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Theorem List for Metamath Proof Explorer - 20701-20800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremlss0cl 20701 The zero vector belongs to every subspace. (Contributed by NM, 12-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ 0 ∈ π‘ˆ)
 
Theoremlsssn0 20702 The singleton of the zero vector is a subspace. (Contributed by NM, 13-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ { 0 } ∈ 𝑆)
 
Theoremlss0ss 20703 The zero subspace is included in every subspace. (sh0le 30960 analog.) (Contributed by NM, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑆) β†’ { 0 } βŠ† 𝑋)
 
Theoremlssle0 20704 No subspace is smaller than the zero subspace. (shle0 30962 analog.) (Contributed by NM, 20-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑆) β†’ (𝑋 βŠ† { 0 } ↔ 𝑋 = { 0 }))
 
Theoremlssne0 20705* A nonzero subspace has a nonzero vector. (shne0i 30968 analog.) (Contributed by NM, 20-Apr-2014.) (Proof shortened by Mario Carneiro, 8-Jan-2015.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (𝑋 ∈ 𝑆 β†’ (𝑋 β‰  { 0 } ↔ βˆƒπ‘¦ ∈ 𝑋 𝑦 β‰  0 ))
 
Theoremlssvneln0 20706 A vector 𝑋 which doesn't belong to a subspace π‘ˆ is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 19-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 β‰  0 )
 
Theoremlssneln0 20707 A vector 𝑋 which doesn't belong to a subspace π‘ˆ is nonzero. (Contributed by NM, 14-May-2015.) (Revised by AV, 17-Jul-2022.) (Proof shortened by AV, 19-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ Β¬ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ 𝑋 ∈ (𝑉 βˆ– { 0 }))
 
Theoremlssssr 20708* Conclude subspace ordering from nonzero vector membership. (ssrdv 3987 analog.) (Contributed by NM, 17-Aug-2014.) (Revised by AV, 13-Jul-2022.)
0 = (0gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑇 βŠ† 𝑉)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   ((πœ‘ ∧ π‘₯ ∈ (𝑉 βˆ– { 0 })) β†’ (π‘₯ ∈ 𝑇 β†’ π‘₯ ∈ π‘ˆ))    β‡’   (πœ‘ β†’ 𝑇 βŠ† π‘ˆ)
 
Theoremlssvacl 20709 Closure of vector addition in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
+ = (+gβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) ∧ (𝑋 ∈ π‘ˆ ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 + π‘Œ) ∈ π‘ˆ)
 
Theoremlssvscl 20710 Closure of scalar product in a subspace. (Contributed by NM, 11-Jan-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) ∧ (𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ)) β†’ (𝑋 Β· π‘Œ) ∈ π‘ˆ)
 
Theoremlssvnegcl 20711 Closure of negative vectors in a subspace. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (invgβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜π‘‹) ∈ π‘ˆ)
 
Theoremlsssubg 20712 All subspaces are subgroups. (Contributed by Stefan O'Rear, 11-Dec-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ π‘ˆ ∈ (SubGrpβ€˜π‘Š))
 
Theoremlsssssubg 20713 All subspaces are subgroups. (Contributed by Mario Carneiro, 19-Apr-2016.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 βŠ† (SubGrpβ€˜π‘Š))
 
Theoremislss3 20714 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 = (π‘Š β†Ύs π‘ˆ)    &   π‘‰ = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ βŠ† 𝑉 ∧ 𝑋 ∈ LMod)))
 
Theoremlsslmod 20715 A submodule is a module. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ 𝑋 ∈ LMod)
 
Theoremlsslss 20716 The subspaces of a subspace are the smaller subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘‡ = (LSubSpβ€˜π‘‹)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (𝑉 ∈ 𝑇 ↔ (𝑉 ∈ 𝑆 ∧ 𝑉 βŠ† π‘ˆ)))
 
Theoremislss4 20717* 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 = (Scalarβ€˜π‘Š)    &   π΅ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝑆 ↔ (π‘ˆ ∈ (SubGrpβ€˜π‘Š) ∧ βˆ€π‘Ž ∈ 𝐡 βˆ€π‘ ∈ π‘ˆ (π‘Ž Β· 𝑏) ∈ π‘ˆ)))
 
Theoremlss1d 20718* 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 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ {𝑣 ∣ βˆƒπ‘˜ ∈ 𝐾 𝑣 = (π‘˜ Β· 𝑋)} ∈ 𝑆)
 
Theoremlssintcl 20719 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 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝐴 βŠ† 𝑆 ∧ 𝐴 β‰  βˆ…) β†’ ∩ 𝐴 ∈ 𝑆)
 
Theoremlssincl 20720 The intersection of two subspaces is a subspace. (Contributed by NM, 7-Mar-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑇 ∈ 𝑆 ∧ π‘ˆ ∈ 𝑆) β†’ (𝑇 ∩ π‘ˆ) ∈ 𝑆)
 
Theoremlssmre 20721 The subspaces of a module comprise a Moore system on the vectors of the module. (Contributed by Stefan O'Rear, 31-Jan-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 ∈ (Mooreβ€˜π΅))
 
Theoremlssacs 20722 Submodules are an algebraic closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑆 ∈ (ACSβ€˜π΅))
 
Theoremprdsvscacl 20723* Pointwise scalar multiplication is closed in products of modules. (Contributed by Stefan O'Rear, 10-Jan-2015.)
π‘Œ = (𝑆Xs𝑅)    &   π΅ = (Baseβ€˜π‘Œ)    &    Β· = ( ·𝑠 β€˜π‘Œ)    &   πΎ = (Baseβ€˜π‘†)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ π‘Š)    &   (πœ‘ β†’ 𝑅:𝐼⟢LMod)    &   (πœ‘ β†’ 𝐹 ∈ 𝐾)    &   (πœ‘ β†’ 𝐺 ∈ 𝐡)    &   ((πœ‘ ∧ π‘₯ ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘₯)) = 𝑆)    β‡’   (πœ‘ β†’ (𝐹 Β· 𝐺) ∈ 𝐡)
 
Theoremprdslmodd 20724* The product of a family of left modules is a left module. (Contributed by Stefan O'Rear, 10-Jan-2015.)
π‘Œ = (𝑆Xs𝑅)    &   (πœ‘ β†’ 𝑆 ∈ Ring)    &   (πœ‘ β†’ 𝐼 ∈ 𝑉)    &   (πœ‘ β†’ 𝑅:𝐼⟢LMod)    &   ((πœ‘ ∧ 𝑦 ∈ 𝐼) β†’ (Scalarβ€˜(π‘…β€˜π‘¦)) = 𝑆)    β‡’   (πœ‘ β†’ π‘Œ ∈ LMod)
 
Theorempwslmod 20725 A structure power of a left module is a left module. (Contributed by Mario Carneiro, 11-Jan-2015.)
π‘Œ = (𝑅 ↑s 𝐼)    β‡’   ((𝑅 ∈ LMod ∧ 𝐼 ∈ 𝑉) β†’ π‘Œ ∈ LMod)
 
Syntaxclspn 20726 Extend class notation with span of a set of vectors.
class LSpan
 
Definitiondf-lsp 20727* Define span of a set of vectors of a left module or left vector space. (Contributed by NM, 8-Dec-2013.)
LSpan = (𝑀 ∈ V ↦ (𝑠 ∈ 𝒫 (Baseβ€˜π‘€) ↦ ∩ {𝑑 ∈ (LSubSpβ€˜π‘€) ∣ 𝑠 βŠ† 𝑑}))
 
Theoremlspfval 20728* The span function for a left vector space (or a left module). (df-span 30829 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ 𝑋 β†’ 𝑁 = (𝑠 ∈ 𝒫 𝑉 ↦ ∩ {𝑑 ∈ 𝑆 ∣ 𝑠 βŠ† 𝑑}))
 
Theoremlspf 20729 The span function on a left module maps subsets to subspaces. (Contributed by Stefan O'Rear, 12-Dec-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ 𝑁:𝒫 π‘‰βŸΆπ‘†)
 
Theoremlspval 20730* The span of a set of vectors (in a left module). (spanval 30853 analog.) (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) = ∩ {𝑑 ∈ 𝑆 ∣ π‘ˆ βŠ† 𝑑})
 
Theoremlspcl 20731 The span of a set of vectors is a subspace. (spancl 30856 analog.) (Contributed by NM, 9-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) ∈ 𝑆)
 
Theoremlspsncl 20732 The span of a singleton is a subspace (frequently used special case of lspcl 20731). (Contributed by NM, 17-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ 𝑆)
 
Theoremlspprcl 20733 The span of a pair is a subspace (frequently used special case of lspcl 20731). (Contributed by NM, 11-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) ∈ 𝑆)
 
Theoremlsptpcl 20734 The span of an unordered triple is a subspace (frequently used special case of lspcl 20731). (Contributed by NM, 22-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ 𝑍 ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ, 𝑍}) ∈ 𝑆)
 
Theoremlspsnsubg 20735 The span of a singleton is an additive subgroup (frequently used special case of lspcl 20731). (Contributed by Mario Carneiro, 21-Apr-2016.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) ∈ (SubGrpβ€˜π‘Š))
 
Theorem00lsp 20736 fvco4i 6991 lemma for linear spans. (Contributed by Stefan O'Rear, 4-Apr-2015.)
βˆ… = (LSpanβ€˜βˆ…)
 
Theoremlspid 20737 The span of a subspace is itself. (spanid 30867 analog.) (Contributed by NM, 15-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆) β†’ (π‘β€˜π‘ˆ) = π‘ˆ)
 
Theoremlspssv 20738 A span is a set of vectors. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜π‘ˆ) βŠ† 𝑉)
 
Theoremlspss 20739 Span preserves subset ordering. (spanss 30868 analog.) (Contributed by NM, 11-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† (π‘β€˜π‘ˆ))
 
Theoremlspssid 20740 A set of vectors is a subset of its span. (spanss2 30865 analog.) (Contributed by NM, 6-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ π‘ˆ βŠ† (π‘β€˜π‘ˆ))
 
Theoremlspidm 20741 The span of a set of vectors is idempotent. (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜(π‘β€˜π‘ˆ)) = (π‘β€˜π‘ˆ))
 
Theoremlspun 20742 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 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑇 βŠ† 𝑉 ∧ π‘ˆ βŠ† 𝑉) β†’ (π‘β€˜(𝑇 βˆͺ π‘ˆ)) = (π‘β€˜((π‘β€˜π‘‡) βˆͺ (π‘β€˜π‘ˆ))))
 
Theoremlspssp 20743 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 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑇 βŠ† π‘ˆ) β†’ (π‘β€˜π‘‡) βŠ† π‘ˆ)
 
Theoremmrclsp 20744 Moore closure generalizes module span. (Contributed by Stefan O'Rear, 31-Jan-2015.)
π‘ˆ = (LSubSpβ€˜π‘Š)    &   πΎ = (LSpanβ€˜π‘Š)    &   πΉ = (mrClsβ€˜π‘ˆ)    β‡’   (π‘Š ∈ LMod β†’ 𝐾 = 𝐹)
 
Theoremlspsnss 20745 The span of the singleton of a subspace member is included in the subspace. (spansnss 31091 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 4-Sep-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝑆 ∧ 𝑋 ∈ π‘ˆ) β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
 
Theoremlspsnel3 20746 A member of the span of the singleton of a vector is a member of a subspace containing the vector. (elspansn3 31092 analog.) (Contributed by NM, 4-Jul-2014.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋}))    β‡’   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)
 
Theoremlspprss 20747 The span of a pair of vectors in a subspace belongs to the subspace. (Contributed by NM, 12-Jan-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    &   (πœ‘ β†’ π‘Œ ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋, π‘Œ}) βŠ† π‘ˆ)
 
Theoremlspsnid 20748 A vector belongs to the span of its singleton. (spansnid 31083 analog.) (Contributed by NM, 9-Apr-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ 𝑋 ∈ (π‘β€˜{𝑋}))
 
Theoremlspsnel6 20749 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 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ↔ (𝑋 ∈ 𝑉 ∧ (π‘β€˜{𝑋}) βŠ† π‘ˆ)))
 
Theoremlspsnel5 20750 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 ∈ π‘ˆ ↔ (π‘β€˜{𝑋}) βŠ† π‘ˆ))
 
Theoremlspsnel5a 20751 Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 20-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    &   (πœ‘ β†’ 𝑋 ∈ π‘ˆ)    β‡’   (πœ‘ β†’ (π‘β€˜{𝑋}) βŠ† π‘ˆ)
 
Theoremlspprid1 20752 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ 𝑋 ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprid2 20753 A member of a pair of vectors belongs to their span. (Contributed by NM, 14-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ π‘Œ ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlspprvacl 20754 The sum of two vectors belongs to their span. (Contributed by NM, 20-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) ∈ (π‘β€˜{𝑋, π‘Œ}))
 
Theoremlssats2 20755* A way to express atomisticity (a subspace is the union of its atoms). (Contributed by NM, 3-Feb-2015.)
𝑆 = (LSubSpβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ π‘ˆ ∈ 𝑆)    β‡’   (πœ‘ β†’ π‘ˆ = βˆͺ π‘₯ ∈ π‘ˆ (π‘β€˜{π‘₯}))
 
Theoremlspsneli 20756 A scalar product with a vector belongs to the span of its singleton. (spansnmul 31084 analog.) (Contributed by NM, 2-Jul-2014.)
𝑉 = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝐴 ∈ 𝐾)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    β‡’   (πœ‘ β†’ (𝐴 Β· 𝑋) ∈ (π‘β€˜{𝑋}))
 
Theoremlspsn 20757* Span of the singleton of a vector. (Contributed by NM, 14-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{𝑋}) = {𝑣 ∣ βˆƒπ‘˜ ∈ 𝐾 𝑣 = (π‘˜ Β· 𝑋)})
 
Theoremlspsnel 20758* Member of span of the singleton of a vector. (elspansn 31086 analog.) (Contributed by NM, 22-Feb-2014.) (Revised by Mario Carneiro, 19-Jun-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘ˆ ∈ (π‘β€˜{𝑋}) ↔ βˆƒπ‘˜ ∈ 𝐾 π‘ˆ = (π‘˜ Β· 𝑋)))
 
Theoremlspsnvsi 20759 Span of a scalar product of a singleton. (Contributed by NM, 23-Apr-2014.) (Proof shortened by Mario Carneiro, 4-Sep-2014.)
𝐹 = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    &   π‘‰ = (Baseβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(𝑅 Β· 𝑋)}) βŠ† (π‘β€˜{𝑋}))
 
Theoremlspsnss2 20760* Comparable spans of singletons must have proportional vectors. See lspsneq 20880 for equal span version. (Contributed by NM, 7-Jun-2015.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘† = (Scalarβ€˜π‘Š)    &   πΎ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ ((π‘β€˜{𝑋}) βŠ† (π‘β€˜{π‘Œ}) ↔ βˆƒπ‘˜ ∈ 𝐾 𝑋 = (π‘˜ Β· π‘Œ)))
 
Theoremlspsnneg 20761 Negation does not change the span of a singleton. (Contributed by NM, 24-Apr-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘€ = (invgβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ (π‘β€˜{(π‘€β€˜π‘‹)}) = (π‘β€˜{𝑋}))
 
Theoremlspsnsub 20762 Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜{(𝑋 βˆ’ π‘Œ)}) = (π‘β€˜{(π‘Œ βˆ’ 𝑋)}))
 
Theoremlspsn0 20763 Span of the singleton of the zero vector. (spansn0 31061 analog.) (Contributed by NM, 15-Jan-2014.) (Proof shortened by Mario Carneiro, 19-Jun-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜{ 0 }) = { 0 })
 
Theoremlsp0 20764 Span of the empty set. (Contributed by Mario Carneiro, 5-Sep-2014.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ (π‘β€˜βˆ…) = { 0 })
 
Theoremlspuni0 20765 Union of the span of the empty set. (Contributed by NM, 14-Mar-2015.)
0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   (π‘Š ∈ LMod β†’ βˆͺ (π‘β€˜βˆ…) = 0 )
 
Theoremlspun0 20766 The span of a union with the zero subspace. (Contributed by NM, 22-May-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 βŠ† 𝑉)    β‡’   (πœ‘ β†’ (π‘β€˜(𝑋 βˆͺ { 0 })) = (π‘β€˜π‘‹))
 
Theoremlspsneq0 20767 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 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ 𝑋 ∈ 𝑉) β†’ ((π‘β€˜{𝑋}) = { 0 } ↔ 𝑋 = 0 ))
 
Theoremlspsneq0b 20768 Equal singleton spans imply both arguments are zero or both are nonzero. (Contributed by NM, 21-Mar-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) = (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 = 0 ↔ π‘Œ = 0 ))
 
Theoremlmodindp1 20769 Two independent (non-colinear) vectors have nonzero sum. (Contributed by NM, 22-Apr-2015.)
𝑉 = (Baseβ€˜π‘Š)    &    + = (+gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘Š)    &   (πœ‘ β†’ π‘Š ∈ LMod)    &   (πœ‘ β†’ 𝑋 ∈ 𝑉)    &   (πœ‘ β†’ π‘Œ ∈ 𝑉)    &   (πœ‘ β†’ (π‘β€˜{𝑋}) β‰  (π‘β€˜{π‘Œ}))    β‡’   (πœ‘ β†’ (𝑋 + π‘Œ) β‰  0 )
 
Theoremlsslsp 20770 Spans in submodules correspond to spans in the containing module. (Contributed by Stefan O'Rear, 12-Dec-2014.) TODO: Shouldn't we swap π‘€β€˜πΊ and π‘β€˜πΊ since we are computing a property of π‘β€˜πΊ? (Like we say sin 0 = 0 and not 0 = sin 0.) - NM 15-Mar-2015.
𝑋 = (π‘Š β†Ύs π‘ˆ)    &   π‘€ = (LSpanβ€˜π‘Š)    &   π‘ = (LSpanβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿 ∧ 𝐺 βŠ† π‘ˆ) β†’ (π‘€β€˜πΊ) = (π‘β€˜πΊ))
 
Theoremlss0v 20771 The zero vector in a submodule equals the zero vector in the including module. (Contributed by NM, 15-Mar-2015.)
𝑋 = (π‘Š β†Ύs π‘ˆ)    &    0 = (0gβ€˜π‘Š)    &   π‘ = (0gβ€˜π‘‹)    &   πΏ = (LSubSpβ€˜π‘Š)    β‡’   ((π‘Š ∈ LMod ∧ π‘ˆ ∈ 𝐿) β†’ 𝑍 = 0 )
 
Theoremlsspropd 20772* 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.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))    β‡’   (πœ‘ β†’ (LSubSpβ€˜πΎ) = (LSubSpβ€˜πΏ))
 
Theoremlsppropd 20773* 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.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐡 βŠ† π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ π‘Š ∧ 𝑦 ∈ π‘Š)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) ∈ π‘Š)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑃 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯( ·𝑠 β€˜πΎ)𝑦) = (π‘₯( ·𝑠 β€˜πΏ)𝑦))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΎ)))    &   (πœ‘ β†’ 𝑃 = (Baseβ€˜(Scalarβ€˜πΏ)))    &   (πœ‘ β†’ 𝐾 ∈ 𝑋)    &   (πœ‘ β†’ 𝐿 ∈ π‘Œ)    β‡’   (πœ‘ β†’ (LSpanβ€˜πΎ) = (LSpanβ€˜πΏ))
 
10.5.3  Homomorphisms and isomorphisms of left modules
 
Syntaxclmhm 20774 Extend class notation with the generator of left module hom-sets.
class LMHom
 
Syntaxclmim 20775 The class of left module isomorphism sets.
class LMIso
 
Syntaxclmic 20776 The class of the left module isomorphism relation.
class β‰ƒπ‘š
 
Definitiondf-lmhm 20777* 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, 𝑑 ∈ LMod ↦ {𝑓 ∈ (𝑠 GrpHom 𝑑) ∣ [(Scalarβ€˜π‘ ) / 𝑀]((Scalarβ€˜π‘‘) = 𝑀 ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘€)βˆ€π‘¦ ∈ (Baseβ€˜π‘ )(π‘“β€˜(π‘₯( ·𝑠 β€˜π‘ )𝑦)) = (π‘₯( ·𝑠 β€˜π‘‘)(π‘“β€˜π‘¦)))})
 
Definitiondf-lmim 20778* 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, 𝑑 ∈ LMod ↦ {𝑔 ∈ (𝑠 LMHom 𝑑) ∣ 𝑔:(Baseβ€˜π‘ )–1-1-ontoβ†’(Baseβ€˜π‘‘)})
 
Definitiondf-lmic 20779 Two modules are said to be isomorphic iff they are connected by at least one isomorphism. (Contributed by Stefan O'Rear, 25-Jan-2015.)
β‰ƒπ‘š = (β—‘ LMIso β€œ (V βˆ– 1o))
 
Theoremreldmlmhm 20780 Lemma for module homomorphisms. (Contributed by Stefan O'Rear, 31-Dec-2014.)
Rel dom LMHom
 
Theoremlmimfn 20781 Lemma for module isomorphisms. (Contributed by Stefan O'Rear, 23-Aug-2015.)
LMIso Fn (LMod Γ— LMod)
 
Theoremislmhm 20782* 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 = (Scalarβ€˜π‘†)    &   πΏ = (Scalarβ€˜π‘‡)    &   π΅ = (Baseβ€˜πΎ)    &   πΈ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘†)    &    Γ— = ( ·𝑠 β€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
 
Theoremislmhm3 20783* Property of a module homomorphism, similar to ismhm 18707. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝐾 = (Scalarβ€˜π‘†)    &   πΏ = (Scalarβ€˜π‘‡)    &   π΅ = (Baseβ€˜πΎ)    &   πΈ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘†)    &    Γ— = ( ·𝑠 β€˜π‘‡)    β‡’   ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) β†’ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐡 βˆ€π‘¦ ∈ 𝐸 (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))))
 
Theoremlmhmlem 20784 Non-quantified consequences of a left module homomorphism. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalarβ€˜π‘†)    &   πΏ = (Scalarβ€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ 𝐿 = 𝐾)))
 
Theoremlmhmsca 20785 A homomorphism of left modules constrains both modules to the same ring of scalars. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalarβ€˜π‘†)    &   πΏ = (Scalarβ€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐿 = 𝐾)
 
Theoremlmghm 20786 A homomorphism of left modules is a homomorphism of groups. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))
 
Theoremlmhmlmod2 20787 A homomorphism of left modules has a left module as codomain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑇 ∈ LMod)
 
Theoremlmhmlmod1 20788 A homomorphism of left modules has a left module as domain. (Contributed by Stefan O'Rear, 1-Jan-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝑆 ∈ LMod)
 
Theoremlmhmf 20789 A homomorphism of left modules is a function. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐡 = (Baseβ€˜π‘†)    &   πΆ = (Baseβ€˜π‘‡)    β‡’   (𝐹 ∈ (𝑆 LMHom 𝑇) β†’ 𝐹:𝐡⟢𝐢)
 
Theoremlmhmlin 20790 A homomorphism of left modules is 𝐾-linear. (Contributed by Stefan O'Rear, 1-Jan-2015.)
𝐾 = (Scalarβ€˜π‘†)    &   π΅ = (Baseβ€˜πΎ)    &   πΈ = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘†)    &    Γ— = ( ·𝑠 β€˜π‘‡)    β‡’   ((𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐸) β†’ (πΉβ€˜(𝑋 Β· π‘Œ)) = (𝑋 Γ— (πΉβ€˜π‘Œ)))
 
Theoremlmodvsinv 20791 Multiplication of a vector by a negated scalar. (Contributed by Stefan O'Rear, 28-Feb-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (invgβ€˜π‘Š)    &   π‘€ = (invgβ€˜πΉ)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐡) β†’ ((π‘€β€˜π‘…) Β· 𝑋) = (π‘β€˜(𝑅 Β· 𝑋)))
 
Theoremlmodvsinv2 20792 Multiplying a negated vector by a scalar. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐡 = (Baseβ€˜π‘Š)    &   πΉ = (Scalarβ€˜π‘Š)    &    Β· = ( ·𝑠 β€˜π‘Š)    &   π‘ = (invgβ€˜π‘Š)    &   πΎ = (Baseβ€˜πΉ)    β‡’   ((π‘Š ∈ LMod ∧ 𝑅 ∈ 𝐾 ∧ 𝑋 ∈ 𝐡) β†’ (𝑅 Β· (π‘β€˜π‘‹)) = (π‘β€˜(𝑅 Β· 𝑋)))
 
Theoremislmhm2 20793* A one-equation proof of linearity of a left module homomorphism, similar to df-lss 20687. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐡 = (Baseβ€˜π‘†)    &   πΆ = (Baseβ€˜π‘‡)    &   πΎ = (Scalarβ€˜π‘†)    &   πΏ = (Scalarβ€˜π‘‡)    &   πΈ = (Baseβ€˜πΎ)    &    + = (+gβ€˜π‘†)    &    ⨣ = (+gβ€˜π‘‡)    &    Β· = ( ·𝑠 β€˜π‘†)    &    Γ— = ( ·𝑠 β€˜π‘‡)    β‡’   ((𝑆 ∈ LMod ∧ 𝑇 ∈ LMod) β†’ (𝐹 ∈ (𝑆 LMHom 𝑇) ↔ (𝐹:𝐡⟢𝐢 ∧ 𝐿 = 𝐾 ∧ βˆ€π‘₯ ∈ 𝐸 βˆ€π‘¦ ∈ 𝐡 βˆ€π‘§ ∈ 𝐡 (πΉβ€˜((π‘₯ Β· 𝑦) + 𝑧)) = ((π‘₯ Γ— (πΉβ€˜π‘¦)) ⨣ (πΉβ€˜π‘§)))))
 
Theoremislmhmd 20794* Deduction for a module homomorphism. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑋 = (Baseβ€˜π‘†)    &    Β· = ( ·𝑠 β€˜π‘†)    &    Γ— = ( ·𝑠 β€˜π‘‡)    &   πΎ = (Scalarβ€˜π‘†)    &   π½ = (Scalarβ€˜π‘‡)    &   π‘ = (Baseβ€˜πΎ)    &   (πœ‘ β†’ 𝑆 ∈ LMod)    &   (πœ‘ β†’ 𝑇 ∈ LMod)    &   (πœ‘ β†’ 𝐽 = 𝐾)    &   (πœ‘ β†’ 𝐹 ∈ (𝑆 GrpHom 𝑇))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑁 ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜(π‘₯ Β· 𝑦)) = (π‘₯ Γ— (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ 𝐹 ∈ (𝑆 LMHom 𝑇))
 
Theorem0lmhm 20795 The constant zero linear function between two modules. (Contributed by Stefan O'Rear, 5-Sep-2015.)
0 = (0gβ€˜π‘)    &   π΅ = (Baseβ€˜π‘€)    &   π‘† = (Scalarβ€˜π‘€)    &   π‘‡ = (Scalarβ€˜π‘)    β‡’   ((𝑀 ∈ LMod ∧ 𝑁 ∈ LMod ∧ 𝑆 = 𝑇) β†’ (𝐡 Γ— { 0 }) ∈ (𝑀 LMHom 𝑁))
 
Theoremidlmhm 20796 The identity function on a module is linear. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐡 = (Baseβ€˜π‘€)    β‡’   (𝑀 ∈ LMod β†’ ( I β†Ύ 𝐡) ∈ (𝑀 LMHom 𝑀))
 
Theoreminvlmhm 20797 The negative function on a module is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐼 = (invgβ€˜π‘€)    β‡’   (𝑀 ∈ LMod β†’ 𝐼 ∈ (𝑀 LMHom 𝑀))
 
Theoremlmhmco 20798 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)) β†’ (𝐹 ∘ 𝐺) ∈ (𝑀 LMHom 𝑂))
 
Theoremlmhmplusg 20799 The pointwise sum of two linear functions is linear. (Contributed by Stefan O'Rear, 5-Sep-2015.)
+ = (+gβ€˜π‘)    β‡’   ((𝐹 ∈ (𝑀 LMHom 𝑁) ∧ 𝐺 ∈ (𝑀 LMHom 𝑁)) β†’ (𝐹 ∘f + 𝐺) ∈ (𝑀 LMHom 𝑁))
 
Theoremlmhmvsca 20800 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 = (Baseβ€˜π‘€)    &    Β· = ( ·𝑠 β€˜π‘)    &   π½ = (Scalarβ€˜π‘)    &   πΎ = (Baseβ€˜π½)    β‡’   ((𝐽 ∈ CRing ∧ 𝐴 ∈ 𝐾 ∧ 𝐹 ∈ (𝑀 LMHom 𝑁)) β†’ ((𝑉 Γ— {𝐴}) ∘f Β· 𝐹) ∈ (𝑀 LMHom 𝑁))
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