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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ocvin 21001 | An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ πΏ = (LSubSpβπ) & β’ 0 = (0gβπ) β β’ ((π β PreHil β§ π β πΏ) β (π β© ( β₯ βπ)) = { 0 }) | ||
Theorem | ocvsscon 21002 | Two ways to say that π and π are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ((π β PreHil β§ π β π β§ π β π) β (π β ( β₯ βπ) β π β ( β₯ βπ))) | ||
Theorem | ocvlsp 21003 | The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ π = (LSpanβπ) β β’ ((π β PreHil β§ π β π) β ( β₯ β(πβπ)) = ( β₯ βπ)) | ||
Theorem | ocv0 21004 | The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) β β’ ( β₯ ββ ) = π | ||
Theorem | ocvz 21005 | The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β ( β₯ β{ 0 }) = π) | ||
Theorem | ocv1 21006 | The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ π = (Baseβπ) & β’ β₯ = (ocvβπ) & β’ 0 = (0gβπ) β β’ (π β PreHil β ( β₯ βπ) = { 0 }) | ||
Theorem | unocv 21007 | The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ β₯ = (ocvβπ) β β’ ( β₯ β(π΄ βͺ π΅)) = (( β₯ βπ΄) β© ( β₯ βπ΅)) | ||
Theorem | iunocv 21008* | The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ β₯ = (ocvβπ) & β’ π = (Baseβπ) β β’ ( β₯ ββͺ π₯ β π΄ π΅) = (π β© β© π₯ β π΄ ( β₯ βπ΅)) | ||
Theorem | cssval 21009* | 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 21010 | 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 21011 | 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 21012 | 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 21013 | 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 21014 | 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 21015 | 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 21016 | 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 21017 | 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 21018 | 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 21019 | 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 21020 | 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 17404: 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 17469. (Contributed by Mario Carneiro, 13-Oct-2015.) |
β’ π = (Baseβπ) & β’ πΆ = (ClSubSpβπ) β β’ (π β PreHil β πΆ β (Mooreβπ)) | ||
Theorem | mrccss 21021 | 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 21022 | 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 21023 | 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 21024 | Obsolete proof of thlbas 21023 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 21025 | 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 21026 | Obsolete proof of thlle 21025 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 21027 | 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 21028 | 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 21029 | Extend class notation with orthogonal projection function. |
class proj | ||
Syntax | chil 21030 | Extend class notation with class of all Hilbert spaces. |
class Hil | ||
Syntax | cobs 21031 | Extend class notation with the set of orthonormal bases. |
class OBasis | ||
Definition | df-pj 21032* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19348, 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 21033 | 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 21034* | 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 21035* | 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 21036 | 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 21037 | 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 21038* | 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 21039 | 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 21040 | 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 21041 | 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 21042 | 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 21043 | 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 21044 | 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 21045 | 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 21046 | 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 21047 | 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 21048* | 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 21049* | 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 21050 | 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 21051 | 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 21052 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
β’ (π΅ β (OBasisβπ) β π β PreHil) | ||
Theorem | obsss 21053 | 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 21054 | 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 21055 | 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 21056 | 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 21057 | 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 21058 | 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 21059 | 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 19891), but the existence of a unity element is always assumed (our rings are unital, see df-ring 19890). 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 21060 | Class of module direct sum generator. |
class βm | ||
Definition | df-dsmm 21061* | 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 21062 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
β’ Rel dom βm | ||
Theorem | dsmmval 21063* | 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 21064* | 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 21065 | 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 21066* | Base set of the direct sum module using the fndmin 6991 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 21067 | 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 21068* | 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 21069 | 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 21070 | 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 21071 | 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 21072 | 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 21073* | 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 21074* | 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 21076. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 21076 (see lmisfree 21171), the two definitions are essentially equivalent. The free modules as defined by df-frlm 21076 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 21075 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 21076* | 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 21061 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 21077 | 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 21078 | 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 21079 | 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 21080 | 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 21081 | 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 21082 | 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 21083 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21080). (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 21084* | 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 21085 | 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 21086 | 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 21087 | 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 21088 | 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 21089 | 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 21090 | 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 21091 | 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 21092 | 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 21093 | 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 21094 | 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 21095 | 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 21096 | 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 21097 | 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 21098* | 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 21099* | 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 21100* | 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))))) |
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