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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempjfval2 20101* Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))
 
Theorempjval 20102 Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))
 
Theorempjdm2 20103 A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &    = (LSSum‘𝑊)    &   𝐾 = (proj‘𝑊)       (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇 ( 𝑇)) = 𝑉)))
 
Theorempjff 20104 A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)       (𝑊 ∈ PreHil → 𝐾:dom 𝐾⟶(𝑊 LMHom 𝑊))
 
Theorempjf 20105 A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       (𝑇 ∈ dom 𝐾 → (𝐾𝑇):𝑉𝑉)
 
Theorempjf2 20106 A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾𝑇):𝑉𝑇)
 
Theorempjfo 20107 A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾𝑇):𝑉onto𝑇)
 
Theorempjcss 20108 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → dom 𝐾𝐶)
 
Theoremocvpj 20109 The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( 𝑇) ∈ dom 𝐾)
 
Theoremishil 20110 The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝐾 = (proj‘𝐻)    &   𝐶 = (CSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
 
Theoremishil2 20111* The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
𝑉 = (Base‘𝐻)    &    = (LSSum‘𝐻)    &    = (ocv‘𝐻)    &   𝐶 = (CSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠𝐶 (𝑠 ( 𝑠)) = 𝑉))
 
Theoremisobs 20112* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)    &   𝑌 = (0g𝑊)       (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))
 
Theoremobsip 20113 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
 
Theoremobsipid 20114 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
, = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = 1 )
 
Theoremobsrcl 20115 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
(𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
 
Theoremobsss 20116 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)
 
Theoremobsne0 20117 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → 𝐴0 )
 
Theoremobsocv 20118 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)    &    = (ocv‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })
 
Theoremobs2ocv 20119 The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( ‘( 𝐵)) = 𝑉)
 
Theoremobselocv 20120 A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶𝐵𝐴𝐵) → (𝐴 ∈ ( 𝐶) ↔ ¬ 𝐴𝐶))
 
Theoremobs2ss 20121 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶𝐵) → 𝐶 = 𝐵)
 
Theoremobslbs 20122 An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
𝐽 = (LBasis‘𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → (𝐵𝐽 ↔ (𝑁𝐵) ∈ 𝐶))
 
PART 11  BASIC LINEAR ALGEBRA

According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." However, dealing with modules (over rings) instead of vector spaces (over fields) allows for a more general approach. Therefore, "vectors" are regarded as members (elements of the base set) of a (free) module over a ring (see df-frlm 20139) in the following. By this, the number of entries in a vector is determined by the size of the index set of the direct sum building the free module the vector is belonging to. Since every vector space is isomorphic to a free module (see lvecisfrlm 20230), the theorems stated for free modules are also valid for vector spaces.

Until not explicitly stated, the underlying ring needs not to be commutative (see df-cring 18596), but the existence of a multiplicative neutral element is always presumed (the ring is a unital ring, see also df-ring 18595). In this sense, linear equations, matrices and determinants are usually regarded as "over a ring" in this part.

 
11.1  Vectors and free modules
 
11.1.1  Direct sum of left modules

According to Wikipedia ("Direct sum of modules", 28-Mar-2019, https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and { Mi: i ∈ I } a family of left R-modules indexed by the set I. The direct sum of {Mi} is then defined to be the set of all sequences (αi) where αi ∈ Mi and αi = 0 for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.)". In this definition, "cofinitely many" means "almost all" or "for all but finitely many". Furthemore, "This set inherits the module structure via componentwise addition and scalar multiplication. Explicitly, two such sequences α and β can be added by writing (α + β)i = αi + βi for all i (note that this is again zero for all but finitely many indices), and such a sequence can be multiplied with an element r from R by defining r(α)i = (rα)i for all i.".
In [Lang] p. 128, the definition of the direct sum of left modules is based on direct sums of abelian groups ("We define on [the direct sum of abelian groups Mi] M a structure of A-module: If (xi)i ∈ I is an element of M, i.e. a familiy of elements xi ∈ Mi such that xi = 0 for almost all i, and if a ∈ A, then we define a(xi)i ∈ I = (axi)i ∈ I, that is we define multiplication by a componentwise.") which itself is based on the direct product of abelian groups ([Lang] p. 36: "Let {Ai}i ∈ I be a family of abelian groups. We define their direct sum A ... to be the subset of the direct product ... consisting of all families (xi)i ∈ I with xi ∈ Ai such that xi = 0 for all but a finite number of indices i").
In short, the direct sum of a familiy of (left) modules {Mi}i ∈ I is the restriction of the direct product of {Mi}i ∈ I to the elements with index function having finite support, as formalized by the definition df-dsmm 20124.

 
Syntaxcdsmm 20123 Class of module direct sum generator.
class m
 
Definitiondf-dsmm 20124* The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))
 
Theoremreldmdsmm 20125 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m
 
Theoremdsmmval 20126* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉 → (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵))
 
Theoremdsmmbase 20127* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉𝐵 = (Base‘(𝑆m 𝑅)))
 
Theoremdsmmval2 20128 Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐵 = (Base‘(𝑆m 𝑅))       (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)
 
Theoremdsmmbas2 20129* Base set of the direct sum module using the fndmin 6364 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}       ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 𝑅)))
 
Theoremdsmmfi 20130 For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.)
((𝑅 Fn 𝐼𝐼 ∈ Fin) → (𝑆m 𝑅) = (𝑆Xs𝑅))
 
Theoremdsmmelbas 20131* Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐶 = (𝑆m 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐻 = (Base‘𝐶)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → (𝑋𝐻 ↔ (𝑋𝐵 ∧ {𝑎𝐼 ∣ (𝑋𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))
 
Theoremdsmm0cl 20132 The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑃)       (𝜑0𝐻)
 
Theoremdsmmacl 20133 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐽𝐻)    &   (𝜑𝐾𝐻)    &    + = (+g𝑃)       (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)
 
Theoremprdsinvgd2 20134 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝑁𝑋)‘𝐽) = ((invg‘(𝑅𝐽))‘(𝑋𝐽)))
 
Theoremdsmmsubg 20135 The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝐻 ∈ (SubGrp‘𝑃))
 
Theoremdsmmlss 20136* The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝑃 = (𝑆Xs𝑅)    &   𝑈 = (LSubSp‘𝑃)    &   𝐻 = (Base‘(𝑆m 𝑅))       (𝜑𝐻𝑈)
 
Theoremdsmmlmod 20137* The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝐶 = (𝑆m 𝑅)       (𝜑𝐶 ∈ LMod)
 
11.1.2  Free modules

According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules.". The same definition is used in [Lang] p. 135: "By a free module we shall mean a module which admits a basis, or the zero module.".

In the following, however, a free module is defined as direct sum of a family consisting of the same ring regarded as a (left) module over itself, see df-frlm 20139. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20139 (see lmisfree 20229), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20139 are also taken for the motivation of free modules by [Lang] p. 135.

 
Syntaxcfrlm 20138 Class of free module generator.
class freeLMod
 
Definitiondf-frlm 20139* The 𝑖-dimensional free module over a ring 𝑟 is the product of 𝑖-many copies of the ring with componentwise addition and multiplication. If 𝑖 is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))
 
Theoremfrlmval 20140 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
 
Theoremfrlmlmod 20141 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ LMod)
 
Theoremfrlmpws 20142 The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))
 
Theoremfrlmlss 20143 The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐵𝑈)
 
Theoremfrlmpwsfi 20144 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼))
 
Theoremfrlmsca 20145 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝑅 = (Scalar‘𝐹))
 
Theoremfrlm0 20146 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20143). (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → (𝐼 × { 0 }) = (0g𝐹))
 
Theoremfrlmbas 20147* Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐵 = {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 }       ((𝑅𝑉𝐼𝑊) → 𝐵 = (Base‘𝐹))
 
Theoremfrlmelbas 20148 Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼𝑊) → (𝑋𝐵 ↔ (𝑋 ∈ (𝑁𝑚 𝐼) ∧ 𝑋 finSupp 0 )))
 
Theoremfrlmrcl 20149 If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)       (𝑋𝐵𝑅 ∈ V)
 
Theoremfrlmbasfsupp 20150 Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 finSupp 0 )
 
Theoremfrlmbasmap 20151 Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 ∈ (𝑁𝑚 𝐼))
 
Theoremfrlmbasf 20152 Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋:𝐼𝑁)
 
Theoremfrlmfibas 20153 The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑁𝑚 𝐼) = (Base‘𝐹))
 
Theoremelfrlmbasn0 20154 If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑁 = (Base‘𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑉𝐼 ≠ ∅) → (𝑋𝐵𝑋 ≠ ∅))
 
Theoremfrlmplusgval 20155 Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    + = (+g𝑅)    &    = (+g𝑌)       (𝜑 → (𝐹 𝐺) = (𝐹𝑓 + 𝐺))
 
Theoremfrlmsubgval 20156 Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐹𝐵)    &   (𝜑𝐺𝐵)    &    = (-g𝑅)    &   𝑀 = (-g𝑌)       (𝜑 → (𝐹𝑀𝐺) = (𝐹𝑓 𝐺))
 
Theoremfrlmvscafval 20157 Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → (𝐴 𝑋) = ((𝐼 × {𝐴}) ∘𝑓 · 𝑋))
 
Theoremfrlmvscaval 20158 Scalar multiplication in a free module at a coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &   𝐾 = (Base‘𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝐴𝐾)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)    &    = ( ·𝑠𝑌)    &    · = (.r𝑅)       (𝜑 → ((𝐴 𝑋)‘𝐽) = (𝐴 · (𝑋𝐽)))
 
Theoremfrlmgsum 20159* Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑌)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑥𝐼𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑥𝐼𝑈)) finSupp 0 )       (𝜑 → (𝑌 Σg (𝑦𝐽 ↦ (𝑥𝐼𝑈))) = (𝑥𝐼 ↦ (𝑅 Σg (𝑦𝐽𝑈))))
 
Theoremfrlmsplit2 20160* Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝑈)    &   𝑍 = (𝑅 freeLMod 𝑉)    &   𝐵 = (Base‘𝑌)    &   𝐶 = (Base‘𝑍)    &   𝐹 = (𝑥𝐵 ↦ (𝑥𝑉))       ((𝑅 ∈ Ring ∧ 𝑈𝑋𝑉𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
 
Theoremfrlmsslss 20161* A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝑈 = (LSubSp‘𝑌)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥𝐽) = (𝐽 × { 0 })}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)
 
Theoremfrlmsslss2 20162* A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝑈 = (LSubSp‘𝑌)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → 𝐶𝑈)
 
Theoremfrlmbas3 20163 An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.)
𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   𝐵 = (Base‘𝑅)    &   𝑉 = (Base‘𝐹)       (((𝑅𝑊𝑋𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼𝑁𝐽𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵)
 
Theoremmpt2frlmd 20164* Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.)
𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   𝑉 = (Base‘𝐹)    &   ((𝑖 = 𝑎𝑗 = 𝑏) → 𝐴 = 𝐵)    &   ((𝜑𝑖𝑁𝑗𝑀) → 𝐴𝑋)    &   ((𝜑𝑎𝑁𝑏𝑀) → 𝐵𝑌)    &   (𝜑 → (𝑁𝑈𝑀𝑊𝑍𝑉))       (𝜑 → (𝑍 = (𝑎𝑁, 𝑏𝑀𝐵) ↔ ∀𝑖𝑁𝑗𝑀 (𝑖𝑍𝑗) = 𝐴))
 
Theoremfrlmip 20165* The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝐼𝑊𝑅𝑉) → (𝑓 ∈ (𝐵𝑚 𝐼), 𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑅 Σg (𝑥𝐼 ↦ ((𝑓𝑥) · (𝑔𝑥))))) = (·𝑖𝑌))
 
Theoremfrlmipval 20166 The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)       (((𝐼𝑊𝑅𝑋) ∧ (𝐹𝑉𝐺𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹𝑓 · 𝐺)))
 
Theoremfrlmphllem 20167* Lemma for frlmphl 20168. (Contributed by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       ((𝜑𝑔𝑉𝑉) → (𝑥𝐼 ↦ ((𝑔𝑥) · (𝑥))) finSupp 0 )
 
Theoremfrlmphl 20168* Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝑉 = (Base‘𝑌)    &    , = (·𝑖𝑌)    &   𝑂 = (0g𝑌)    &    0 = (0g𝑅)    &    = (*𝑟𝑅)    &   (𝜑𝑅 ∈ Field)    &   ((𝜑𝑔𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ((𝜑𝑥𝐵) → ( 𝑥) = 𝑥)    &   (𝜑𝐼𝑊)       (𝜑𝑌 ∈ PreHil)
 
11.1.3  Standard basis (unit vectors)

According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces.

 
Syntaxcuvc 20169 Class of basic unit vectors for an explicit free module.
class unitVec
 
Definitiondf-uvc 20170* ((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗𝑖 ↦ (𝑘𝑖 ↦ if(𝑘 = 𝑗, (1r𝑟), (0g𝑟)))))
 
Theoremuvcfval 20171* Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅𝑉𝐼𝑊) → 𝑈 = (𝑗𝐼 ↦ (𝑘𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
 
Theoremuvcval 20172* Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑅𝑉𝐼𝑊𝐽𝐼) → (𝑈𝐽) = (𝑘𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
 
Theoremuvcvval 20173 Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
 
Theoremuvcvvcl 20174 A coodinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       (((𝑅𝑉𝐼𝑊𝐽𝐼) ∧ 𝐾𝐼) → ((𝑈𝐽)‘𝐾) ∈ { 0 , 1 })
 
Theoremuvcvvcl2 20175 A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐾𝐼)       (𝜑 → ((𝑈𝐽)‘𝐾) ∈ 𝐵)
 
Theoremuvcvv1 20176 The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &    1 = (1r𝑅)       (𝜑 → ((𝑈𝐽)‘𝐽) = 1 )
 
Theoremuvcvv0 20177 The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   (𝜑𝑅𝑉)    &   (𝜑𝐼𝑊)    &   (𝜑𝐽𝐼)    &   (𝜑𝐾𝐼)    &   (𝜑𝐽𝐾)    &    0 = (0g𝑅)       (𝜑 → ((𝑈𝐽)‘𝐾) = 0 )
 
Theoremuvcff 20178 Domain and range of the unit vector generator; ring condition required to be sure 1 and 0 are actually in the ring. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝑈:𝐼𝐵)
 
Theoremuvcf1 20179 In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)       ((𝑅 ∈ NzRing ∧ 𝐼𝑊) → 𝑈:𝐼1-1𝐵)
 
Theoremuvcresum 20180 Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
𝑈 = (𝑅 unitVec 𝐼)    &   𝑌 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝑌)    &    · = ( ·𝑠𝑌)       ((𝑅 ∈ Ring ∧ 𝐼𝑊𝑋𝐵) → 𝑋 = (𝑌 Σg (𝑋𝑓 · 𝑈)))
 
Theoremfrlmssuvc1 20181* A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐹)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐿𝐽)    &   (𝜑𝑋𝐾)       (𝜑 → (𝑋 · (𝑈𝐿)) ∈ 𝐶)
 
Theoremfrlmssuvc2 20182* A nonzero scalar multiple of a unit vector not included in a support-restriction subspace is not included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐹)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑉)    &   (𝜑𝐽𝐼)    &   (𝜑𝐿 ∈ (𝐼𝐽))    &   (𝜑𝑋 ∈ (𝐾 ∖ { 0 }))       (𝜑 → ¬ (𝑋 · (𝑈𝐿)) ∈ 𝐶)
 
Theoremfrlmsslsp 20183* A subset of a free module obtained by restricting the support set is spanned by the relevant unit vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Revised by AV, 24-Jun-2019.)
𝑌 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐾 = (LSpan‘𝑌)    &   𝐵 = (Base‘𝑌)    &    0 = (0g𝑅)    &   𝐶 = {𝑥𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}       ((𝑅 ∈ Ring ∧ 𝐼𝑉𝐽𝐼) → (𝐾‘(𝑈𝐽)) = 𝐶)
 
Theoremfrlmlbs 20184 The unit vectors comprise a basis for a free module. (Contributed by Stefan O'Rear, 6-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐽 = (LBasis‘𝐹)       ((𝑅 ∈ Ring ∧ 𝐼𝑉) → ran 𝑈𝐽)
 
Theoremfrlmup1 20185* Any assignment of unit vectors to target vectors can be extended (uniquely) to a homomorphism from a free module to an arbitrary other module on the same base ring. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐶 = (Base‘𝑇)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)       (𝜑𝐸 ∈ (𝐹 LMHom 𝑇))
 
Theoremfrlmup2 20186* The evaluation map has the intended behavior on the unit vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐶 = (Base‘𝑇)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)    &   (𝜑𝑌𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)       (𝜑 → (𝐸‘(𝑈𝑌)) = (𝐴𝑌))
 
Theoremfrlmup3 20187* The range of such an evaluation map is the finite linear combinations of the target vectors and also the span of the target vectors. (Contributed by Stefan O'Rear, 6-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &   𝐵 = (Base‘𝐹)    &   𝐶 = (Base‘𝑇)    &    · = ( ·𝑠𝑇)    &   𝐸 = (𝑥𝐵 ↦ (𝑇 Σg (𝑥𝑓 · 𝐴)))    &   (𝜑𝑇 ∈ LMod)    &   (𝜑𝐼𝑋)    &   (𝜑𝑅 = (Scalar‘𝑇))    &   (𝜑𝐴:𝐼𝐶)    &   𝐾 = (LSpan‘𝑇)       (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴))
 
Theoremfrlmup4 20188* Universal property of the free module by existential uniquenes. (Contributed by Stefan O'Rear, 7-Mar-2015.)
𝑅 = (Scalar‘𝑇)    &   𝐹 = (𝑅 freeLMod 𝐼)    &   𝑈 = (𝑅 unitVec 𝐼)    &   𝐶 = (Base‘𝑇)       ((𝑇 ∈ LMod ∧ 𝐼𝑋𝐴:𝐼𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚𝑈) = 𝐴)
 
Theoremellspd 20189* The elements of the span of an indexed collection of basic vectors are those vectors which can be written as finite linear combinations of basic vectors. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by AV, 24-Jun-2019.)
𝑁 = (LSpan‘𝑀)    &   𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐼 ∈ V)       (𝜑 → (𝑋 ∈ (𝑁‘(𝐹𝐼)) ↔ ∃𝑓 ∈ (𝐾𝑚 𝐼)(𝑓 finSupp 0𝑋 = (𝑀 Σg (𝑓𝑓 · 𝐹)))))
 
Theoremelfilspd 20190* Simplified version of ellspd 20189 when the spanning set is finite: all linear combinations are then acceptable. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
𝑁 = (LSpan‘𝑀)    &   𝐵 = (Base‘𝑀)    &   𝐾 = (Base‘𝑆)    &   𝑆 = (Scalar‘𝑀)    &    0 = (0g𝑆)    &    · = ( ·𝑠𝑀)    &   (𝜑𝐹:𝐼𝐵)    &   (𝜑𝑀 ∈ LMod)    &   (𝜑𝐼 ∈ Fin)       (𝜑 → (𝑋 ∈ (𝑁‘(𝐹𝐼)) ↔ ∃𝑓 ∈ (𝐾𝑚 𝐼)𝑋 = (𝑀 Σg (𝑓𝑓 · 𝐹))))
 
11.1.4  Independent sets and families

According to the definition in [Lang] p. 129: "A subset S of a module M is said to be linearly independent (over A) if whenever we have a linear combination ∑x ∈ S axx which is equal to 0, then ax = 0 for all x ∈ S.", and according to the Definition in [Lang] p. 130: "a familiy {xi}i ∈ I of elements of M is said to be linearly independent (over A) if whenever we have a linear combination ∑i ∈ I aixi = 0, then ai = 0 for all i.". These definitions correspond to the definitions df-linds 20194 resp. df-lindf 20193, where it is claimed that a nonzero summand can be extracted ( ∑i ∈ {I \ { j } }aixi = -ajxj ) and be represented as a linear combination of the remaining elements of the family.
TODO: After introducing a definition of "linear combination", it should be shown that these definitions are actually equivalent.

 
Syntaxclindf 20191 The class relationship of independent families in a module.
class LIndF
 
Syntaxclinds 20192 The class generator of independent sets in a module.
class LIndS
 
Definitiondf-lindf 20193* An independent family is a family of vectors, no nonzero multiple of which can be expressed as a linear combination of other elements of the family. This is almost, but not quite, the same as a function into an independent set.

This is a defined concept because it matters in many cases whether independence is taken at a set or family level. For instance, a number is transcedental iff its nonzero powers are linearly independent. Is 1 transcedental? It has only one nonzero power.

We can almost define family independence as a family of unequal elements with independent range, as islindf3 20213, but taking that as primitive would lead to unpleasant corner case behavior with the zero ring.

This is equivalent to the common definition of having no nontrivial representations of zero (islindf4 20225) and only one representation for each element of the range (islindf5 20226). (Contributed by Stefan O'Rear, 24-Feb-2015.)

LIndF = {⟨𝑓, 𝑤⟩ ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]𝑥 ∈ dom 𝑓𝑘 ∈ ((Base‘𝑠) ∖ {(0g𝑠)}) ¬ (𝑘( ·𝑠𝑤)(𝑓𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))}
 
Definitiondf-linds 20194* An independent set is a set which is independent as a family. See also islinds3 20221 and islinds4 20222. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
 
Theoremrellindf 20195 The independent-family predicate is a proper relation and can be used with brrelexi 5192. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Rel LIndF
 
Theoremislinds 20196 Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       (𝑊𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊)))
 
Theoremlinds1 20197 An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       (𝑋 ∈ (LIndS‘𝑊) → 𝑋𝐵)
 
Theoremlinds2 20198 An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
(𝑋 ∈ (LIndS‘𝑊) → ( I ↾ 𝑋) LIndF 𝑊)
 
Theoremislindf 20199* Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑊𝑌𝐹𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹𝐵 ∧ ∀𝑥 ∈ dom 𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥}))))))
 
Theoremislinds2 20200* Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       (𝑊𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹𝐵 ∧ ∀𝑥𝐹𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥})))))
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