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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | obsocv 21501 | An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ 0 = (0gβπ) & β’ β₯ = (ocvβπ) β β’ (π΅ β (OBasisβπ) β ( β₯ βπ΅) = { 0 }) | ||
Theorem | obs2ocv 21502 | The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (Baseβπ) β β’ (π΅ β (OBasisβπ) β ( β₯ β( β₯ βπ΅)) = π) | ||
Theorem | obselocv 21503 | 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.) |
β’ β₯ = (ocvβπ) β β’ ((π΅ β (OBasisβπ) β§ πΆ β π΅ β§ π΄ β π΅) β (π΄ β ( β₯ βπΆ) β Β¬ π΄ β πΆ)) | ||
Theorem | obs2ss 21504 | A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ ((π΅ β (OBasisβπ) β§ πΆ β (OBasisβπ) β§ πΆ β π΅) β πΆ = π΅) | ||
Theorem | obslbs 21505 | 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.) |
β’ π½ = (LBasisβπ) & β’ π = (LSpanβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π΅ β (OBasisβπ) β (π΅ β π½ β (πβπ΅) β πΆ)) | ||
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 20131), but the existence of a unity element is always assumed (our rings are unital, see df-ring 20130). 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 β 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 (Ξ±i) where Ξ±i β Mi
and Ξ±i = 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 Ξ± and
Ξ² can be added by writing (Ξ± + Ξ²)i =
Ξ±i + Ξ²i 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(Ξ±)i =
(rΞ±)i for all i.".
| ||
Syntax | cdsmm 21506 | Class of module direct sum generator. |
class βm | ||
Definition | df-dsmm 21507* | 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.) |
β’ βm = (π β V, π β V β¦ ((π Xsπ) βΎs {π β Xπ₯ β dom π(Baseβ(πβπ₯)) β£ {π₯ β dom π β£ (πβπ₯) β (0gβ(πβπ₯))} β Fin})) | ||
Theorem | reldmdsmm 21508 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
β’ Rel dom βm | ||
Theorem | dsmmval 21509* | Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
β’ π΅ = {π β (Baseβ(πXsπ )) β£ {π₯ β dom π β£ (πβπ₯) β (0gβ(π βπ₯))} β Fin} β β’ (π β π β (π βm π ) = ((πXsπ ) βΎs π΅)) | ||
Theorem | dsmmbase 21510* | Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
β’ π΅ = {π β (Baseβ(πXsπ )) β£ {π₯ β dom π β£ (πβπ₯) β (0gβ(π βπ₯))} β Fin} β β’ (π β π β π΅ = (Baseβ(π βm π ))) | ||
Theorem | dsmmval2 21511 | Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ π΅ = (Baseβ(π βm π )) β β’ (π βm π ) = ((πXsπ ) βΎs π΅) | ||
Theorem | dsmmbas2 21512* | Base set of the direct sum module using the fndmin 7046 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ π = (πXsπ ) & β’ π΅ = {π β (Baseβπ) β£ dom (π β (0g β π )) β Fin} β β’ ((π Fn πΌ β§ πΌ β π) β π΅ = (Baseβ(π βm π ))) | ||
Theorem | dsmmfi 21513 | 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.) |
β’ ((π Fn πΌ β§ πΌ β Fin) β (π βm π ) = (πXsπ )) | ||
Theorem | dsmmelbas 21514* | Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
β’ π = (πXsπ ) & β’ πΆ = (π βm π ) & β’ π΅ = (Baseβπ) & β’ π» = (BaseβπΆ) & β’ (π β πΌ β π) & β’ (π β π Fn πΌ) β β’ (π β (π β π» β (π β π΅ β§ {π β πΌ β£ (πβπ) β (0gβ(π βπ))} β Fin))) | ||
Theorem | dsmm0cl 21515 | 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.) |
β’ π = (πXsπ ) & β’ π» = (Baseβ(π βm π )) & β’ (π β πΌ β π) & β’ (π β π β π) & β’ (π β π :πΌβΆMnd) & β’ 0 = (0gβπ) β β’ (π β 0 β π») | ||
Theorem | dsmmacl 21516 | The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
β’ π = (πXsπ ) & β’ π» = (Baseβ(π βm π )) & β’ (π β πΌ β π) & β’ (π β π β π) & β’ (π β π :πΌβΆMnd) & β’ (π β π½ β π») & β’ (π β πΎ β π») & β’ + = (+gβπ) β β’ (π β (π½ + πΎ) β π») | ||
Theorem | prdsinvgd2 21517 | Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
β’ π = (πXsπ ) & β’ (π β πΌ β π) & β’ (π β π β π) & β’ (π β π :πΌβΆGrp) & β’ π΅ = (Baseβπ) & β’ π = (invgβπ) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) β β’ (π β ((πβπ)βπ½) = ((invgβ(π βπ½))β(πβπ½))) | ||
Theorem | dsmmsubg 21518 | 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.) |
β’ π = (πXsπ ) & β’ π» = (Baseβ(π βm π )) & β’ (π β πΌ β π) & β’ (π β π β π) & β’ (π β π :πΌβΆGrp) β β’ (π β π» β (SubGrpβπ)) | ||
Theorem | dsmmlss 21519* | 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.) |
β’ (π β πΌ β π) & β’ (π β π β Ring) & β’ (π β π :πΌβΆLMod) & β’ ((π β§ π₯ β πΌ) β (Scalarβ(π βπ₯)) = π) & β’ π = (πXsπ ) & β’ π = (LSubSpβπ) & β’ π» = (Baseβ(π βm π )) β β’ (π β π» β π) | ||
Theorem | dsmmlmod 21520* | 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.) |
β’ (π β πΌ β π) & β’ (π β π β Ring) & β’ (π β π :πΌβΆLMod) & β’ ((π β§ π₯ β πΌ) β (Scalarβ(π βπ₯)) = π) & β’ πΆ = (π βm π ) β β’ (π β πΆ β 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 nonfree 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 21522. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 21522 (see lmisfree 21617), the two definitions are essentially equivalent. The free modules as defined by df-frlm 21522 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 21521 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 21522* | 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 21507 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ freeLMod = (π β V, π β V β¦ (π βm (π Γ {(ringLModβπ)}))) | ||
Theorem | frlmval 21523 | Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β π) β πΉ = (π βm (πΌ Γ {(ringLModβπ )}))) | ||
Theorem | frlmlmod 21524 | The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β Ring β§ πΌ β π) β πΉ β LMod) | ||
Theorem | frlmpws 21525 | The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) β β’ ((π β π β§ πΌ β π) β πΉ = (((ringLModβπ ) βs πΌ) βΎs π΅)) | ||
Theorem | frlmlss 21526 | The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ π = (LSubSpβ((ringLModβπ ) βs πΌ)) β β’ ((π β Ring β§ πΌ β π) β π΅ β π) | ||
Theorem | frlmpwsfi 21527 | The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β Fin) β πΉ = ((ringLModβπ ) βs πΌ)) | ||
Theorem | frlmsca 21528 | The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β π β§ πΌ β π) β π = (ScalarβπΉ)) | ||
Theorem | frlm0 21529 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21526). (Contributed by Stefan O'Rear, 4-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ 0 = (0gβπ ) β β’ ((π β Ring β§ πΌ β π) β (πΌ Γ { 0 }) = (0gβπΉ)) | ||
Theorem | frlmbas 21530* | Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π΅ = {π β (π βm πΌ) β£ π finSupp 0 } β β’ ((π β π β§ πΌ β π) β π΅ = (BaseβπΉ)) | ||
Theorem | frlmelbas 21531 | Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ 0 = (0gβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((π β π β§ πΌ β π) β (π β π΅ β (π β (π βm πΌ) β§ π finSupp 0 ))) | ||
Theorem | frlmrcl 21532 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) β β’ (π β π΅ β π β V) | ||
Theorem | frlmbasfsupp 21533 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ 0 = (0gβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π finSupp 0 ) | ||
Theorem | frlmbasmap 21534 | Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π β (π βm πΌ)) | ||
Theorem | frlmbasf 21535 | Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ π β π΅) β π:πΌβΆπ) | ||
Theorem | frlmlvec 21536 | The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.) |
β’ πΉ = (π freeLMod πΌ) β β’ ((π β DivRing β§ πΌ β π) β πΉ β LVec) | ||
Theorem | frlmfibas 21537 | The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) β β’ ((π β π β§ πΌ β Fin) β (π βm πΌ) = (BaseβπΉ)) | ||
Theorem | elfrlmbasn0 21538 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (Baseβπ ) & β’ π΅ = (BaseβπΉ) β β’ ((πΌ β π β§ πΌ β β ) β (π β π΅ β π β β )) | ||
Theorem | frlmplusgval 21539 | Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπ) β β’ (π β (πΉ β πΊ) = (πΉ βf + πΊ)) | ||
Theorem | frlmsubgval 21540 | Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β πΉ β π΅) & β’ (π β πΊ β π΅) & β’ β = (-gβπ ) & β’ π = (-gβπ) β β’ (π β (πΉππΊ) = (πΉ βf β πΊ)) | ||
Theorem | frlmvscafval 21541 | Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) & β’ β = ( Β·π βπ) & β’ Β· = (.rβπ ) β β’ (π β (π΄ β π) = ((πΌ Γ {π΄}) βf Β· π)) | ||
Theorem | frlmvplusgvalc 21542 | Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) β β’ (π β ((π β π)βπ½) = ((πβπ½) + (πβπ½))) | ||
Theorem | frlmvscaval 21543 | Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ ) & β’ (π β πΌ β π) & β’ (π β π΄ β πΎ) & β’ (π β π β π΅) & β’ (π β π½ β πΌ) & β’ β = ( Β·π βπ) & β’ Β· = (.rβπ ) β β’ (π β ((π΄ β π)βπ½) = (π΄ Β· (πβπ½))) | ||
Theorem | frlmplusgvalb 21544* | Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ (π β π β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) β β’ (π β (π = (π β π) β βπ β πΌ (πβπ) = ((πβπ) + (πβπ)))) | ||
Theorem | frlmvscavalb 21545* | Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ πΎ = (Baseβπ ) & β’ (π β π΄ β πΎ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ ) β β’ (π β (π = (π΄ β π) β βπ β πΌ (πβπ) = (π΄ Β· (πβπ)))) | ||
Theorem | frlmvplusgscavalb 21546* | Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ (π β πΌ β π) & β’ (π β π β π΅) & β’ (π β π β π΅) & β’ (π β π β Ring) & β’ πΎ = (Baseβπ ) & β’ (π β π΄ β πΎ) & β’ β = ( Β·π βπΉ) & β’ Β· = (.rβπ ) & β’ (π β π β π΅) & β’ + = (+gβπ ) & β’ β = (+gβπΉ) & β’ (π β πΆ β πΎ) β β’ (π β (π = ((π΄ β π) β (πΆ β π)) β βπ β πΌ (πβπ) = ((π΄ Β· (πβπ)) + (πΆ Β· (πβπ))))) | ||
Theorem | frlmgsum 21547* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ) & β’ (π β πΌ β π) & β’ (π β π½ β π) & β’ (π β π β Ring) & β’ ((π β§ π¦ β π½) β (π₯ β πΌ β¦ π) β π΅) & β’ (π β (π¦ β π½ β¦ (π₯ β πΌ β¦ π)) finSupp 0 ) β β’ (π β (π Ξ£g (π¦ β π½ β¦ (π₯ β πΌ β¦ π))) = (π₯ β πΌ β¦ (π Ξ£g (π¦ β π½ β¦ π)))) | ||
Theorem | frlmsplit2 21548* | Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.) |
β’ π = (π freeLMod π) & β’ π = (π freeLMod π) & β’ π΅ = (Baseβπ) & β’ πΆ = (Baseβπ) & β’ πΉ = (π₯ β π΅ β¦ (π₯ βΎ π)) β β’ ((π β Ring β§ π β π β§ π β π) β πΉ β (π LMHom π)) | ||
Theorem | frlmsslss 21549* | A subset of a free module obtained by restricting the support set is a submodule. π½ is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.) |
β’ π = (π freeLMod πΌ) & β’ π = (LSubSpβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ βΎ π½) = (π½ Γ { 0 })} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β πΆ β π) | ||
Theorem | frlmsslss2 21550* | A subset of a free module obtained by restricting the support set is a submodule. π½ is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π = (LSubSpβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β πΆ β π) | ||
Theorem | frlmbas3 21551 | 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.) |
β’ πΉ = (π freeLMod (π Γ π)) & β’ π΅ = (Baseβπ ) & β’ π = (BaseβπΉ) β β’ (((π β π β§ π β π) β§ (π β Fin β§ π β Fin) β§ (πΌ β π β§ π½ β π)) β (πΌππ½) β π΅) | ||
Theorem | mpofrlmd 21552* | 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.) |
β’ πΉ = (π freeLMod (π Γ π)) & β’ π = (BaseβπΉ) & β’ ((π = π β§ π = π) β π΄ = π΅) & β’ ((π β§ π β π β§ π β π) β π΄ β π) & β’ ((π β§ π β π β§ π β π) β π΅ β π) & β’ (π β (π β π β§ π β π β§ π β π)) β β’ (π β (π = (π β π, π β π β¦ π΅) β βπ β π βπ β π (πππ) = π΄)) | ||
Theorem | frlmip 21553* | The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) β β’ ((πΌ β π β§ π β π) β (π β (π΅ βm πΌ), π β (π΅ βm πΌ) β¦ (π Ξ£g (π₯ β πΌ β¦ ((πβπ₯) Β· (πβπ₯))))) = (Β·πβπ)) | ||
Theorem | frlmipval 21554 | The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) β β’ (((πΌ β π β§ π β π) β§ (πΉ β π β§ πΊ β π)) β (πΉ , πΊ) = (π Ξ£g (πΉ βf Β· πΊ))) | ||
Theorem | frlmphllem 21555* | Lemma for frlmphl 21556. (Contributed by AV, 21-Jul-2019.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ π = (0gβπ) & β’ 0 = (0gβπ ) & β’ β = (*πβπ ) & β’ (π β π β Field) & β’ ((π β§ π β π β§ (π , π) = 0 ) β π = π) & β’ ((π β§ π₯ β π΅) β ( β βπ₯) = π₯) & β’ (π β πΌ β π) β β’ ((π β§ π β π β§ β β π) β (π₯ β πΌ β¦ ((πβπ₯) Β· (ββπ₯))) finSupp 0 ) | ||
Theorem | frlmphl 21556* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ ) & β’ Β· = (.rβπ ) & β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ π = (0gβπ) & β’ 0 = (0gβπ ) & β’ β = (*πβπ ) & β’ (π β π β Field) & β’ ((π β§ π β π β§ (π , π) = 0 ) β π = π) & β’ ((π β§ π₯ β π΅) β ( β βπ₯) = π₯) & β’ (π β πΌ β π) β β’ (π β π β 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 21557 | Class of basic unit vectors for an explicit free module. |
class unitVec | ||
Definition | df-uvc 21558* | ((π unitVec πΌ)βπ) is the unit vector in (π freeLMod πΌ) along the π axis. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ unitVec = (π β V, π β V β¦ (π β π β¦ (π β π β¦ if(π = π, (1rβπ), (0gβπ))))) | ||
Theorem | uvcfval 21559* | Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ πΌ β π) β π = (π β πΌ β¦ (π β πΌ β¦ if(π = π, 1 , 0 )))) | ||
Theorem | uvcval 21560* | Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ ((π β π β§ πΌ β π β§ π½ β πΌ) β (πβπ½) = (π β πΌ β¦ if(π = π½, 1 , 0 ))) | ||
Theorem | uvcvval 21561 | Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ (((π β π β§ πΌ β π β§ π½ β πΌ) β§ πΎ β πΌ) β ((πβπ½)βπΎ) = if(πΎ = π½, 1 , 0 )) | ||
Theorem | uvcvvcl 21562 | A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ 1 = (1rβπ ) & β’ 0 = (0gβπ ) β β’ (((π β π β§ πΌ β π β§ π½ β πΌ) β§ πΎ β πΌ) β ((πβπ½)βπΎ) β { 0 , 1 }) | ||
Theorem | uvcvvcl2 21563 | A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ π΅ = (Baseβπ ) & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΎ β πΌ) β β’ (π β ((πβπ½)βπΎ) β π΅) | ||
Theorem | uvcvv1 21564 | The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ 1 = (1rβπ ) β β’ (π β ((πβπ½)βπ½) = 1 ) | ||
Theorem | uvcvv0 21565 | The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.) |
β’ π = (π unitVec πΌ) & β’ (π β π β π) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΎ β πΌ) & β’ (π β π½ β πΎ) & β’ 0 = (0gβπ ) β β’ (π β ((πβπ½)βπΎ) = 0 ) | ||
Theorem | uvcff 21566 | Domain and codomain 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.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β Ring β§ πΌ β π) β π:πΌβΆπ΅) | ||
Theorem | uvcf1 21567 | In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) β β’ ((π β NzRing β§ πΌ β π) β π:πΌβ1-1βπ΅) | ||
Theorem | uvcresum 21568 | 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.) |
β’ π = (π unitVec πΌ) & β’ π = (π freeLMod πΌ) & β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) β β’ ((π β Ring β§ πΌ β π β§ π β π΅) β π = (π Ξ£g (π βf Β· π))) | ||
Theorem | frlmssuvc1 21569* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΎ = (Baseβπ ) & β’ Β· = ( Β·π βπΉ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΏ β π½) & β’ (π β π β πΎ) β β’ (π β (π Β· (πβπΏ)) β πΆ) | ||
Theorem | frlmssuvc2 21570* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΎ = (Baseβπ ) & β’ Β· = ( Β·π βπΉ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} & β’ (π β π β Ring) & β’ (π β πΌ β π) & β’ (π β π½ β πΌ) & β’ (π β πΏ β (πΌ β π½)) & β’ (π β π β (πΎ β { 0 })) β β’ (π β Β¬ (π Β· (πβπΏ)) β πΆ) | ||
Theorem | frlmsslsp 21571* | 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.) |
β’ π = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ πΎ = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ 0 = (0gβπ ) & β’ πΆ = {π₯ β π΅ β£ (π₯ supp 0 ) β π½} β β’ ((π β Ring β§ πΌ β π β§ π½ β πΌ) β (πΎβ(π β π½)) = πΆ) | ||
Theorem | frlmlbs 21572 | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ π½ = (LBasisβπΉ) β β’ ((π β Ring β§ πΌ β π) β ran π β π½) | ||
Theorem | frlmup1 21573* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) β β’ (π β πΈ β (πΉ LMHom π)) | ||
Theorem | frlmup2 21574* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) & β’ (π β π β πΌ) & β’ π = (π unitVec πΌ) β β’ (π β (πΈβ(πβπ)) = (π΄βπ)) | ||
Theorem | frlmup3 21575* | 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.) |
β’ πΉ = (π freeLMod πΌ) & β’ π΅ = (BaseβπΉ) & β’ πΆ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΈ = (π₯ β π΅ β¦ (π Ξ£g (π₯ βf Β· π΄))) & β’ (π β π β LMod) & β’ (π β πΌ β π) & β’ (π β π = (Scalarβπ)) & β’ (π β π΄:πΌβΆπΆ) & β’ πΎ = (LSpanβπ) β β’ (π β ran πΈ = (πΎβran π΄)) | ||
Theorem | frlmup4 21576* | Universal property of the free module by existential uniqueness. (Contributed by Stefan O'Rear, 7-Mar-2015.) |
β’ π = (Scalarβπ) & β’ πΉ = (π freeLMod πΌ) & β’ π = (π unitVec πΌ) & β’ πΆ = (Baseβπ) β β’ ((π β LMod β§ πΌ β π β§ π΄:πΌβΆπΆ) β β!π β (πΉ LMHom π)(π β π) = π΄) | ||
Theorem | ellspd 21577* | 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.) (Revised by AV, 11-Apr-2024.) |
β’ π = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ (π β πΉ:πΌβΆπ΅) & β’ (π β π β LMod) & β’ (π β πΌ β π) β β’ (π β (π β (πβ(πΉ β πΌ)) β βπ β (πΎ βm πΌ)(π finSupp 0 β§ π = (π Ξ£g (π βf Β· πΉ))))) | ||
Theorem | elfilspd 21578* | Simplified version of ellspd 21577 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.) |
β’ π = (LSpanβπ) & β’ π΅ = (Baseβπ) & β’ πΎ = (Baseβπ) & β’ π = (Scalarβπ) & β’ 0 = (0gβπ) & β’ Β· = ( Β·π βπ) & β’ (π β πΉ:πΌβΆπ΅) & β’ (π β π β LMod) & β’ (π β πΌ β Fin) β β’ (π β (π β (πβ(πΉ β πΌ)) β βπ β (πΎ βm πΌ)π = (π Ξ£g (π βf Β· πΉ)))) | ||
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 βxβSaxx which is equal to 0, then ax = 0 for all x β S", and according to the Definition in [Lang] p. 130: "a familiy {xi}iβI of elements of M is said to be linearly independent (over A) if whenever we have a linear combination βiβIaixi = 0, then ai = 0 for all i β I." These definitions correspond to Definitions df-linds 21582 and df-lindf 21581 respectively, where it is claimed that a nonzero summand can be extracted (βiβ{I\{j}}aixi = -ajxj) and be represented as a linear combination of the remaining elements of the family. | ||
Syntax | clindf 21579 | The class relationship of independent families in a module. |
class LIndF | ||
Syntax | clinds 21580 | The class generator of independent sets in a module. |
class LIndS | ||
Definition | df-lindf 21581* |
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 21601, 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 21613) and only one representation for each element of the range (islindf5 21614). (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ LIndF = {β¨π, π€β© β£ (π:dom πβΆ(Baseβπ€) β§ [(Scalarβπ€) / π ]βπ₯ β dom πβπ β ((Baseβπ ) β {(0gβπ )}) Β¬ (π( Β·π βπ€)(πβπ₯)) β ((LSpanβπ€)β(π β (dom π β {π₯}))))} | ||
Definition | df-linds 21582* | An independent set is a set which is independent as a family. See also islinds3 21609 and islinds4 21610. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ LIndS = (π€ β V β¦ {π β π« (Baseβπ€) β£ ( I βΎ π ) LIndF π€}) | ||
Theorem | rellindf 21583 | The independent-family predicate is a proper relation and can be used with brrelex1i 5732. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ Rel LIndF | ||
Theorem | islinds 21584 | Property of an independent set of vectors in terms of an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β π β (π β (LIndSβπ) β (π β π΅ β§ ( I βΎ π) LIndF π))) | ||
Theorem | linds1 21585 | An independent set of vectors is a set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ (π β (LIndSβπ) β π β π΅) | ||
Theorem | linds2 21586 | An independent set of vectors is independent as a family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ (π β (LIndSβπ) β ( I βΎ π) LIndF π) | ||
Theorem | islindf 21587* | Property of an independent family of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ ((π β π β§ πΉ β π) β (πΉ LIndF π β (πΉ:dom πΉβΆπ΅ β§ βπ₯ β dom πΉβπ β (π β { 0 }) Β¬ (π Β· (πΉβπ₯)) β (πΎβ(πΉ β (dom πΉ β {π₯})))))) | ||
Theorem | islinds2 21588* | Expanded property of an independent set of vectors. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ (π β π β (πΉ β (LIndSβπ) β (πΉ β π΅ β§ βπ₯ β πΉ βπ β (π β { 0 }) Β¬ (π Β· π₯) β (πΎβ(πΉ β {π₯}))))) | ||
Theorem | islindf2 21589* | Property of an independent family of vectors with prior constrained domain and codomain. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ Β· = ( Β·π βπ) & β’ πΎ = (LSpanβπ) & β’ π = (Scalarβπ) & β’ π = (Baseβπ) & β’ 0 = (0gβπ) β β’ ((π β π β§ πΌ β π β§ πΉ:πΌβΆπ΅) β (πΉ LIndF π β βπ₯ β πΌ βπ β (π β { 0 }) Β¬ (π Β· (πΉβπ₯)) β (πΎβ(πΉ β (πΌ β {π₯}))))) | ||
Theorem | lindff 21590 | Functional property of a linearly independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) β β’ ((πΉ LIndF π β§ π β π) β πΉ:dom πΉβΆπ΅) | ||
Theorem | lindfind 21591 | 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.) |
β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ πΏ = (Scalarβπ) & β’ 0 = (0gβπΏ) & β’ πΎ = (BaseβπΏ) β β’ (((πΉ LIndF π β§ πΈ β dom πΉ) β§ (π΄ β πΎ β§ π΄ β 0 )) β Β¬ (π΄ Β· (πΉβπΈ)) β (πβ(πΉ β (dom πΉ β {πΈ})))) | ||
Theorem | lindsind 21592 | 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.) |
β’ Β· = ( Β·π βπ) & β’ π = (LSpanβπ) & β’ πΏ = (Scalarβπ) & β’ 0 = (0gβπΏ) & β’ πΎ = (BaseβπΏ) β β’ (((πΉ β (LIndSβπ) β§ πΈ β πΉ) β§ (π΄ β πΎ β§ π΄ β 0 )) β Β¬ (π΄ Β· πΈ) β (πβ(πΉ β {πΈ}))) | ||
Theorem | lindfind2 21593 | 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.) |
β’ πΎ = (LSpanβπ) & β’ πΏ = (Scalarβπ) β β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ LIndF π β§ πΈ β dom πΉ) β Β¬ (πΉβπΈ) β (πΎβ(πΉ β (dom πΉ β {πΈ})))) | ||
Theorem | lindsind2 21594 | 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.) |
β’ πΎ = (LSpanβπ) & β’ πΏ = (Scalarβπ) β β’ (((π β LMod β§ πΏ β NzRing) β§ πΉ β (LIndSβπ) β§ πΈ β πΉ) β Β¬ πΈ β (πΎβ(πΉ β {πΈ}))) | ||
Theorem | lindff1 21595 | A linearly independent family over a nonzero ring has no repeated elements. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ π΅ = (Baseβπ) & β’ πΏ = (Scalarβπ) β β’ ((π β LMod β§ πΏ β NzRing β§ πΉ LIndF π) β πΉ:dom πΉβ1-1βπ΅) | ||
Theorem | lindfrn 21596 | The range of an independent family is an independent set. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π) β ran πΉ β (LIndSβπ)) | ||
Theorem | f1lindf 21597 | Rearranging and deleting elements from an independent family gives an independent family. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π β§ πΊ:πΎβ1-1βdom πΉ) β (πΉ β πΊ) LIndF π) | ||
Theorem | lindfres 21598 | Any restriction of an independent family is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ LIndF π) β (πΉ βΎ π) LIndF π) | ||
Theorem | lindsss 21599 | Any subset of an independent set is independent. (Contributed by Stefan O'Rear, 24-Feb-2015.) |
β’ ((π β LMod β§ πΉ β (LIndSβπ) β§ πΊ β πΉ) β πΊ β (LIndSβπ)) | ||
Theorem | f1linds 21600 | A family constructed from non-repeated elements of an independent set is independent. (Contributed by Stefan O'Rear, 26-Feb-2015.) |
β’ ((π β LMod β§ π β (LIndSβπ) β§ πΉ:π·β1-1βπ) β πΉ LIndF π) |
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