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Theorem List for Metamath Proof Explorer - 42701-42800   *Has distinct variable group(s)
TypeLabelDescription
Statement

TheoremdmatALTbasel 42701* An element of the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over a ring 𝑅, i.e. an 𝑁 x 𝑁 diagonal matrix over the ring 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMatALT 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (𝑀 ∈ (Base‘𝐷) ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))

Theoremdmatbas 42702 The set of all 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅 is the base set of the algebra of 𝑁 x 𝑁 diagonal matrices over (the ring) 𝑅. (Contributed by AV, 8-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = (Base‘(𝑁 DMatALT 𝑅)))

20.35.16.2  Linear combinations

According to Wikipedia ("Linear combination", 29-Mar-2019, https://en.wikipedia.org/wiki/Linear_combination) "In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g., a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics." In linear algebra, these "terms" are "vectors" (elements from vector spaces or left modules), and the constants are elements of the underlying field resp. ring. This corresponds to the definition in [Lang] p. 129: "Let M be a module over a ring A and let S be a subset of M. By a linear combination of elements of S (with coefficients in A) one means a sum ∑x ∈S axx where {ax} is a set of elements of A, ...". In the definition in [Lang] p. 129, it is additionally claimed that "..., almost all of which [elements of A] are equal to 0.". This is not necessarily required in the following definition df-linc 42705, but it is essential if additions and scalar multiplications of linear combinations are considered. Therefore, we define the set of all linear combinations with finite support in df-lco 42706, so that we can show that such sets are submodules of the corresponding modules, see lincolss 42733.
Remark:According to Wikipedia ("Linear span", 28-Apr-2019, https://en.wikipedia.org/wiki/Linear_span) "In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space [or module] is the intersection of all linear subspaces which each contain every vector in that set.", and "Alternately, the span of [a set] S may be defined as the set of all finite linear combinations of elements (vectors) of S". Whereas spans are defined according to the first approach in df-lsp 19174, the set of all linear combinations as defined by df-lco 42706 follows the alternative approach. That both definitions are equivalent is shown by lspeqlco 42738.

Syntaxclinc 42703 Extend class notation with the operation constructing a linear combination (of vectors from a left module).
class linC

Syntaxclinco 42704 Extend class notation with the operation constructing a set of linear combinations (of vectors from a left module) with finite support.
class LinCo

Definitiondf-linc 42705* Define the operation constructing a linear combination. Although this definition is taylored for linear combinations of vectors from left modules, it can be used for any structure having a Base, Scalar s and a scalar multiplication ·𝑠. (Contributed by AV, 29-Mar-2019.)
linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑚)𝑥)))))

Definitiondf-lco 42706* Define the operation constructing the set of all linear combinations for a set of vectors. (Contributed by AV, 31-Mar-2019.) (Revised by AV, 28-Jul-2019.)
LinCo = (𝑚 ∈ V, 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ {𝑐 ∈ (Base‘𝑚) ∣ ∃𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣)(𝑠 finSupp (0g‘(Scalar‘𝑚)) ∧ 𝑐 = (𝑠( linC ‘𝑚)𝑣))})

Theoremlincop 42707* A linear combination as operation. (Contributed by AV, 30-Mar-2019.)
(𝑀𝑋 → ( linC ‘𝑀) = (𝑠 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑀) ↦ (𝑀 Σg (𝑥𝑣 ↦ ((𝑠𝑥)( ·𝑠𝑀)𝑥)))))

Theoremlincval 42708* The value of a linear combination. (Contributed by AV, 30-Mar-2019.)
((𝑀𝑋𝑆 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑆( linC ‘𝑀)𝑉) = (𝑀 Σg (𝑥𝑉 ↦ ((𝑆𝑥)( ·𝑠𝑀)𝑥))))

Theoremdflinc2 42709* Alternative definition of linear combinations using the function operation. (Contributed by AV, 1-Apr-2019.)
linC = (𝑚 ∈ V ↦ (𝑠 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑣), 𝑣 ∈ 𝒫 (Base‘𝑚) ↦ (𝑚 Σg (𝑠𝑓 ( ·𝑠𝑚)( I ↾ 𝑣)))))

Theoremlcoop 42710* A linear combination as operation. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = {𝑐𝐵 ∣ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝑐 = (𝑠( linC ‘𝑀)𝑉))})

Theoremlcoval 42711* The value of a linear combination. (Contributed by AV, 5-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)       ((𝑀𝑋𝑉 ∈ 𝒫 𝐵) → (𝐶 ∈ (𝑀 LinCo 𝑉) ↔ (𝐶𝐵 ∧ ∃𝑠 ∈ (𝑅𝑚 𝑉)(𝑠 finSupp (0g𝑆) ∧ 𝐶 = (𝑠( linC ‘𝑀)𝑉)))))

Theoremlincfsuppcl 42712 A linear combination of vectors (with finite support) is a vector. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ (𝑉𝑊𝑉𝐵) ∧ (𝐹 ∈ (𝑆𝑚 𝑉) ∧ 𝐹 finSupp 0 )) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝐵)

Theoremlinccl 42713 A linear combination of vectors is a vector. (Contributed by AV, 31-Mar-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))       ((𝑀 ∈ LMod ∧ (𝑉 ∈ Fin ∧ 𝑉𝐵𝑆 ∈ (𝑅𝑚 𝑉))) → (𝑆( linC ‘𝑀)𝑉) ∈ 𝐵)

Theoremlincval0 42714 The value of an empty linear combination. (Contributed by AV, 12-Apr-2019.)
(𝑀𝑋 → (∅( linC ‘𝑀)∅) = (0g𝑀))

Theoremlincvalsng 42715 The linear combination over a singleton. (Contributed by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → ({⟨𝑉, 𝑌⟩} ( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))

Theoremlincvalsn 42716 The linear combination over a singleton. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 25-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)    &   𝐹 = {⟨𝑉, 𝑌⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵𝑌𝑅) → (𝐹( linC ‘𝑀){𝑉}) = (𝑌 · 𝑉))

Theoremlincvalpr 42717 The linear combination over an unordered pair. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &   𝐹 = {⟨𝑉, 𝑋⟩, ⟨𝑊, 𝑌⟩}       (((𝑀 ∈ LMod ∧ 𝑉𝑊) ∧ (𝑉𝐵𝑋𝑅) ∧ (𝑊𝐵𝑌𝑅)) → (𝐹( linC ‘𝑀){𝑉, 𝑊}) = ((𝑋 · 𝑉) + (𝑌 · 𝑊)))

Theoremlincval1 42718 The linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &   𝐹 = {⟨𝑉, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀))

Theoremlcosn0 42719 Properties of a linear combination over a singleton mapping to 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &   𝐹 = {⟨𝑉, (0g𝑆)⟩}       ((𝑀 ∈ LMod ∧ 𝑉𝐵) → (𝐹 ∈ (𝑅𝑚 {𝑉}) ∧ 𝐹 finSupp (0g𝑆) ∧ (𝐹( linC ‘𝑀){𝑉}) = (0g𝑀)))

Theoremlincvalsc0 42720* The linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Theoremlcoc0 42721* Properties of a linear combination where all scalars are 0. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉0 )    &   𝑅 = (Base‘𝑆)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹 ∈ (𝑅𝑚 𝑉) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑉) = 𝑍))

Theoremlinc0scn0 42722* If a set contains the zero element of a module, there is a linear combination being 0 where not all scalars are 0. (Contributed by AV, 13-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    1 = (1r𝑆)    &   𝑍 = (0g𝑀)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑍, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝐹( linC ‘𝑀)𝑉) = 𝑍)

Theoremlincdifsn 42723 A vector is a linear combination of a set containing this vector. (Contributed by AV, 21-Apr-2019.) (Revised by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    · = ( ·𝑠𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑅)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) ∧ (𝐹 ∈ (𝑆𝑚 𝑉) ∧ 𝐹 finSupp 0 ) ∧ 𝐺 = (𝐹 ↾ (𝑉 ∖ {𝑋}))) → (𝐹( linC ‘𝑀)𝑉) = ((𝐺( linC ‘𝑀)(𝑉 ∖ {𝑋})) + ((𝐹𝑋) · 𝑋)))

Theoremlinc1 42724* A vector is a linear combination of a set containing this vector. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    1 = (1r𝑆)    &   𝐹 = (𝑥𝑉 ↦ if(𝑥 = 𝑋, 1 , 0 ))       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵𝑋𝑉) → (𝐹( linC ‘𝑀)𝑉) = 𝑋)

Theoremlincellss 42725 A linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 28-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉𝑆) → ((𝐹 ∈ ((Base‘(Scalar‘𝑀)) ↑𝑚 𝑉) ∧ 𝐹 finSupp (0g‘(Scalar‘𝑀))) → (𝐹( linC ‘𝑀)𝑉) ∈ 𝑆))

Theoremlco0 42726 The set of empty linear combinations over a monoid is the singleton with the identity element of the monoid. (Contributed by AV, 12-Apr-2019.)
(𝑀 ∈ Mnd → (𝑀 LinCo ∅) = {(0g𝑀)})

Theoremlcoel0 42727 The zero vector is always a linear combination. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (0g𝑀) ∈ (𝑀 LinCo 𝑉))

Theoremlincsum 42728 The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Revised by AV, 28-Jul-2019.)
+ = (+g𝑀)    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑌 = (𝐵( linC ‘𝑀)𝑉)    &   𝑆 = (Scalar‘𝑀)    &   𝑅 = (Base‘𝑆)    &    = (+g𝑆)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝐵 ∈ (𝑅𝑚 𝑉)) ∧ (𝐴 finSupp (0g𝑆) ∧ 𝐵 finSupp (0g𝑆))) → (𝑋 + 𝑌) = ((𝐴𝑓 𝐵)( linC ‘𝑀)𝑉))

Theoremlincscm 42729* A linear combinations multiplied with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 9-Apr-2019.) (Revised by AV, 28-Jul-2019.)
= ( ·𝑠𝑀)    &    · = (.r‘(Scalar‘𝑀))    &   𝑋 = (𝐴( linC ‘𝑀)𝑉)    &   𝑅 = (Base‘(Scalar‘𝑀))    &   𝐹 = (𝑥𝑉 ↦ (𝑆 · (𝐴𝑥)))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐴 ∈ (𝑅𝑚 𝑉) ∧ 𝑆𝑅) ∧ 𝐴 finSupp (0g‘(Scalar‘𝑀))) → (𝑆 𝑋) = (𝐹( linC ‘𝑀)𝑉))

Theoremlincsumcl 42730 The sum of two linear combinations is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 4-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
+ = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶 ∈ (𝑀 LinCo 𝑉) ∧ 𝐷 ∈ (𝑀 LinCo 𝑉))) → (𝐶 + 𝐷) ∈ (𝑀 LinCo 𝑉))

Theoremlincscmcl 42731 The multiplication of a linear combination with a scalar is a linear combination, see also the proof in [Lang] p. 129. (Contributed by AV, 11-Apr-2019.) (Proof shortened by AV, 28-Jul-2019.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ 𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉)) → (𝐶 · 𝐷) ∈ (𝑀 LinCo 𝑉))

Theoremlincsumscmcl 42732 The sum of a linear combination and a multiplication of a linear combination with a scalar is a linear combination. (Contributed by AV, 11-Apr-2019.)
· = ( ·𝑠𝑀)    &   𝑅 = (Base‘(Scalar‘𝑀))    &    + = (+g𝑀)       (((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) ∧ (𝐶𝑅𝐷 ∈ (𝑀 LinCo 𝑉) ∧ 𝐵 ∈ (𝑀 LinCo 𝑉))) → ((𝐶 · 𝐷) + 𝐵) ∈ (𝑀 LinCo 𝑉))

Theoremlincolss 42733 According to the statement in [Lang] p. 129, the set (LSubSp‘𝑀) of all linear combinations of a set of vectors V is a submodule (generated by V) of the module M. The elements of V are called generators of (LSubSp‘𝑀). (Contributed by AV, 12-Apr-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → (𝑀 LinCo 𝑉) ∈ (LSubSp‘𝑀))

Theoremellcoellss 42734* Every linear combination of a subset of a linear subspace is also contained in the linear subspace. (Contributed by AV, 20-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑆 ∈ (LSubSp‘𝑀) ∧ 𝑉𝑆) → ∀𝑥 ∈ (𝑀 LinCo 𝑉)𝑥𝑆)

Theoremlcoss 42735 A set of vectors of a module is a subset of the set of all linear combinations of the set. (Contributed by AV, 18-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 (Base‘𝑀)) → 𝑉 ⊆ (𝑀 LinCo 𝑉))

Theoremlspsslco 42736 Lemma for lspeqlco 42738. (Contributed by AV, 17-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → ((LSpan‘𝑀)‘𝑉) ⊆ (𝑀 LinCo 𝑉))

Theoremlcosslsp 42737 Lemma for lspeqlco 42738. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) ⊆ ((LSpan‘𝑀)‘𝑉))

Theoremlspeqlco 42738 Equivalence of a span of a set of vectors of a left module defined as the intersection of all linear subspaces which each contain every vector in that set ( see df-lsp 19174) and as the set of all linear combinations of the vectors of the set with finite support. (Contributed by AV, 20-Apr-2019.)
𝐵 = (Base‘𝑀)       ((𝑀 ∈ LMod ∧ 𝑉 ∈ 𝒫 𝐵) → (𝑀 LinCo 𝑉) = ((LSpan‘𝑀)‘𝑉))

20.35.16.3  Linear independency

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over [the ring] A) if whenever we have a linear combination ∑x ∈S axx which is equal to 0, then ax=0 for all x∈S.". This definition does not care for the finiteness of the set S (because the definition of a linear combination in [Lang] p.129 does already assure that only a finite number of coefficients can be 0 in the sum). Our definition df-lininds 42741 does also neither claim that the subset must be finite, nor that almost all coefficients within the linear combination are 0. If this is required, it must be explicitly stated as precondition in the corresponding theorems.

Usually, the linear independency is defined for vector spaces, see Wikipedia ("Linear independence", 15-Apr-2019, https://en.wikipedia.org/wiki/Linear_independence): "In the theory of vector spaces, a set of vectors is said to be linearly dependent if at least one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be linearly independent.". Furthermore, "In order to allow the number of linearly independent vectors in a vector space to be countably infinite, it is useful to define linear dependence as follows. More generally, let V be a vector space over a field K, and let {vi | i∈I} be a family of elements of V. The family is linearly dependent over K if there exists a finite family {aj | j∈J} of elements of K, all nonzero, such that ∑j∈J ajvj=0. A set X of elements of V is linearly independent if the corresponding family{x}x∈X is linearly independent".
Remark 1: There are already definitions of (linearly) independent families (df-lindf 20347) and (linearly) independent sets (df-linds 20348). These definitions are based on the principle "of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements" or (see lbsind2 19283) "every element is not in the span of the remainder of the [set]". The equivalence of the definitions df-linds 20348 and df-lininds 42741 for (linear) independency for (left) modules is shown in lindslininds 42763.
Remark 2: Subsets of the base set of a (left) module are linearly dependent if they are not linearly indepent (see df-lindeps 42743) or, according to Wikipedia, "if at least one of the vectors in the set can be defined as a linear combination of the others", see islindeps2 42782. The reversed implication is not valid for arbitrary modules (but for arbitrary vector spaces), because it requires a division by a coefficient. Therefore, the definition of Wikipedia is equivalent with our definition for (left) vector spaces (see isldepslvec2 42784) and not for (left) modules in general.

Syntaxclininds 42739 Extend class notation with the relation between a module and its linearly independent subsets.
class linIndS

Syntaxclindeps 42740 Extend class notation with the relation between a module and its linearly dependent subsets.
class linDepS

Definitiondf-lininds 42741* Define the relation between a module and its linearly independent subsets. (Contributed by AV, 12-Apr-2019.) (Revised by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑𝑚 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g𝑚)) → ∀𝑥𝑠 (𝑓𝑥) = (0g‘(Scalar‘𝑚))))}

Theoremrellininds 42742 The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)
Rel linIndS

Definitiondf-lindeps 42743* Define the relation between a module and its linearly dependent subsets. (Contributed by AV, 26-Apr-2019.)
linDepS = {⟨𝑠, 𝑚⟩ ∣ ¬ 𝑠 linIndS 𝑚}

Theoremlinindsv 42744 The classes of the module and its linearly independent subsets are sets. (Contributed by AV, 13-Apr-2019.)
(𝑆 linIndS 𝑀 → (𝑆 ∈ V ∧ 𝑀 ∈ V))

Theoremislininds 42745* The property of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆𝑉𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))

Theoremlinindsi 42746* The implications of being a linearly independent subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       (𝑆 linIndS 𝑀 → (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )))

Theoremlinindslinci 42747* The implications of being a linearly independent subset and a linear combination of this subset being 0. (Contributed by AV, 24-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆 linIndS 𝑀 ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ 𝐹 finSupp 0 ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍)) → ∀𝑥𝑆 (𝐹𝑥) = 0 )

Theoremislinindfis 42748* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑆 ∈ Fin ∧ 𝑀𝑊) → (𝑆 linIndS 𝑀 ↔ (𝑆 ∈ 𝒫 𝐵 ∧ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))

Theoremislinindfiss 42749* The property of being a linearly independent finite subset. (Contributed by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀𝑊𝑆 ∈ Fin ∧ 𝑆 ∈ 𝒫 𝐵) → (𝑆 linIndS 𝑀 ↔ ∀𝑓 ∈ (𝐸𝑚 𝑆)((𝑓( linC ‘𝑀)𝑆) = 𝑍 → ∀𝑥𝑆 (𝑓𝑥) = 0 )))

Theoremlinindscl 42750 A linearly independent set is a subset of (the base set of) a module. (Contributed by AV, 13-Apr-2019.)
(𝑆 linIndS 𝑀𝑆 ∈ 𝒫 (Base‘𝑀))

Theoremlindepsnlininds 42751 A linearly dependent subset is not a linearly independent subset. (Contributed by AV, 26-Apr-2019.)
((𝑆𝑉𝑀𝑊) → (𝑆 linDepS 𝑀 ↔ ¬ 𝑆 linIndS 𝑀))

Theoremislindeps 42752* The property of being a linearly dependent subset. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀𝑊𝑆 ∈ 𝒫 𝐵) → (𝑆 linDepS 𝑀 ↔ ∃𝑓 ∈ (𝐸𝑚 𝑆)(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍 ∧ ∃𝑥𝑆 (𝑓𝑥) ≠ 0 )))

Theoremlincext1 42753* Property 1 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 29-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))) → 𝐹 ∈ (𝐸𝑚 𝑆))

Theoremlincext2 42754* Property 2 of an extension of a linear combination. (Contributed by AV, 20-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋}))) ∧ 𝐺 finSupp 0 ) → 𝐹 finSupp 0 )

Theoremlincext3 42755* Property 3 of an extension of a linear combination. (Contributed by AV, 23-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐹 = (𝑧𝑆 ↦ if(𝑧 = 𝑋, (𝑁𝑌), (𝐺𝑧)))       (((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵) ∧ (𝑌𝐸𝑋𝑆𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋}))) ∧ (𝐺 finSupp 0 ∧ (𝑌( ·𝑠𝑀)𝑋) = (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})))) → (𝐹( linC ‘𝑀)𝑆) = 𝑍)

Theoremlindslinindsimp1 42756* Implication 1 for lindslininds 42763. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 )) → (𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠})))))

Theoremlindslinindimp2lem1 42757* Lemma 1 for lindslinindsimp2 42762. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝑌𝐵)

Theoremlindslinindimp2lem2 42758* Lemma 2 for lindslinindsimp2 42762. (Contributed by AV, 25-Apr-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆𝑓 ∈ (𝐵𝑚 𝑆))) → 𝐺 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥})))

Theoremlindslinindimp2lem3 42759* Lemma 3 for lindslinindsimp2 42762. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 )) → 𝐺 finSupp 0 )

Theoremlindslinindimp2lem4 42760* Lemma 4 for lindslinindsimp2 42762. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑌 = ((invg𝑅)‘(𝑓𝑥))    &   𝐺 = (𝑓 ↾ (𝑆 ∖ {𝑥}))       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆) ∧ (𝑓 ∈ (𝐵𝑚 𝑆) ∧ 𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (𝑀 Σg (𝑦 ∈ (𝑆 ∖ {𝑥}) ↦ ((𝑓𝑦)( ·𝑠𝑀)𝑦))) = (𝑌( ·𝑠𝑀)𝑥))

Theoremlindslinindsimp2lem5 42761* Lemma 5 for lindslinindsimp2 42762. (Contributed by AV, 25-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       (((𝑆𝑉𝑀 ∈ LMod) ∧ (𝑆 ⊆ (Base‘𝑀) ∧ 𝑥𝑆)) → ((𝑓 ∈ (𝐵𝑚 𝑆) ∧ (𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍)) → (∀𝑦 ∈ (𝐵 ∖ { 0 })∀𝑔 ∈ (𝐵𝑚 (𝑆 ∖ {𝑥}))(¬ 𝑔 finSupp 0 ∨ ¬ (𝑦( ·𝑠𝑀)𝑥) = (𝑔( linC ‘𝑀)(𝑆 ∖ {𝑥}))) → (𝑓𝑥) = 0 )))

Theoremlindslinindsimp2 42762* Implication 2 for lindslininds 42763. (Contributed by AV, 26-Apr-2019.) (Revised by AV, 30-Jul-2019.)
𝑅 = (Scalar‘𝑀)    &   𝐵 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)       ((𝑆𝑉𝑀 ∈ LMod) → ((𝑆 ⊆ (Base‘𝑀) ∧ ∀𝑠𝑆𝑦 ∈ (𝐵 ∖ { 0 }) ¬ (𝑦( ·𝑠𝑀)𝑠) ∈ ((LSpan‘𝑀)‘(𝑆 ∖ {𝑠}))) → (𝑆 ∈ 𝒫 (Base‘𝑀) ∧ ∀𝑓 ∈ (𝐵𝑚 𝑆)((𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)𝑆) = 𝑍) → ∀𝑥𝑆 (𝑓𝑥) = 0 ))))

Theoremlindslininds 42763 Equivalence of definitions df-linds 20348 and df-lininds 42741 for (linear) independency for (left) modules. (Contributed by AV, 26-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
((𝑆𝑉𝑀 ∈ LMod) → (𝑆 linIndS 𝑀𝑆 ∈ (LIndS‘𝑀)))

Theoremlinds0 42764 The empty set is always a linearly independet subset. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
(𝑀𝑉 → ∅ linIndS 𝑀)

Theoremel0ldep 42765 A set containing the zero element of a module is always linearly dependent, if the underlying ring has at least two elements. (Contributed by AV, 13-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
(((𝑀 ∈ LMod ∧ 1 < (♯‘(Base‘(Scalar‘𝑀)))) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Theoremel0ldepsnzr 42766 A set containing the zero element of a module over a nonzero ring is always linearly dependent. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
(((𝑀 ∈ LMod ∧ (Scalar‘𝑀) ∈ NzRing) ∧ 𝑆 ∈ 𝒫 (Base‘𝑀) ∧ (0g𝑀) ∈ 𝑆) → 𝑆 linDepS 𝑀)

Theoremlindsrng01 42767 Any subset of a module is always linearly independent if the underlying ring has at most one element. Since the underlying ring cannot be the empty set (see lmodsn0 19078), this means that the underlying ring has only one element, so it is a zero ring. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)       ((𝑀 ∈ LMod ∧ ((♯‘𝐸) = 0 ∨ (♯‘𝐸) = 1) ∧ 𝑆 ∈ 𝒫 𝐵) → 𝑆 linIndS 𝑀)

Theoremlindszr 42768 Any subset of a module over a zero ring is always linearly independent. (Contributed by AV, 27-Apr-2019.)
((𝑀 ∈ LMod ∧ ¬ (Scalar‘𝑀) ∈ NzRing ∧ 𝑆 ∈ 𝒫 (Base‘𝑀)) → 𝑆 linIndS 𝑀)

Theoremsnlindsntorlem 42769* Lemma for snlindsntor 42770. (Contributed by AV, 15-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑓 ∈ (𝑆𝑚 {𝑋})((𝑓( linC ‘𝑀){𝑋}) = 𝑍 → (𝑓𝑋) = 0 ) → ∀𝑠𝑆 ((𝑠 · 𝑋) = 𝑍𝑠 = 0 )))

Theoremsnlindsntor 42770* A singleton is linearly independent iff it does not contain a torsion element. According to Wikipedia ("Torsion (algebra)", 15-Apr-2019, https://en.wikipedia.org/wiki/Torsion_(algebra)): "An element m of a module M over a ring R is called a torsion element of the module if there exists a regular element r of the ring (an element that is neither a left nor a right zero divisor) that annihilates m, i.e., (𝑟 · 𝑚) = 0. In an integral domain (a commutative ring without zero divisors), every nonzero element is regular, so a torsion element of a module over an integral domain is one annihilated by a nonzero element of the integral domain." Analogously, the definition in [Lang] p. 147 states that "An element x of [a module] E [over a ring R] is called a torsion element if there exists 𝑎𝑅, 𝑎 ≠ 0, such that 𝑎 · 𝑥 = 0. This definition includes the zero element of the module. Some authors, however, exclude the zero element from the definition of torsion elements. (Contributed by AV, 14-Apr-2019.) (Revised by AV, 27-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ 𝑋𝐵) → (∀𝑠 ∈ (𝑆 ∖ { 0 })(𝑠 · 𝑋) ≠ 𝑍 ↔ {𝑋} linIndS 𝑀))

Theoremldepsprlem 42771 Lemma for ldepspr 42772. (Contributed by AV, 16-Apr-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)    &    1 = (1r𝑅)    &   𝑁 = (invg𝑅)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝐴𝑆)) → (𝑋 = (𝐴 · 𝑌) → (( 1 · 𝑋)(+g𝑀)((𝑁𝐴) · 𝑌)) = 𝑍))

Theoremldepspr 42772 If a vector is a scalar multiple of another vector, the (unordered pair containing the) two vectors are linearly dependent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 27-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝑆 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &    · = ( ·𝑠𝑀)       ((𝑀 ∈ LMod ∧ (𝑋𝐵𝑌𝐵𝑋𝑌)) → ((𝐴𝑆𝑋 = (𝐴 · 𝑌)) → {𝑋, 𝑌} linDepS 𝑀))

Theoremlincresunit3lem3 42773 Lemma 3 for lincresunit3 42780. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝑁 = (invg𝑅)    &    · = ( ·𝑠𝑀)       (((𝑀 ∈ LMod ∧ 𝑋𝐵𝑌𝐵) ∧ 𝐴𝑈) → (((𝑁𝐴) · 𝑋) = ((𝑁𝐴) · 𝑌) ↔ 𝑋 = 𝑌))

Theoremlincresunitlem1 42774 Lemma 1 for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → (𝐼‘(𝑁‘(𝐹𝑋))) ∈ 𝐸)

Theoremlincresunitlem2 42775 Lemma for properties of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       ((((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) ∧ 𝑌𝑆) → ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑌)) ∈ 𝐸)

Theoremlincresunit1 42776* Property 1 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈)) → 𝐺 ∈ (𝐸𝑚 (𝑆 ∖ {𝑋})))

Theoremlincresunit2 42777* Property 2 of a specially modified restriction of a linear combination containing a unit as scalar. (Contributed by AV, 18-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → 𝐺 finSupp 0 )

Theoremlincresunit3lem1 42778* Lemma 1 for lincresunit3 42780. (Contributed by AV, 17-May-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝑧 ∈ (𝑆 ∖ {𝑋}))) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)((𝐺𝑧)( ·𝑠𝑀)𝑧)) = ((𝐹𝑧)( ·𝑠𝑀)𝑧))

Theoremlincresunit3lem2 42779* Lemma 2 for lincresunit3 42780. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 )) → ((𝑁‘(𝐹𝑋))( ·𝑠𝑀)(𝑀 Σg (𝑧 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐺𝑧)( ·𝑠𝑀)𝑧)))) = ((𝐹 ↾ (𝑆 ∖ {𝑋}))( linC ‘𝑀)(𝑆 ∖ {𝑋})))

Theoremlincresunit3 42780* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LMod ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ∈ 𝑈𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)

Theoremlincreslvec3 42781* Property 3 of a specially modified restriction of a linear combination in a vector space. (Contributed by AV, 18-May-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    0 = (0g𝑅)    &   𝑍 = (0g𝑀)    &   𝑁 = (invg𝑅)    &   𝐼 = (invr𝑅)    &    · = (.r𝑅)    &   𝐺 = (𝑠 ∈ (𝑆 ∖ {𝑋}) ↦ ((𝐼‘(𝑁‘(𝐹𝑋))) · (𝐹𝑠)))       (((𝑆 ∈ 𝒫 𝐵𝑀 ∈ LVec ∧ 𝑋𝑆) ∧ (𝐹 ∈ (𝐸𝑚 𝑆) ∧ (𝐹𝑋) ≠ 0𝐹 finSupp 0 ) ∧ (𝐹( linC ‘𝑀)𝑆) = 𝑍) → (𝐺( linC ‘𝑀)(𝑆 ∖ {𝑋})) = 𝑋)

Theoremislindeps2 42782* Conditions for being a linearly dependent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (∃𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) → 𝑆 linDepS 𝑀))

Theoremislininds2 42783* Implication of being a linearly independent subset of a (left) module over a nonzero ring. (Contributed by AV, 29-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LMod ∧ 𝑆 ∈ 𝒫 𝐵𝑅 ∈ NzRing) → (𝑆 linIndS 𝑀 → ∀𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(¬ 𝑓 finSupp 0 ∨ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) ≠ 𝑠)))

Theoremisldepslvec2 42784* Alternative definition of being a linearly dependent subset of a (left) vector space. In this case, the reverse implication of islindeps2 42782 holds, so that both definitions are equivalent (see theorem 1.6 in [Roman] p. 46 and the note in [Roman] p. 112: if a nontrivial linear combination of elements (where not all of the coefficients are 0) in an R-vector space is 0, then and only then each of the elements is a linear combination of the others. (Contributed by AV, 30-Apr-2019.) (Proof shortened by AV, 30-Jul-2019.)
𝐵 = (Base‘𝑀)    &   𝑍 = (0g𝑀)    &   𝑅 = (Scalar‘𝑀)    &   𝐸 = (Base‘𝑅)    &    0 = (0g𝑅)       ((𝑀 ∈ LVec ∧ 𝑆 ∈ 𝒫 𝐵) → (∃𝑠𝑆𝑓 ∈ (𝐸𝑚 (𝑆 ∖ {𝑠}))(𝑓 finSupp 0 ∧ (𝑓( linC ‘𝑀)(𝑆 ∖ {𝑠})) = 𝑠) ↔ 𝑆 linDepS 𝑀))

Theoremlindssnlvec 42785 A singleton not containing the zero element of a vector space is always linearly independent. (Contributed by AV, 16-Apr-2019.) (Revised by AV, 28-Apr-2019.)
((𝑀 ∈ LVec ∧ 𝑆 ∈ (Base‘𝑀) ∧ 𝑆 ≠ (0g𝑀)) → {𝑆} linIndS 𝑀)

20.35.16.4  Simple left modules and the ` ZZ `-module

Theoremlmod1lem1 42786* Lemma 1 for lmod1 42791. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)𝐼) ∈ {𝐼})

Theoremlmod1lem2 42787* Lemma 2 for lmod1 42791. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring ∧ 𝑟 ∈ (Base‘𝑅)) → (𝑟( ·𝑠𝑀)(𝐼(+g𝑀)𝐼)) = ((𝑟( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem3 42788* Lemma 3 for lmod1 42791. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(+g‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = ((𝑞( ·𝑠𝑀)𝐼)(+g𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem4 42789* Lemma 4 for lmod1 42791. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       (((𝐼𝑉𝑅 ∈ Ring) ∧ (𝑞 ∈ (Base‘𝑅) ∧ 𝑟 ∈ (Base‘𝑅))) → ((𝑞(.r‘(Scalar‘𝑀))𝑟)( ·𝑠𝑀)𝐼) = (𝑞( ·𝑠𝑀)(𝑟( ·𝑠𝑀)𝐼)))

Theoremlmod1lem5 42790* Lemma 5 for lmod1 42791. (Contributed by AV, 28-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → ((1r‘(Scalar‘𝑀))( ·𝑠𝑀)𝐼) = 𝐼)

Theoremlmod1 42791* The (smallest) structure representing a zero module over an arbitrary ring. (Contributed by AV, 29-Apr-2019.)
𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), (𝑥 ∈ (Base‘𝑅), 𝑦 ∈ {𝐼} ↦ 𝑦)⟩})       ((𝐼𝑉𝑅 ∈ Ring) → 𝑀 ∈ LMod)

Theoremlmod1zr 42792 The (smallest) structure representing a zero module over a zero ring. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∈ LMod)

Theoremlmod1zrnlvec 42793 There is a (left) module (a zero module) which is not a (left) vector space. (Contributed by AV, 29-Apr-2019.)
𝑅 = {⟨(Base‘ndx), {𝑍}⟩, ⟨(+g‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩, ⟨(.r‘ndx), {⟨⟨𝑍, 𝑍⟩, 𝑍⟩}⟩}    &   𝑀 = ({⟨(Base‘ndx), {𝐼}⟩, ⟨(+g‘ndx), {⟨⟨𝐼, 𝐼⟩, 𝐼⟩}⟩, ⟨(Scalar‘ndx), 𝑅⟩} ∪ {⟨( ·𝑠 ‘ndx), {⟨⟨𝑍, 𝐼⟩, 𝐼⟩}⟩})       ((𝐼𝑉𝑍𝑊) → 𝑀 ∉ LVec)

Theoremlmodn0 42794 Left modules exist. (Contributed by AV, 29-Apr-2019.)
LMod ≠ ∅

Theoremzlmodzxzequa 42795 Example of an equation within the -module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set). (Contributed by AV, 22-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    = ( ·𝑠𝑍)    &    = (-g𝑍)    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       ((2 𝐴) (3 𝐵)) = 0

Theoremzlmodzxznm 42796 Example of a linearly dependent set whose elements are not linear combinations of the others, see note in [Roman] p. 112). (Contributed by AV, 23-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    = ( ·𝑠𝑍)    &    = (-g𝑍)    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       𝑖 ∈ ℤ ((𝑖 𝐴) ≠ 𝐵 ∧ (𝑖 𝐵) ≠ 𝐴)

Theoremzlmodzxzldeplem 42797 A and B are not equal. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}       𝐴𝐵

Theoremzlmodzxzequap 42798 Example of an equation within the -module ℤ × ℤ (see example in [Roman] p. 112 for a linearly dependent set), written as a sum. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &    0 = {⟨0, 0⟩, ⟨1, 0⟩}    &    + = (+g𝑍)    &    = ( ·𝑠𝑍)       ((2 𝐴) + (-3 𝐵)) = 0

Theoremzlmodzxzldeplem1 42799 Lemma 1 for zlmodzxzldep 42803. (Contributed by AV, 24-May-2019.) (Revised by AV, 10-Jun-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       𝐹 ∈ (ℤ ↑𝑚 {𝐴, 𝐵})

Theoremzlmodzxzldeplem2 42800 Lemma 2 for zlmodzxzldep 42803. (Contributed by AV, 24-May-2019.) (Revised by AV, 30-Jul-2019.)
𝑍 = (ℤring freeLMod {0, 1})    &   𝐴 = {⟨0, 3⟩, ⟨1, 6⟩}    &   𝐵 = {⟨0, 2⟩, ⟨1, 4⟩}    &   𝐹 = {⟨𝐴, 2⟩, ⟨𝐵, -3⟩}       𝐹 finSupp 0

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