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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | obs2ocv 20801 | 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) | ||
Theorem | obselocv 20802 | 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) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ¬ 𝐴 ∈ 𝐶)) | ||
Theorem | obs2ss 20803 | A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵) | ||
Theorem | obslbs 20804 | 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 = (ClSubSp‘𝑊) ⇒ ⊢ (𝐵 ∈ (OBasis‘𝑊) → (𝐵 ∈ 𝐽 ↔ (𝑁‘𝐵) ∈ 𝐶)) | ||
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." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 19231), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring 19230). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space. | ||
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.".
| ||
Syntax | cdsmm 20805 | Class of module direct sum generator. |
class ⊕m | ||
Definition | df-dsmm 20806* | 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 {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin})) | ||
Theorem | reldmdsmm 20807 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
⊢ Rel dom ⊕m | ||
Theorem | dsmmval 20808* | Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} ⇒ ⊢ (𝑅 ∈ 𝑉 → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)) | ||
Theorem | dsmmbase 20809* | Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin} ⇒ ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕m 𝑅))) | ||
Theorem | dsmmval2 20810 | 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‘(𝑆 ⊕m 𝑅)) ⇒ ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵) | ||
Theorem | dsmmbas2 20811* | Base set of the direct sum module using the fndmin 6808 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin} ⇒ ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅))) | ||
Theorem | dsmmfi 20812 | 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 𝐼 ∧ 𝐼 ∈ Fin) → (𝑆 ⊕m 𝑅) = (𝑆Xs𝑅)) | ||
Theorem | dsmmelbas 20813* | 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 = (𝑆 ⊕m 𝑅) & ⊢ 𝐵 = (Base‘𝑃) & ⊢ 𝐻 = (Base‘𝐶) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 Fn 𝐼) ⇒ ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin))) | ||
Theorem | dsmm0cl 20814 | 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‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ 0 = (0g‘𝑃) ⇒ ⊢ (𝜑 → 0 ∈ 𝐻) | ||
Theorem | dsmmacl 20815 | The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Mnd) & ⊢ (𝜑 → 𝐽 ∈ 𝐻) & ⊢ (𝜑 → 𝐾 ∈ 𝐻) & ⊢ + = (+g‘𝑃) ⇒ ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻) | ||
Theorem | prdsinvgd2 20816 | Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ 𝑌 = (𝑆Xs𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝑁 = (invg‘𝑌) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽))) | ||
Theorem | dsmmsubg 20817 | 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‘(𝑆 ⊕m 𝑅)) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ 𝑉) & ⊢ (𝜑 → 𝑅:𝐼⟶Grp) ⇒ ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃)) | ||
Theorem | dsmmlss 20818* | 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.) |
⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) & ⊢ 𝑃 = (𝑆Xs𝑅) & ⊢ 𝑈 = (LSubSp‘𝑃) & ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅)) ⇒ ⊢ (𝜑 → 𝐻 ∈ 𝑈) | ||
Theorem | dsmmlmod 20819* | 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.) |
⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) & ⊢ 𝐶 = (𝑆 ⊕m 𝑅) ⇒ ⊢ (𝜑 → 𝐶 ∈ LMod) | ||
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, a free module is defined as the direct sum of copies of a ring regarded as a left module over itself, see df-frlm 20821. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20821 (see lmisfree 20916), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20821 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 20820 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 20821* | Define the function associating with a ring and a set the direct sum indexed by that set of copies of that ring regarded as a left module over itself. Recall from df-dsmm 20806 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)}))) | ||
Theorem | frlmval 20822 | Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)}))) | ||
Theorem | frlmlmod 20823 | The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod) | ||
Theorem | frlmpws 20824 | 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) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵)) | ||
Theorem | frlmlss 20825 | 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 ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈) | ||
Theorem | frlmpwsfi 20826 | The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼)) | ||
Theorem | frlmsca 20827 | The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹)) | ||
Theorem | frlm0 20828 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20825). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹)) | ||
Theorem | frlmbas 20829* | 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 = (0g‘𝑅) & ⊢ 𝐵 = {𝑘 ∈ (𝑁 ↑m 𝐼) ∣ 𝑘 finSupp 0 } ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘𝐹)) | ||
Theorem | frlmelbas 20830 | 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 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝑁 ↑m 𝐼) ∧ 𝑋 finSupp 0 ))) | ||
Theorem | frlmrcl 20831 | 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) ⇒ ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V) | ||
Theorem | frlmbasfsupp 20832 | 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 = (0g‘𝑅) & ⊢ 𝐵 = (Base‘𝐹) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 finSupp 0 ) | ||
Theorem | frlmbasmap 20833 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝑁 ↑m 𝐼)) | ||
Theorem | frlmbasf 20834 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝑁) | ||
Theorem | frlmlvec 20835 | The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) ⇒ ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LVec) | ||
Theorem | frlmfibas 20836 | 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) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑁 ↑m 𝐼) = (Base‘𝐹)) | ||
Theorem | elfrlmbasn0 20837 | 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) ⇒ ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ≠ ∅) → (𝑋 ∈ 𝐵 → 𝑋 ≠ ∅)) | ||
Theorem | frlmplusgval 20838 | 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) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺)) | ||
Theorem | frlmsubgval 20839 | Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐹 ∈ 𝐵) & ⊢ (𝜑 → 𝐺 ∈ 𝐵) & ⊢ − = (-g‘𝑅) & ⊢ 𝑀 = (-g‘𝑌) ⇒ ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘f − 𝐺)) | ||
Theorem | frlmvscafval 20840 | 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) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋)) | ||
Theorem | frlmvplusgvalc 20841 | Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝐹) ⇒ ⊢ (𝜑 → ((𝑋 ✚ 𝑌)‘𝐽) = ((𝑋‘𝐽) + (𝑌‘𝐽))) | ||
Theorem | frlmvscaval 20842 | Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑌) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ ∙ = ( ·𝑠 ‘𝑌) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽))) | ||
Theorem | frlmplusgvalb 20843* | Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝐹) ⇒ ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖)))) | ||
Theorem | frlmvscavalb 20844* | Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ ∙ = ( ·𝑠 ‘𝐹) & ⊢ · = (.r‘𝑅) ⇒ ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖)))) | ||
Theorem | frlmvplusgscavalb 20845* | Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝐹) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐵) & ⊢ (𝜑 → 𝑍 ∈ 𝐵) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ 𝐾 = (Base‘𝑅) & ⊢ (𝜑 → 𝐴 ∈ 𝐾) & ⊢ ∙ = ( ·𝑠 ‘𝐹) & ⊢ · = (.r‘𝑅) & ⊢ (𝜑 → 𝑌 ∈ 𝐵) & ⊢ + = (+g‘𝑅) & ⊢ ✚ = (+g‘𝐹) & ⊢ (𝜑 → 𝐶 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖))))) | ||
Theorem | frlmgsum 20846* | 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 = (0g‘𝑌) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ∈ 𝑊) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵) & ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 ) ⇒ ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈)))) | ||
Theorem | frlmsplit2 20847* | 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 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉)) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍)) | ||
Theorem | frlmsslss 20848* | 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 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ 𝐽) = (𝐽 × { 0 })} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmsslss2 20849* | 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 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈) | ||
Theorem | frlmbas3 20850 | 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) ⇒ ⊢ (((𝑅 ∈ 𝑊 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵) | ||
Theorem | mpofrlmd 20851* | Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.) |
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀)) & ⊢ 𝑉 = (Base‘𝐹) & ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵) & ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋) & ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌) & ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉)) ⇒ ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴)) | ||
Theorem | frlmip 20852* | The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) ⇒ ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌)) | ||
Theorem | frlmipval 20853 | The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) ⇒ ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹 ∘f · 𝐺))) | ||
Theorem | frlmphllem 20854* | Lemma for frlmphl 20855. (Contributed by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 ) | ||
Theorem | frlmphl 20855* | 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) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑌 ∈ PreHil) | ||
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. | ||
Syntax | cuvc 20856 | Class of basic unit vectors for an explicit free module. |
class unitVec | ||
Definition | df-uvc 20857* | ((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 ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟))))) | ||
Theorem | uvcfval 20858* | Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 )))) | ||
Theorem | uvcval 20859* | Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 ))) | ||
Theorem | uvcvval 20860 | Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 )) | ||
Theorem | uvcvvcl 20861 | A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 1 = (1r‘𝑅) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 }) | ||
Theorem | uvcvvcl2 20862 | A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵) | ||
Theorem | uvcvv1 20863 | The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ 1 = (1r‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 ) | ||
Theorem | uvcvv0 20864 | The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ (𝜑 → 𝑅 ∈ 𝑉) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝐽 ∈ 𝐼) & ⊢ (𝜑 → 𝐾 ∈ 𝐼) & ⊢ (𝜑 → 𝐽 ≠ 𝐾) & ⊢ 0 = (0g‘𝑅) ⇒ ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 ) | ||
Theorem | uvcff 20865 | 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 ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵) | ||
Theorem | uvcf1 20866 | 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 ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵) | ||
Theorem | uvcresum 20867 | 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) & ⊢ · = ( ·𝑠 ‘𝑌) ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘f · 𝑈))) | ||
Theorem | frlmssuvc1 20868* | 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) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ 𝐽) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) ⇒ ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmssuvc2 20869* | 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) & ⊢ · = ( ·𝑠 ‘𝐹) & ⊢ 0 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} & ⊢ (𝜑 → 𝑅 ∈ Ring) & ⊢ (𝜑 → 𝐼 ∈ 𝑉) & ⊢ (𝜑 → 𝐽 ⊆ 𝐼) & ⊢ (𝜑 → 𝐿 ∈ (𝐼 ∖ 𝐽)) & ⊢ (𝜑 → 𝑋 ∈ (𝐾 ∖ { 0 })) ⇒ ⊢ (𝜑 → ¬ (𝑋 · (𝑈‘𝐿)) ∈ 𝐶) | ||
Theorem | frlmsslsp 20870* | 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 = (0g‘𝑅) & ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽} ⇒ ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → (𝐾‘(𝑈 “ 𝐽)) = 𝐶) | ||
Theorem | frlmlbs 20871 | 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 ∧ 𝐼 ∈ 𝑉) → ran 𝑈 ∈ 𝐽) | ||
Theorem | frlmup1 20872* | 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) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) ⇒ ⊢ (𝜑 → 𝐸 ∈ (𝐹 LMHom 𝑇)) | ||
Theorem | frlmup2 20873* | 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) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ (𝜑 → 𝑌 ∈ 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) ⇒ ⊢ (𝜑 → (𝐸‘(𝑈‘𝑌)) = (𝐴‘𝑌)) | ||
Theorem | frlmup3 20874* | 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) & ⊢ · = ( ·𝑠 ‘𝑇) & ⊢ 𝐸 = (𝑥 ∈ 𝐵 ↦ (𝑇 Σg (𝑥 ∘f · 𝐴))) & ⊢ (𝜑 → 𝑇 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ 𝑋) & ⊢ (𝜑 → 𝑅 = (Scalar‘𝑇)) & ⊢ (𝜑 → 𝐴:𝐼⟶𝐶) & ⊢ 𝐾 = (LSpan‘𝑇) ⇒ ⊢ (𝜑 → ran 𝐸 = (𝐾‘ran 𝐴)) | ||
Theorem | frlmup4 20875* | Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
⊢ 𝑅 = (Scalar‘𝑇) & ⊢ 𝐹 = (𝑅 freeLMod 𝐼) & ⊢ 𝑈 = (𝑅 unitVec 𝐼) & ⊢ 𝐶 = (Base‘𝑇) ⇒ ⊢ ((𝑇 ∈ LMod ∧ 𝐼 ∈ 𝑋 ∧ 𝐴:𝐼⟶𝐶) → ∃!𝑚 ∈ (𝐹 LMHom 𝑇)(𝑚 ∘ 𝑈) = 𝐴) | ||
Theorem | ellspd 20876* | 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 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ V) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)(𝑓 finSupp 0 ∧ 𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹))))) | ||
Theorem | elfilspd 20877* | Simplified version of ellspd 20876 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 = (0g‘𝑆) & ⊢ · = ( ·𝑠 ‘𝑀) & ⊢ (𝜑 → 𝐹:𝐼⟶𝐵) & ⊢ (𝜑 → 𝑀 ∈ LMod) & ⊢ (𝜑 → 𝐼 ∈ Fin) ⇒ ⊢ (𝜑 → (𝑋 ∈ (𝑁‘(𝐹 “ 𝐼)) ↔ ∃𝑓 ∈ (𝐾 ↑m 𝐼)𝑋 = (𝑀 Σg (𝑓 ∘f · 𝐹)))) | ||
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∈Saxx 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∈Iaixi = 0, then ai = 0 for all i ∈ I." These definitions correspond to the definitions df-linds 20881 and df-lindf 20880 respectively, 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. | ||
Syntax | clindf 20878 | The class relationship of independent families in a module. |
class LIndF | ||
Syntax | clinds 20879 | The class generator of independent sets in a module. |
class LIndS | ||
Definition | df-lindf 20880* |
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 20900, 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 20912) and only one representation for each element of the range (islindf5 20913). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndF = {〈𝑓, 𝑤〉 ∣ (𝑓:dom 𝑓⟶(Base‘𝑤) ∧ [(Scalar‘𝑤) / 𝑠]∀𝑥 ∈ dom 𝑓∀𝑘 ∈ ((Base‘𝑠) ∖ {(0g‘𝑠)}) ¬ (𝑘( ·𝑠 ‘𝑤)(𝑓‘𝑥)) ∈ ((LSpan‘𝑤)‘(𝑓 “ (dom 𝑓 ∖ {𝑥}))))} | ||
Definition | df-linds 20881* | An independent set is a set which is independent as a family. See also islinds3 20908 and islinds4 20909. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ LIndS = (𝑤 ∈ V ↦ {𝑠 ∈ 𝒫 (Base‘𝑤) ∣ ( I ↾ 𝑠) LIndF 𝑤}) | ||
Theorem | rellindf 20882 | The independent-family predicate is a proper relation and can be used with brrelex1i 5602. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ Rel LIndF | ||
Theorem | islinds 20883 | Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → (𝑋 ∈ (LIndS‘𝑊) ↔ (𝑋 ⊆ 𝐵 ∧ ( I ↾ 𝑋) LIndF 𝑊))) | ||
Theorem | linds1 20884 | An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ (𝑋 ∈ (LIndS‘𝑊) → 𝑋 ⊆ 𝐵) | ||
Theorem | linds2 20885 | 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) | ||
Theorem | islindf 20886* | Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐹 ∈ 𝑋) → (𝐹 LIndF 𝑊 ↔ (𝐹:dom 𝐹⟶𝐵 ∧ ∀𝑥 ∈ dom 𝐹∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝑥})))))) | ||
Theorem | islinds2 20887* | Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ (𝑊 ∈ 𝑌 → (𝐹 ∈ (LIndS‘𝑊) ↔ (𝐹 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐹 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · 𝑥) ∈ (𝐾‘(𝐹 ∖ {𝑥}))))) | ||
Theorem | islindf2 20888* | Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝐾 = (LSpan‘𝑊) & ⊢ 𝑆 = (Scalar‘𝑊) & ⊢ 𝑁 = (Base‘𝑆) & ⊢ 0 = (0g‘𝑆) ⇒ ⊢ ((𝑊 ∈ 𝑌 ∧ 𝐼 ∈ 𝑋 ∧ 𝐹:𝐼⟶𝐵) → (𝐹 LIndF 𝑊 ↔ ∀𝑥 ∈ 𝐼 ∀𝑘 ∈ (𝑁 ∖ { 0 }) ¬ (𝑘 · (𝐹‘𝑥)) ∈ (𝐾‘(𝐹 “ (𝐼 ∖ {𝑥}))))) | ||
Theorem | lindff 20889 | Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ 𝐵 = (Base‘𝑊) ⇒ ⊢ ((𝐹 LIndF 𝑊 ∧ 𝑊 ∈ 𝑌) → 𝐹:dom 𝐹⟶𝐵) | ||
Theorem | lindfind 20890 | 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.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 LIndF 𝑊 ∧ 𝐸 ∈ dom 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · (𝐹‘𝐸)) ∈ (𝑁‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind 20891 | 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.) |
⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐿 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐿) & ⊢ 𝐾 = (Base‘𝐿) ⇒ ⊢ (((𝐹 ∈ (LIndS‘𝑊) ∧ 𝐸 ∈ 𝐹) ∧ (𝐴 ∈ 𝐾 ∧ 𝐴 ≠ 0 )) → ¬ (𝐴 · 𝐸) ∈ (𝑁‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindfind2 20892 | 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 𝑊 ∧ 𝐸 ∈ dom 𝐹) → ¬ (𝐹‘𝐸) ∈ (𝐾‘(𝐹 “ (dom 𝐹 ∖ {𝐸})))) | ||
Theorem | lindsind2 20893 | 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) ∧ 𝐸 ∈ 𝐹) → ¬ 𝐸 ∈ (𝐾‘(𝐹 ∖ {𝐸}))) | ||
Theorem | lindff1 20894 | 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 𝐹–1-1→𝐵) | ||
Theorem | lindfrn 20895 | 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)) | ||
Theorem | f1lindf 20896 | Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ 𝐺:𝐾–1-1→dom 𝐹) → (𝐹 ∘ 𝐺) LIndF 𝑊) | ||
Theorem | lindfres 20897 | Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊) → (𝐹 ↾ 𝑋) LIndF 𝑊) | ||
Theorem | lindsss 20898 | Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
⊢ ((𝑊 ∈ LMod ∧ 𝐹 ∈ (LIndS‘𝑊) ∧ 𝐺 ⊆ 𝐹) → 𝐺 ∈ (LIndS‘𝑊)) | ||
Theorem | f1linds 20899 | 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:𝐷–1-1→𝑆) → 𝐹 LIndF 𝑊) | ||
Theorem | islindf3 20900 | 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 𝐹–1-1→V ∧ ran 𝐹 ∈ (LIndS‘𝑊)))) |
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