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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cssval 21601* | The set of closed subspaces of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β π β πΆ = {π β£ π = ( β₯ β( β₯ βπ ))}) | ||
Theorem | iscss 21602 | The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β π β (π β πΆ β π = ( β₯ β( β₯ βπ)))) | ||
Theorem | cssi 21603 | Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β πΆ β π = ( β₯ β( β₯ βπ))) | ||
Theorem | cssss 21604 | A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β πΆ β π β π) | ||
Theorem | iscss2 21605 | It is sufficient to prove that the double orthocomplement is a subset of the target set to show that the set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π) β (π β πΆ β ( β₯ β( β₯ βπ)) β π)) | ||
Theorem | ocvcss 21606 | The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π) β ( β₯ βπ) β πΆ) | ||
Theorem | cssincl 21607 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) β β’ ((π β PreHil β§ π΄ β πΆ β§ π΅ β πΆ) β (π΄ β© π΅) β πΆ) | ||
Theorem | css0 21608 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β { 0 } β πΆ) | ||
Theorem | css1 21609 | The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β π β πΆ) | ||
Theorem | csslss 21610 | A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ πΏ = (LSubSpβπ) β β’ ((π β PreHil β§ π β πΆ) β π β πΏ) | ||
Theorem | lsmcss 21611 | A subset of a pre-Hilbert space whose double orthocomplement has a projection decomposition is a closed subspace. This is the core of the proof that a topologically closed subspace is algebraically closed in a Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ πΆ = (ClSubSpβπ) & β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ β = (LSSumβπ) & β’ (π β π β PreHil) & β’ (π β π β π) & β’ (π β ( β₯ β( β₯ βπ)) β (π β ( β₯ βπ))) β β’ (π β π β πΆ) | ||
Theorem | cssmre 21612 | The closed subspaces of a pre-Hilbert space are a Moore system. Unlike many of our other examples of closure systems, this one is not usually an algebraic closure system df-acs 17560: consider the Hilbert space of sequences ββΆβ with convergent sum; the subspace of all sequences with finite support is the classic example of a non-closed subspace, but for every finite set of sequences of finite support, there is a finite-dimensional (and hence closed) subspace containing all of the sequences, so if closed subspaces were an algebraic closure system this would violate acsfiel 17625. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β πΆ β (Mooreβπ)) | ||
Theorem | mrccss 21613 | The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΉ = (mrClsβπΆ) β β’ ((π β PreHil β§ π β π) β (πΉβπ) = ( β₯ β( β₯ βπ))) | ||
Theorem | thlval 21614 | Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β₯ = (ocvβπ) β β’ (π β π β πΎ = (πΌ sSet β¨(ocβndx), β₯ β©)) | ||
Theorem | thlbas 21615 | Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) β β’ πΆ = (BaseβπΎ) | ||
Theorem | thlbasOLD 21616 | Obsolete proof of thlbas 21615 as of 11-Nov-2024. Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) β β’ πΆ = (BaseβπΎ) | ||
Theorem | thlle 21617 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β€ = (leβπΌ) β β’ β€ = (leβπΎ) | ||
Theorem | thlleOLD 21618 | Obsolete proof of thlle 21617 as of 11-Nov-2024. Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ πΌ = (toIncβπΆ) & β’ β€ = (leβπΌ) β β’ β€ = (leβπΎ) | ||
Theorem | thlleval 21619 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ πΆ = (ClSubSpβπ) & β’ β€ = (leβπΎ) β β’ ((π β πΆ β§ π β πΆ) β (π β€ π β π β π)) | ||
Theorem | thloc 21620 | Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
β’ πΎ = (toHLβπ) & β’ β₯ = (ocvβπ) β β’ β₯ = (ocβπΎ) | ||
Syntax | cpj 21621 | Extend class notation with orthogonal projection function. |
class proj | ||
Syntax | chil 21622 | Extend class notation with class of all Hilbert spaces. |
class Hil | ||
Syntax | cobs 21623 | Extend class notation with the set of orthonormal bases. |
class OBasis | ||
Definition | df-pj 21624* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19583, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ proj = (β β V β¦ ((π₯ β (LSubSpββ) β¦ (π₯(proj1ββ)((ocvββ)βπ₯))) β© (V Γ ((Baseββ) βm (Baseββ))))) | ||
Definition | df-hil 21625 | Define class of all Hilbert spaces. Based on Proposition 4.5, p. 176, Gudrun Kalmbach, Quantum Measures and Spaces, Kluwer, Dordrecht, 1998. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 16-Oct-2015.) |
β’ Hil = {β β PreHil β£ dom (projββ) = (ClSubSpββ)} | ||
Definition | df-obs 21626* | Define the set of all orthonormal bases for a pre-Hilbert space. An orthonormal basis is a set of mutually orthogonal vectors with norm 1 and such that the linear span is dense in the whole space. (As this is an "algebraic" definition, before we have topology available, we express this denseness by saying that the double orthocomplement is the whole space, or equivalently, the single orthocomplement is trivial.) (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ OBasis = (β β PreHil β¦ {π β π« (Baseββ) β£ (βπ₯ β π βπ¦ β π (π₯(Β·πββ)π¦) = if(π₯ = π¦, (1rβ(Scalarββ)), (0gβ(Scalarββ))) β§ ((ocvββ)βπ) = {(0gββ)})}) | ||
Theorem | pjfval 21627* | The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ πΎ = ((π₯ β πΏ β¦ (π₯π( β₯ βπ₯))) β© (V Γ (π βm π))) | ||
Theorem | pjdm 21628 | A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ (π β dom πΎ β (π β πΏ β§ (ππ( β₯ βπ)):πβΆπ)) | ||
Theorem | pjpm 21629 | The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ πΎ = (projβπ) β β’ πΎ β ((π βm π) βpm πΏ) | ||
Theorem | pjfval2 21630* | Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ πΎ = (π₯ β dom πΎ β¦ (π₯π( β₯ βπ₯))) | ||
Theorem | pjval 21631 | Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (proj1βπ) & β’ πΎ = (projβπ) β β’ (π β dom πΎ β (πΎβπ) = (ππ( β₯ βπ))) | ||
Theorem | pjdm2 21632 | A subspace is in the domain of the projection function iff the subspace admits a projection decomposition of the whole space. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΏ = (LSubSpβπ) & β’ β₯ = (ocvβπ) & β’ β = (LSSumβπ) & β’ πΎ = (projβπ) β β’ (π β PreHil β (π β dom πΎ β (π β πΏ β§ (π β ( β₯ βπ)) = π))) | ||
Theorem | pjff 21633 | A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) β β’ (π β PreHil β πΎ:dom πΎβΆ(π LMHom π)) | ||
Theorem | pjf 21634 | A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ (π β dom πΎ β (πΎβπ):πβΆπ) | ||
Theorem | pjf2 21635 | A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβΆπ) | ||
Theorem | pjfo 21636 | A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ π = (Baseβπ) β β’ ((π β PreHil β§ π β dom πΎ) β (πΎβπ):πβontoβπ) | ||
Theorem | pjcss 21637 | A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β dom πΎ β πΆ) | ||
Theorem | ocvpj 21638 | The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ πΎ = (projβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β dom πΎ) β ( β₯ βπ) β dom πΎ) | ||
Theorem | ishil 21639 | The predicate "is a Hilbert space" (over a *-division ring). A Hilbert space is a pre-Hilbert space such that all closed subspaces have a projection decomposition. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
β’ πΎ = (projβπ») & β’ πΆ = (ClSubSpβπ») β β’ (π» β Hil β (π» β PreHil β§ dom πΎ = πΆ)) | ||
Theorem | ishil2 21640* | The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.) |
β’ π = (Baseβπ») & β’ β = (LSSumβπ») & β’ β₯ = (ocvβπ») & β’ πΆ = (ClSubSpβπ») β β’ (π» β Hil β (π» β PreHil β§ βπ β πΆ (π β ( β₯ βπ )) = π)) | ||
Theorem | isobs 21641* | The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) & β’ β₯ = (ocvβπ) & β’ π = (0gβπ) β β’ (π΅ β (OBasisβπ) β (π β PreHil β§ π΅ β π β§ (βπ₯ β π΅ βπ¦ β π΅ (π₯ , π¦) = if(π₯ = π¦, 1 , 0 ) β§ ( β₯ βπ΅) = {π}))) | ||
Theorem | obsip 21642 | The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) & β’ 0 = (0gβπΉ) β β’ ((π΅ β (OBasisβπ) β§ π β π΅ β§ π β π΅) β (π , π) = if(π = π, 1 , 0 )) | ||
Theorem | obsipid 21643 | A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ , = (Β·πβπ) & β’ πΉ = (Scalarβπ) & β’ 1 = (1rβπΉ) β β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β (π΄ , π΄) = 1 ) | ||
Theorem | obsrcl 21644 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ (π΅ β (OBasisβπ) β π β PreHil) | ||
Theorem | obsss 21645 | An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) β β’ (π΅ β (OBasisβπ) β π΅ β π) | ||
Theorem | obsne0 21646 | A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ 0 = (0gβπ) β β’ ((π΅ β (OBasisβπ) β§ π΄ β π΅) β π΄ β 0 ) | ||
Theorem | obsocv 21647 | 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 21648 | 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 21649 | 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 21650 | 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 21651 | 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 20167), but the existence of a unity element is always assumed (our rings are unital, see df-ring 20166). 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 21652 | Class of module direct sum generator. |
class βm | ||
Definition | df-dsmm 21653* | 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 21654 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
β’ Rel dom βm | ||
Theorem | dsmmval 21655* | 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 21656* | 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 21657 | 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 21658* | Base set of the direct sum module using the fndmin 7048 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 21659 | 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 21660* | 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 21661 | 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 21662 | 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 21663 | 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 21664 | 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 21665* | 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 21666* | 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 21668. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 21668 (see lmisfree 21763), the two definitions are essentially equivalent. The free modules as defined by df-frlm 21668 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 21667 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 21668* | 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 21653 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 21669 | 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 21670 | 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 21671 | 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 21672 | 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 21673 | 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 21674 | 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 21675 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21672). (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 21676* | 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 21677 | 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 21678 | 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 21679 | 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 21680 | 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 21681 | 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 21682 | 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 21683 | 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 21684 | 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 21685 | 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 21686 | 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 21687 | 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 21688 | 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 21689 | 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 21690* | 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 21691* | 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 21692* | 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 21693* | 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 21694* | 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 21695* | 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 21696* | 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 21697 | 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 21698* | 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 21699* | 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 21700 | 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))) |
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