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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremislss 19701* The predicate "is a subspace" (of a left module or left vector space). (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑈𝑆 ↔ (𝑈𝑉𝑈 ≠ ∅ ∧ ∀𝑥𝐵𝑎𝑈𝑏𝑈 ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈))
 
Theoremislssd 19702* Properties that determine a subspace of a left module or left vector space. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
(𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐵 = (Base‘𝐹))    &   (𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝑆 = (LSubSp‘𝑊))    &   (𝜑𝑈𝑉)    &   (𝜑𝑈 ≠ ∅)    &   ((𝜑 ∧ (𝑥𝐵𝑎𝑈𝑏𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈)       (𝜑𝑈𝑆)
 
Theoremlssss 19703 A subspace is a set of vectors. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈𝑉)
 
Theoremlssel 19704 A subspace member is a vector. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑈𝑆𝑋𝑈) → 𝑋𝑉)
 
Theoremlss1 19705 The set of vectors in a left module is a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑉𝑆)
 
Theoremlssuni 19706 The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)       (𝜑 𝑆 = 𝑉)
 
Theoremlssn0 19707 A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝑆 = (LSubSp‘𝑊)       (𝑈𝑆𝑈 ≠ ∅)
 
Theorem00lss 19708 The empty structure has no subspaces (for use with fvco4i 6751). (Contributed by Stefan O'Rear, 31-Mar-2015.)
∅ = (LSubSp‘∅)
 
Theoremlsscl 19709 Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐵 = (Base‘𝐹)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑈𝑆 ∧ (𝑍𝐵𝑋𝑈𝑌𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈)
 
Theoremlssvsubcl 19710 Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
= (-g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ LMod ∧ 𝑈𝑆) ∧ (𝑋𝑈𝑌𝑈)) → (𝑋 𝑌) ∈ 𝑈)
 
Theoremlssvancl1 19711 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. TODO: notice similarity to lspindp3 19903. Can it be used along with lspsnne1 19884, lspsnne2 19885 to shorten this proof? (Contributed by NM, 14-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈)
 
Theoremlssvancl2 19712 Non-closure: if one vector belongs to a subspace but another does not, their sum does not belong. Useful for obtaining a new vector not in a subspace. (Contributed by NM, 20-May-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌𝑉)    &   (𝜑 → ¬ 𝑌𝑈)       (𝜑 → ¬ (𝑌 + 𝑋) ∈ 𝑈)
 
Theoremlss0cl 19713 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 0𝑈)
 
Theoremlsssn0 19714 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → { 0 } ∈ 𝑆)
 
Theoremlss0ss 19715 The zero subspace is included in every subspace. (sh0le 29221 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → { 0 } ⊆ 𝑋)
 
Theoremlssle0 19716 No subspace is smaller than the zero subspace. (shle0 29223 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑆) → (𝑋 ⊆ { 0 } ↔ 𝑋 = { 0 }))
 
Theoremlssne0 19717* A nonzero subspace has a nonzero vector. (shne0i 29229 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)       (𝑋𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦𝑋 𝑦0 ))
 
Theoremlssvneln0 19718 A vector 𝑋 which doesn't belong to a subspace 𝑈 is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 19-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋0 )
 
Theoremlssneln0 19719 A vector 𝑋 which doesn't belong to a subspace 𝑈 is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑉)    &   (𝜑 → ¬ 𝑋𝑈)       (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
 
Theoremlssssr 19720* Conclude subspace ordering from nonzero vector membership. (ssrdv 3959 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.)
0 = (0g𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑇𝑉)    &   (𝜑𝑈𝑆)    &   ((𝜑𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥𝑇𝑥𝑈))       (𝜑𝑇𝑈)
 
Theoremlssvacl 19721 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 19722 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 19723 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 19724 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑈𝑆) → 𝑈 ∈ (SubGrp‘𝑊))
 
Theoremlsssssubg 19725 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSp‘𝑊)       (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊))
 
Theoremislss3 19726 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 19727 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 19728 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 19729* 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 19730* 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 19731 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 19732 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 19733 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 19734 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 19735* 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 19736* 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 19737 A structure power of a left module is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
𝑌 = (𝑅s 𝐼)       ((𝑅 ∈ LMod ∧ 𝐼𝑉) → 𝑌 ∈ LMod)
 
Syntaxclspn 19738 Extend class notation with span of a set of vectors.
class LSpan
 
Definitiondf-lsp 19739* 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 19740* The span function for a left vector space (or a left module). (df-span 29090 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 19741 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 19742* The span of a set of vectors (in a left module). (spanval 29114 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 19743 The span of a set of vectors is a subspace. (spancl 29117 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 19744 The span of a singleton is a subspace (frequently used special case of lspcl 19743). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Base‘𝑊)    &   𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ 𝑆)
 
Theoremlspprcl 19745 The span of a pair is a subspace (frequently used special case of lspcl 19743). (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 19746 The span of an unordered triple is a subspace (frequently used special case of lspcl 19743). (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 19747 The span of a singleton is an additive subgroup (frequently used special case of lspcl 19743). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Base‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ LMod ∧ 𝑋𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊))
 
Theorem00lsp 19748 fvco4i 6751 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
∅ = (LSpan‘∅)
 
Theoremlspid 19749 The span of a subspace is itself. (spanid 29128 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 19750 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 19751 Span preserves subset ordering. (spanss 29129 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 19752 A set of vectors is a subset of its span. (spanss2 29126 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 19753 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 19754 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 19755 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 19756 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 19757 The span of the singleton of a subspace member is included in the subspace. (spansnss 29352 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 19758 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 29353 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSp‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   (𝜑𝑊 ∈ LMod)    &   (𝜑𝑈𝑆)    &   (𝜑𝑋𝑈)    &   (𝜑𝑌 ∈ (𝑁‘{𝑋}))       (𝜑𝑌𝑈)
 
Theoremlspprss 19759 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 19760 A vector belongs to the span of its singleton. (spansnid 29344 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 19761 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 19762 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 19763 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 19764 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 19765 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 19766 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 19767* 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 19768 A scalar product with a vector belongs to the span of its singleton. (spansnmul 29345 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 19769* 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 19770* Member of span of the singleton of a vector. (elspansn 29347 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 19771 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 19772* Comparable spans of singletons must have proportional vectors. See lspsneq 19889 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 19773 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 19774 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 19775 Span of the singleton of the zero vector. (spansn0 29322 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 19776 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = { 0 })
 
Theoremlspuni0 19777 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0g𝑊)    &   𝑁 = (LSpan‘𝑊)       (𝑊 ∈ LMod → (𝑁‘∅) = 0 )
 
Theoremlspun0 19778 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 19779 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 19780 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 19781 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 19782 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 19783 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 19784* 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 19785* 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.) (Revised by AV, 24-Apr-2024.)
(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)))    &   (𝜑𝐾𝑋)    &   (𝜑𝐿𝑌)       (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿))
 
10.5.3  Homomorphisms and isomorphisms of left modules
 
Syntaxclmhm 19786 Extend class notation with the generator of left module hom-sets.
class LMHom
 
Syntaxclmim 19787 The class of left module isomorphism sets.
class LMIso
 
Syntaxclmic 19788 The class of the left module isomorphism relation.
class 𝑚
 
Definitiondf-lmhm 19789* 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 19790* 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 19791 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 19792 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom LMHom
 
Theoremlmimfn 19793 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso Fn (LMod × LMod)
 
Theoremislmhm 19794* 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 19795* Property of a module homomorphism, similar to ismhm 17956. (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 19796 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 19797 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 19798 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 19799 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 19800 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)
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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 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45187
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