18868 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 18869 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 18870 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 18871 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 18872 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 18873 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))

Theoremreslmhm2 18874 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑈 = (𝑇s 𝑋)    &   𝐿 = (LSubSp‘𝑇)       ((𝐹 ∈ (𝑆 LMHom 𝑈) ∧ 𝑇 ∈ LMod ∧ 𝑋𝐿) → 𝐹 ∈ (𝑆 LMHom 𝑇))

Theoremreslmhm2b 18875 Expansion of the codomain of a homomorphism. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
𝑈 = (𝑇s 𝑋)    &   𝐿 = (LSubSp‘𝑇)       ((𝑇 ∈ LMod ∧ 𝑋𝐿 ∧ ran 𝐹𝑋) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ 𝐹 ∈ (𝑆 LMHom 𝑈)))

Theoremlmhmeql 18876 The equalizer of two module homomorphisms is a subspace. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑈 = (LSubSp‘𝑆)       ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺 ∈ (𝑆 LMHom 𝑇)) → dom (𝐹𝐺) ∈ 𝑈)

Theoremlspextmo 18877* A linear function is completely determined (or overdetermined) by its values on a spanning subset. (Contributed by Stefan O'Rear, 7-Mar-2015.) (Revised by NM, 17-Jun-2017.)
𝐵 = (Base‘𝑆)    &   𝐾 = (LSpan‘𝑆)       ((𝑋𝐵 ∧ (𝐾𝑋) = 𝐵) → ∃*𝑔 ∈ (𝑆 LMHom 𝑇)(𝑔𝑋) = 𝐹)

Theorempwsdiaglmhm 18878* Diagonal homomorphism into a structure power. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑅s 𝐼)    &   𝐵 = (Base‘𝑅)    &   𝐹 = (𝑥𝐵 ↦ (𝐼 × {𝑥}))       ((𝑅 ∈ LMod ∧ 𝐼𝑊) → 𝐹 ∈ (𝑅 LMHom 𝑌))

Theorempwssplit0 18879* Splitting for structure powers, part 0: restriction is a function. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊𝑇𝑈𝑋𝑉𝑈) → 𝐹:𝐵𝐶)

Theorempwssplit1 18880* Splitting for structure powers, part 1: restriction is an onto function. The only actual monoid law we need here is that the base set is nonempty. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ Mnd ∧ 𝑈𝑋𝑉𝑈) → 𝐹:𝐵onto𝐶)

Theorempwssplit2 18881* Splitting for structure powers, part 2: restriction is a group homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ Grp ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 GrpHom 𝑍))

Theorempwssplit3 18882* Splitting for structure powers, part 3: restriction is a module homomorphism. (Contributed by Stefan O'Rear, 24-Jan-2015.)
𝑌 = (𝑊s 𝑈)    &   𝑍 = (𝑊s 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑊 ∈ LMod ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))

Theoremislmim 18883 An isomorphism of left modules is a bijective homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹:𝐵1-1-onto𝐶))

Theoremlmimf1o 18884 An isomorphism of left modules is a bijection. (Contributed by Stefan O'Rear, 21-Jan-2015.)
𝐵 = (Base‘𝑅)    &   𝐶 = (Base‘𝑆)       (𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹:𝐵1-1-onto𝐶)

Theoremlmimlmhm 18885 An isomorphism of modules is a homomorphism. (Contributed by Stefan O'Rear, 21-Jan-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 LMHom 𝑆))

Theoremlmimgim 18886 An isomorphism of modules is an isomorphism of groups. (Contributed by Stefan O'Rear, 21-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝐹 ∈ (𝑅 GrpIso 𝑆))

Theoremislmim2 18887 An isomorphism of left modules is a homomorphism whose converse is a homomorphism. (Contributed by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) ↔ (𝐹 ∈ (𝑅 LMHom 𝑆) ∧ 𝐹 ∈ (𝑆 LMHom 𝑅)))

Theoremlmimcnv 18888 The converse of a bijective module homomorphism is a bijective module homomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.) (Revised by Mario Carneiro, 6-May-2015.)
(𝐹 ∈ (𝑆 LMIso 𝑇) → 𝐹 ∈ (𝑇 LMIso 𝑆))

Theorembrlmic 18889 The relation "is isomorphic to" for modules. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆 ↔ (𝑅 LMIso 𝑆) ≠ ∅)

Theorembrlmici 18890 Prove isomorphic by an explicit isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝐹 ∈ (𝑅 LMIso 𝑆) → 𝑅𝑚 𝑆)

Theoremlmiclcl 18891 Isomorphism implies the left side is a module. (Contributed by Stefan O'Rear, 25-Jan-2015.)
(𝑅𝑚 𝑆𝑅 ∈ LMod)

Theoremlmicrcl 18892 Isomorphism implies the right side is a module. (Contributed by Mario Carneiro, 6-May-2015.)
(𝑅𝑚 𝑆𝑆 ∈ LMod)

Theoremlmicsym 18893 Module isomorphism is symmetric. (Contributed by Stefan O'Rear, 26-Feb-2015.)
(𝑅𝑚 𝑆𝑆𝑚 𝑅)

Theoremlmhmpropd 18894* Module homomorphism depends only on the module attributes of structures. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝜑𝐵 = (Base‘𝐽))    &   (𝜑𝐶 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐶 = (Base‘𝑀))    &   (𝜑𝐹 = (Scalar‘𝐽))    &   (𝜑𝐺 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   (𝜑𝐺 = (Scalar‘𝑀))    &   𝑃 = (Base‘𝐹)    &   𝑄 = (Base‘𝐺)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐽)𝑦) = (𝑥(+g𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐶𝑦𝐶)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝑀)𝑦))    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐽)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝑄𝑦𝐶)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝑀)𝑦))       (𝜑 → (𝐽 LMHom 𝐾) = (𝐿 LMHom 𝑀))

10.6.4  Subspace sum; bases for a left module

Syntaxclbs 18895 Extend class notation with the set of bases for a vector space.
class LBasis

Definitiondf-lbs 18896* Define the set of bases to a left module or left vector space. (Contributed by Mario Carneiro, 24-Jun-2014.)
LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ [(LSpan‘𝑤) / 𝑛][(Scalar‘𝑤) / 𝑠]((𝑛𝑏) = (Base‘𝑤) ∧ ∀𝑥𝑏𝑦 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑦( ·𝑠𝑤)𝑥) ∈ (𝑛‘(𝑏 ∖ {𝑥})))})

Theoremislbs 18897* The predicate "𝐵 is a basis for the left module or vector space 𝑊". A subset of the base set is a basis if zero is not in the set, it spans the set, and no nonzero multiple of an element of the basis is in the span of the rest of the family. (Contributed by Mario Carneiro, 24-Jun-2014.) (Revised by Mario Carneiro, 14-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &    0 = (0g𝐹)       (𝑊𝑋 → (𝐵𝐽 ↔ (𝐵𝑉 ∧ (𝑁𝐵) = 𝑉 ∧ ∀𝑥𝐵𝑦 ∈ (𝐾 ∖ { 0 }) ¬ (𝑦 · 𝑥) ∈ (𝑁‘(𝐵 ∖ {𝑥})))))

Theoremlbsss 18898 A basis is a set of vectors. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)       (𝐵𝐽𝐵𝑉)

Theoremlbsel 18899 An element of a basis is a vector. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)       ((𝐵𝐽𝐸𝐵) → 𝐸𝑉)

Theoremlbssp 18900 The span of a basis is the whole space. (Contributed by Mario Carneiro, 24-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝐵𝐽 → (𝑁𝐵) = 𝑉)

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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 42301-42360
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