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Theorem List for Metamath Proof Explorer - 19801-19900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-dsmm 19801* 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 19802 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m
 
Theoremdsmmval 19803* 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 19804* 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 19805 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 19806* Base set of the direct sum module using the fndmin 6121 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 19807 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 19808* 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 19809 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 19810 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 19811 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 19812 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 19813* 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 19814* 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 19816. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 19816 (see lmisfree 19906), the two definitions are essentially equivalent. The free modules as defined by df-frlm 19816 are also taken for the motivation of free modules by [Lang] p. 135.

 
Syntaxcfrlm 19815 Class of free module generator.
class freeLMod
 
Definitiondf-frlm 19816* 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 19817 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))
 
Theoremfrlmlmod 19818 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ LMod)
 
Theoremfrlmpws 19819 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 19820 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 19821 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 19822 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 19823 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 19820). (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → (𝐼 × { 0 }) = (0g𝐹))
 
Theoremfrlmbas 19824* 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 19825 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 19826 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 19827 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 19828 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 19829 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 19830 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 19831 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 19832 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 19833 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 19834 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 19835 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 19836* 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 19837* 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 19838* 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 19839* 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 19840 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 19841* 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 19842* 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 19843 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 19844* Lemma for frlmphl 19845. (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 19845* 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 19846 Class of basic unit vectors for an explicit free module.
class unitVec
 
Definitiondf-uvc 19847* ((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 19848* 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 19849* 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 19850 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 19851 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 19852 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 19853 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 19854 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 19855 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 19856 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 19857 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 19858* 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 19859* 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 19860* 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 19861 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 19862* 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 19863* 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 19864* 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 19865* 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 19866* 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 19867* Simplified version of ellspd 19866 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 19871 resp. df-lindf 19870, 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 19868 The class relationship of independent families in a module.
class LIndF
 
Syntaxclinds 19869 The class generator of independent sets in a module.
class LIndS
 
Definitiondf-lindf 19870* 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 19890, 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 19902) and only one representation for each element of the range (islindf5 19903). (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 19871* An independent set is a set which is independent as a family. See also islinds3 19898 and islinds4 19899. (Contributed by Stefan O'Rear, 24-Feb-2015.)
LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤})
 
Theoremrellindf 19872 The independent-family predicate is a proper relation and can be used with brrelexi 4976. (Contributed by Stefan O'Rear, 24-Feb-2015.)
Rel LIndF
 
Theoremislinds 19873 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 19874 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 19875 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 19876* 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 19877* 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})))))
 
Theoremislindf2 19878* Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐵 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝑁 = (Base‘𝑆)    &    0 = (0g𝑆)       ((𝑊𝑌𝐼𝑋𝐹:𝐼𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥𝐼𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥})))))
 
Theoremlindff 19879 Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)       ((𝐹 LIndF 𝑊𝑊𝑌) → 𝐹:dom 𝐹𝐵)
 
Theoremlindfind 19880 A linearly independent family is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
· = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)    &    0 = (0g𝐿)    &   𝐾 = (Base‘𝐿)       (((𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · (𝐹𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
 
Theoremlindsind 19881 A linearly independent set is independent: no nonzero element multiple can be expressed as a linear combination of the others. (Contributed by Stefan O'Rear, 24-Feb-2015.)
· = ( ·𝑠𝑊)    &   𝑁 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)    &    0 = (0g𝐿)    &   𝐾 = (Base‘𝐿)       (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) ∧ (𝐴𝐾𝐴0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸})))
 
Theoremlindfind2 19882 In a linearly independent family in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐾 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)       (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 LIndF 𝑊𝐸 ∈ dom 𝐹) → ¬ (𝐹𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸}))))
 
Theoremlindsind2 19883 In a linearly independent set in a module over a nonzero ring, no element is contained in the span of any non-containing set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐾 = (LSpan‘𝑊)    &   𝐿 = (Scalar‘𝑊)       (((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸})))
 
Theoremlindff1 19884 A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐿 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing ∧ 𝐹 LIndF 𝑊) → 𝐹:dom 𝐹1-1𝐵)
 
Theoremlindfrn 19885 The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → ran 𝐹 ∈ (LIndS‘𝑊))
 
Theoremf1lindf 19886 Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊𝐺:𝐾1-1→dom 𝐹) → (𝐹𝐺) LIndF 𝑊)
 
Theoremlindfres 19887 Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹𝑋) LIndF 𝑊)
 
Theoremlindsss 19888 Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺𝐹) → 𝐺 ∈ (LIndS‘𝑊))
 
Theoremf1linds 19889 A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
((𝑊 ∈ LMod ∧ 𝑆 ∈ (LIndS‘𝑊) ∧ 𝐹:𝐷1-1𝑆) → 𝐹 LIndF 𝑊)
 
Theoremislindf3 19890 In a nonzero ring, independent families can be equivalently characterized as renamings of independent sets. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐿 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐿 ∈ NzRing) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊))))
 
Theoremlindfmm 19891 Linear independence of a family is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹:𝐼𝐵) → (𝐹 LIndF 𝑆 ↔ (𝐺𝐹) LIndF 𝑇))
 
Theoremlindsmm 19892 Linear independence of a set is unchanged by injective linear functions. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹𝐵) → (𝐹 ∈ (LIndS‘𝑆) ↔ (𝐺𝐹) ∈ (LIndS‘𝑇)))
 
Theoremlindsmm2 19893 The monomorphic image of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐵 = (Base‘𝑆)    &   𝐶 = (Base‘𝑇)       ((𝐺 ∈ (𝑆 LMHom 𝑇) ∧ 𝐺:𝐵1-1𝐶𝐹 ∈ (LIndS‘𝑆)) → (𝐺𝐹) ∈ (LIndS‘𝑇))
 
Theoremlsslindf 19894 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
𝑈 = (LSubSp‘𝑊)    &   𝑋 = (𝑊s 𝑆)       ((𝑊 ∈ LMod ∧ 𝑆𝑈 ∧ ran 𝐹𝑆) → (𝐹 LIndF 𝑋𝐹 LIndF 𝑊))
 
Theoremlsslinds 19895 Linear independence is unchanged by working in a subspace. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝑈 = (LSubSp‘𝑊)    &   𝑋 = (𝑊s 𝑆)       ((𝑊 ∈ LMod ∧ 𝑆𝑈𝐹𝑆) → (𝐹 ∈ (LIndS‘𝑋) ↔ 𝐹 ∈ (LIndS‘𝑊)))
 
Theoremislbs4 19896 A basis is an independent spanning set. This could have been used as alternative definition of a basis: LBasis = (𝑤 ∈ V ↦ {𝑏 ∈ 𝒫 (Base‘𝑤) ∣ (((LSpan‘𝑤) 𝑏) = (Base‘𝑤) ∧ 𝑏 ∈ (LIndS‘𝑤))}). (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐽 = (LBasis‘𝑊)    &   𝐾 = (LSpan‘𝑊)       (𝑋𝐽 ↔ (𝑋 ∈ (LIndS‘𝑊) ∧ (𝐾𝑋) = 𝐵))
 
Theoremlbslinds 19897 A basis is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐽 = (LBasis‘𝑊)       𝐽 ⊆ (LIndS‘𝑊)
 
Theoremislinds3 19898 A subset is linearly independent iff it is a basis of its span. (Contributed by Stefan O'Rear, 25-Feb-2015.)
𝐵 = (Base‘𝑊)    &   𝐾 = (LSpan‘𝑊)    &   𝑋 = (𝑊s (𝐾𝑌))    &   𝐽 = (LBasis‘𝑋)       (𝑊 ∈ LMod → (𝑌 ∈ (LIndS‘𝑊) ↔ 𝑌𝐽))
 
Theoremislinds4 19899* A set is independent in a vector space iff it is a subset of some basis. (AC equivalent) (Contributed by Stefan O'Rear, 24-Feb-2015.)
𝐽 = (LBasis‘𝑊)       (𝑊 ∈ LVec → (𝑌 ∈ (LIndS‘𝑊) ↔ ∃𝑏𝐽 𝑌𝑏))
 
11.1.5  Characterization of free modules
 
Theoremlmimlbs 19900 The isomorphic image of a basis is a basis. (Contributed by Stefan O'Rear, 26-Feb-2015.)
𝐽 = (LBasis‘𝑆)    &   𝐾 = (LBasis‘𝑇)       ((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐵𝐽) → (𝐹𝐵) ∈ 𝐾)
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