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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | islssd 19701* | 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.) |
⊢ (𝜑 → 𝐹 = (Scalar‘𝑊)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐹)) & ⊢ (𝜑 → 𝑉 = (Base‘𝑊)) & ⊢ (𝜑 → + = (+g‘𝑊)) & ⊢ (𝜑 → · = ( ·𝑠 ‘𝑊)) & ⊢ (𝜑 → 𝑆 = (LSubSp‘𝑊)) & ⊢ (𝜑 → 𝑈 ⊆ 𝑉) & ⊢ (𝜑 → 𝑈 ≠ ∅) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑎 ∈ 𝑈 ∧ 𝑏 ∈ 𝑈)) → ((𝑥 · 𝑎) + 𝑏) ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑈 ∈ 𝑆) | ||
Theorem | lssss 19702 | 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) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝑈 ⊆ 𝑉) | ||
Theorem | lssel 19703 | 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) ⇒ ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) | ||
Theorem | lss1 19704 | 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 → 𝑉 ∈ 𝑆) | ||
Theorem | lssuni 19705 | The union of all subspaces is the vector space. (Contributed by NM, 13-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) ⇒ ⊢ (𝜑 → ∪ 𝑆 = 𝑉) | ||
Theorem | lssn0 19706 | A subspace is not empty. (Contributed by NM, 12-Jan-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) |
⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝑈 ≠ ∅) | ||
Theorem | 00lss 19707 | The empty structure has no subspaces (for use with fvco4i 6756). (Contributed by Stefan O'Rear, 31-Mar-2015.) |
⊢ ∅ = (LSubSp‘∅) | ||
Theorem | lsscl 19708 | Closure property of a subspace. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 8-Jan-2015.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑈 ∈ 𝑆 ∧ (𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → ((𝑍 · 𝑋) + 𝑌) ∈ 𝑈) | ||
Theorem | lssvsubcl 19709 | Closure of vector subtraction in a subspace. (Contributed by NM, 31-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ − = (-g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → (𝑋 − 𝑌) ∈ 𝑈) | ||
Theorem | lssvancl1 19710 | 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 19902. Can it be used along with lspsnne1 19883, lspsnne2 19884 to shorten this proof? (Contributed by NM, 14-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → ¬ (𝑋 + 𝑌) ∈ 𝑈) | ||
Theorem | lssvancl2 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. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → ¬ (𝑌 + 𝑋) ∈ 𝑈) | ||
Theorem | lss0cl 19712 | The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 0 ∈ 𝑈) | ||
Theorem | lsssn0 19713 | The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → { 0 } ∈ 𝑆) | ||
Theorem | lss0ss 19714 | The zero subspace is included in every subspace. (sh0le 29211 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆) → { 0 } ⊆ 𝑋) | ||
Theorem | lssle0 19715 | No subspace is smaller than the zero subspace. (shle0 29213 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑆) → (𝑋 ⊆ { 0 } ↔ 𝑋 = { 0 })) | ||
Theorem | lssne0 19716* | A nonzero subspace has a nonzero vector. (shne0i 29219 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑋 ∈ 𝑆 → (𝑋 ≠ { 0 } ↔ ∃𝑦 ∈ 𝑋 𝑦 ≠ 0 )) | ||
Theorem | lssvneln0 19717 | 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 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑋 ≠ 0 ) | ||
Theorem | lssneln0 19718 | 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 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → ¬ 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | ||
Theorem | lssssr 19719* | Conclude subspace ordering from nonzero vector membership. (ssrdv 3972 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑇 ⊆ 𝑉) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑉 ∖ { 0 })) → (𝑥 ∈ 𝑇 → 𝑥 ∈ 𝑈)) ⇒ ⊢ (𝜑 → 𝑇 ⊆ 𝑈) | ||
Theorem | lssvacl 19720 | Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ + = (+g‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝑈 ∧ 𝑌 ∈ 𝑈)) → (𝑋 + 𝑌) ∈ 𝑈) | ||
Theorem | lssvscl 19721 | Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝑈)) → (𝑋 · 𝑌) ∈ 𝑈) | ||
Theorem | lssvnegcl 19722 | Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (invg‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘𝑋) ∈ 𝑈) | ||
Theorem | lsssubg 19723 | All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑈 ∈ (SubGrp‘𝑊)) | ||
Theorem | lsssssubg 19724 | All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.) |
⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → 𝑆 ⊆ (SubGrp‘𝑊)) | ||
Theorem | islss3 19725 | 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 = (𝑊 ↾s 𝑈) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝑆 ↔ (𝑈 ⊆ 𝑉 ∧ 𝑋 ∈ LMod))) | ||
Theorem | lsslmod 19726 | A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ LMod) | ||
Theorem | lsslss 19727 | The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑇 = (LSubSp‘𝑋) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 ⊆ 𝑈))) | ||
Theorem | islss4 19728* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑈 ∈ 𝑆 ↔ (𝑈 ∈ (SubGrp‘𝑊) ∧ ∀𝑎 ∈ 𝐵 ∀𝑏 ∈ 𝑈 (𝑎 · 𝑏) ∈ 𝑈))) | ||
Theorem | lss1d 19729* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)} ∈ 𝑆) | ||
Theorem | lssintcl 19730 | 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 ∧ 𝐴 ⊆ 𝑆 ∧ 𝐴 ≠ ∅) → ∩ 𝐴 ∈ 𝑆) | ||
Theorem | lssincl 19731 | 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 ∧ 𝑇 ∈ 𝑆 ∧ 𝑈 ∈ 𝑆) → (𝑇 ∩ 𝑈) ∈ 𝑆) | ||
Theorem | lssmre 19732 | 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)) | ||
Theorem | lssacs 19733 | 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)) | ||
Theorem | prdsvscacl 19734* | Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ · = ( ·𝑠 ‘𝑌) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ (𝜑 → 𝐹 ∈ 𝐾) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) ⇒ ⊢ (𝜑 → (𝐹 · 𝐺) ∈ 𝐵) | ||
Theorem | prdslmodd 19735* | The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐼) → (Scalar‘(𝑅‘𝑦)) = 𝑆) ⇒ ⊢ (𝜑 → 𝑌 ∈ LMod) | ||
Theorem | pwslmod 19736 | A structure power of a left module is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.) |
⊢ 𝑌 = (𝑅 ↑s 𝐼) ⇒ ⊢ ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉) → 𝑌 ∈ LMod) | ||
Syntax | clspn 19737 | Extend class notation with span of a set of vectors. |
class LSpan | ||
Definition | df-lsp 19738* | 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) ∣ 𝑠 ⊆ 𝑡})) | ||
Theorem | lspfval 19739* | The span function for a left vector space (or a left module). (df-span 29080 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑡 ∈ 𝑆 ∣ 𝑠 ⊆ 𝑡})) | ||
Theorem | lspf 19740 | 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 𝑉⟶𝑆) | ||
Theorem | lspval 19741* | The span of a set of vectors (in a left module). (spanval 29104 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 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) = ∩ {𝑡 ∈ 𝑆 ∣ 𝑈 ⊆ 𝑡}) | ||
Theorem | lspcl 19742 | The span of a set of vectors is a subspace. (spancl 29107 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 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ∈ 𝑆) | ||
Theorem | lspsncl 19743 | The span of a singleton is a subspace (frequently used special case of lspcl 19742). (Contributed by NM, 17-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ 𝑆) | ||
Theorem | lspprcl 19744 | The span of a pair is a subspace (frequently used special case of lspcl 19742). (Contributed by NM, 11-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ∈ 𝑆) | ||
Theorem | lsptpcl 19745 | The span of an unordered triple is a subspace (frequently used special case of lspcl 19742). (Contributed by NM, 22-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → 𝑍 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌, 𝑍}) ∈ 𝑆) | ||
Theorem | lspsnsubg 19746 | The span of a singleton is an additive subgroup (frequently used special case of lspcl 19742). (Contributed by Mario Carneiro, 21-Apr-2016.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (SubGrp‘𝑊)) | ||
Theorem | 00lsp 19747 | fvco4i 6756 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.) |
⊢ ∅ = (LSpan‘∅) | ||
Theorem | lspid 19748 | The span of a subspace is itself. (spanid 29118 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆) → (𝑁‘𝑈) = 𝑈) | ||
Theorem | lspssv 19749 | 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 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘𝑈) ⊆ 𝑉) | ||
Theorem | lspss 19750 | Span preserves subset ordering. (spanss 29119 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ (𝑁‘𝑈)) | ||
Theorem | lspssid 19751 | A set of vectors is a subset of its span. (spanss2 29116 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ⊆ 𝑉) → 𝑈 ⊆ (𝑁‘𝑈)) | ||
Theorem | lspidm 19752 | 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 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑁‘𝑈)) = (𝑁‘𝑈)) | ||
Theorem | lspun 19753 | 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 ∧ 𝑇 ⊆ 𝑉 ∧ 𝑈 ⊆ 𝑉) → (𝑁‘(𝑇 ∪ 𝑈)) = (𝑁‘((𝑁‘𝑇) ∪ (𝑁‘𝑈)))) | ||
Theorem | lspssp 19754 | 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 ∧ 𝑈 ∈ 𝑆 ∧ 𝑇 ⊆ 𝑈) → (𝑁‘𝑇) ⊆ 𝑈) | ||
Theorem | mrclsp 19755 | 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) | ||
Theorem | lspsnss 19756 | The span of the singleton of a subspace member is included in the subspace. (spansnss 29342 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) | ||
Theorem | lspsnel3 19757 | A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 29343 analog.) (Contributed by NM, 4-Jul-2014.) |
⊢ 𝑆 = (LSubSp‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋})) ⇒ ⊢ (𝜑 → 𝑌 ∈ 𝑈) | ||
Theorem | lspprss 19758 | 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) & ⊢ (𝜑 → 𝑌 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋, 𝑌}) ⊆ 𝑈) | ||
Theorem | lspsnid 19759 | A vector belongs to the span of its singleton. (spansnid 29334 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) | ||
Theorem | lspsnel6 19760 | 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) | ||
Theorem | lspsnel5 19761 | 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑁‘{𝑋}) ⊆ 𝑈)) | ||
Theorem | lspsnel5a 19762 | 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) & ⊢ (𝜑 → 𝑋 ∈ 𝑈) ⇒ ⊢ (𝜑 → (𝑁‘{𝑋}) ⊆ 𝑈) | ||
Theorem | lspprid1 19763 | A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑋 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspprid2 19764 | A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → 𝑌 ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lspprvacl 19765 | The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ∈ (𝑁‘{𝑋, 𝑌})) | ||
Theorem | lssats2 19766* | 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) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑈 ∈ 𝑆) ⇒ ⊢ (𝜑 → 𝑈 = ∪ 𝑥 ∈ 𝑈 (𝑁‘{𝑥})) | ||
Theorem | lspsneli 19767 | A scalar product with a vector belongs to the span of its singleton. (spansnmul 29335 analog.) (Contributed by NM, 2-Jul-2014.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝐴 · 𝑋) ∈ (𝑁‘{𝑋})) | ||
Theorem | lspsn 19768* | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) = {𝑣 ∣ ∃𝑘 ∈ 𝐾 𝑣 = (𝑘 · 𝑋)}) | ||
Theorem | lspsnel 19769* | Member of span of the singleton of a vector. (elspansn 29337 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.) |
⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑈 ∈ (𝑁‘{𝑋}) ↔ ∃𝑘 ∈ 𝐾 𝑈 = (𝑘 · 𝑋))) | ||
Theorem | lspsnvsi 19770 | 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) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑅 · 𝑋)}) ⊆ (𝑁‘{𝑋})) | ||
Theorem | lspsnss2 19771* | Comparable spans of singletons must have proportional vectors. See lspsneq 19888 for equal span version. (Contributed by NM, 7-Jun-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌}) ↔ ∃𝑘 ∈ 𝐾 𝑋 = (𝑘 · 𝑌))) | ||
Theorem | lspsnneg 19772 | 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 = (invg‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{(𝑀‘𝑋)}) = (𝑁‘{𝑋})) | ||
Theorem | lspsnsub 19773 | Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) = (𝑁‘{(𝑌 − 𝑋)})) | ||
Theorem | lspsn0 19774 | Span of the singleton of the zero vector. (spansn0 29312 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑁‘{ 0 }) = { 0 }) | ||
Theorem | lsp0 19775 | Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → (𝑁‘∅) = { 0 }) | ||
Theorem | lspuni0 19776 | Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.) |
⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ (𝑊 ∈ LMod → ∪ (𝑁‘∅) = 0 ) | ||
Theorem | lspun0 19777 | The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ⊆ 𝑉) ⇒ ⊢ (𝜑 → (𝑁‘(𝑋 ∪ { 0 })) = (𝑁‘𝑋)) | ||
Theorem | lspsneq0 19778 | 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 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) = { 0 } ↔ 𝑋 = 0 )) | ||
Theorem | lspsneq0b 19779 | Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) = (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 = 0 ↔ 𝑌 = 0 )) | ||
Theorem | lmodindp1 19780 | Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ LMod) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌})) ⇒ ⊢ (𝜑 → (𝑋 + 𝑌) ≠ 0 ) | ||
Theorem | lsslsp 19781 | Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap 𝑀‘𝐺 and 𝑁‘𝐺 since we are computing a property of 𝑁‘𝐺? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015. |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑀 = (LSpan‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑋) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿 ∧ 𝐺 ⊆ 𝑈) → (𝑀‘𝐺) = (𝑁‘𝐺)) | ||
Theorem | lss0v 19782 | The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝑍 = (0g‘𝑋) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝐿) → 𝑍 = 0 ) | ||
Theorem | lsspropd 19783* | 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.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) ⇒ ⊢ (𝜑 → (LSubSp‘𝐾) = (LSubSp‘𝐿)) | ||
Theorem | lsppropd 19784* | 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.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐵 ⊆ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑊 ∧ 𝑦 ∈ 𝑊)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) ∈ 𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑃 ∧ 𝑦 ∈ 𝐵)) → (𝑥( ·𝑠 ‘𝐾)𝑦) = (𝑥( ·𝑠 ‘𝐿)𝑦)) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐾))) & ⊢ (𝜑 → 𝑃 = (Base‘(Scalar‘𝐿))) & ⊢ (𝜑 → 𝐾 ∈ V) & ⊢ (𝜑 → 𝐿 ∈ V) ⇒ ⊢ (𝜑 → (LSpan‘𝐾) = (LSpan‘𝐿)) | ||
Syntax | clmhm 19785 | Extend class notation with the generator of left module hom-sets. |
class LMHom | ||
Syntax | clmim 19786 | The class of left module isomorphism sets. |
class LMIso | ||
Syntax | clmic 19787 | The class of the left module isomorphism relation. |
class ≃𝑚 | ||
Definition | df-lmhm 19788* | 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( ·𝑠 ‘𝑠)𝑦)) = (𝑥( ·𝑠 ‘𝑡)(𝑓‘𝑦)))}) | ||
Definition | df-lmim 19789* | 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)}) | ||
Definition | df-lmic 19790 | 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)) | ||
Theorem | reldmlmhm 19791 | Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.) |
⊢ Rel dom LMHom | ||
Theorem | lmimfn 19792 | Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.) |
⊢ LMIso Fn (LMod × LMod) | ||
Theorem | islmhm 19793* | 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) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) | ||
Theorem | islmhm3 19794* | Property of a module homomorphism, similar to ismhm 17952. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝐾 = (Scalar‘𝑆) & ⊢ 𝐿 = (Scalar‘𝑇) & ⊢ 𝐵 = (Base‘𝐾) & ⊢ 𝐸 = (Base‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑆) & ⊢ × = ( ·𝑠 ‘𝑇) ⇒ ⊢ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) → (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐸 (𝐹‘(𝑥 · 𝑦)) = (𝑥 × (𝐹‘𝑦))))) | ||
Theorem | lmhmlem 19795 | 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))) | ||
Theorem | lmhmsca 19796 | 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) | ||
Theorem | lmghm 19797 | 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)) | ||
Theorem | lmhmlmod2 19798 | 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) | ||
Theorem | lmhmlmod1 19799 | 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) | ||
Theorem | lmhmf 19800 | 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:𝐵⟶𝐶) |
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