P. 212 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	thlbasOLD 21101	Obsolete proof of thlbas 21100 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.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ 𝐶 = (Base‘𝐾)
Theorem	thlle 21102	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) (Proof shortened by AV, 11-Nov-2024.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ≤ = (le‘𝐾)
Theorem	thlleOLD 21103	Obsolete proof of thlle 21102 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.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ 𝐼 = (toInc‘𝐶)    &   ⊢ ≤ = (le‘𝐼)    ⇒   ⊢ ≤ = (le‘𝐾)
Theorem	thlleval 21104	Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    &   ⊢ ≤ = (le‘𝐾)    ⇒   ⊢ ((𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐶) → (𝑆 ≤ 𝑇 ↔ 𝑆 ⊆ 𝑇))
Theorem	thloc 21105	Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
⊢ 𝐾 = (toHL‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ⊥ = (oc‘𝐾)
10.9.3  Orthogonal projection and orthonormal bases
Syntax	cpj 21106	Extend class notation with orthogonal projection function.
class proj
Syntax	chil 21107	Extend class notation with class of all Hilbert spaces.
class Hil
Syntax	cobs 21108	Extend class notation with the set of orthonormal bases.
class OBasis
Definition	df-pj 21109*	Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 19419, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ proj = (ℎ ∈ V ↦ ((𝑥 ∈ (LSubSp‘ℎ) ↦ (𝑥(proj1‘ℎ)((ocv‘ℎ)‘𝑥))) ∩ (V × ((Base‘ℎ) ↑m (Base‘ℎ)))))
Definition	df-hil 21110	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.)
⊢ Hil = {ℎ ∈ PreHil ∣ dom (proj‘ℎ) = (ClSubSp‘ℎ)}
Definition	df-obs 21111*	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.)
⊢ OBasis = (ℎ ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘ℎ) ∣ (∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 (𝑥(·𝑖‘ℎ)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘ℎ)), (0g‘(Scalar‘ℎ))) ∧ ((ocv‘ℎ)‘𝑏) = {(0g‘ℎ)})})
Theorem	pjfval 21112*	The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 = ((𝑥 ∈ 𝐿 ↦ (𝑥𝑃( ⊥ ‘𝑥))) ∩ (V × (𝑉 ↑m 𝑉)))
Theorem	pjdm 21113	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.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇𝑃( ⊥ ‘𝑇)):𝑉⟶𝑉))
Theorem	pjpm 21114	The projection map is a partial function from subspaces of the pre-Hilbert space to total operators. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 ∈ ((𝑉 ↑m 𝑉) ↑pm 𝐿)
Theorem	pjfval2 21115*	Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ 𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( ⊥ ‘𝑥)))
Theorem	pjval 21116	Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑃 = (proj1‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇) = (𝑇𝑃( ⊥ ‘𝑇)))
Theorem	pjdm2 21117	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.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝐿 = (LSubSp‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ ⊕ = (LSSum‘𝑊)    &   ⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → (𝑇 ∈ dom 𝐾 ↔ (𝑇 ∈ 𝐿 ∧ (𝑇 ⊕ ( ⊥ ‘𝑇)) = 𝑉)))
Theorem	pjff 21118	A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → 𝐾:dom 𝐾⟶(𝑊 LMHom 𝑊))
Theorem	pjf 21119	A projection is a function on the base set. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝑇 ∈ dom 𝐾 → (𝐾‘𝑇):𝑉⟶𝑉)
Theorem	pjf2 21120	A projection is a function from the base set to the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉⟶𝑇)
Theorem	pjfo 21121	A projection is a surjection onto the subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → (𝐾‘𝑇):𝑉–onto→𝑇)
Theorem	pjcss 21122	A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝑊 ∈ PreHil → dom 𝐾 ⊆ 𝐶)
Theorem	ocvpj 21123	The orthocomplement of a projection subspace is a projection subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
⊢ 𝐾 = (proj‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝑊 ∈ PreHil ∧ 𝑇 ∈ dom 𝐾) → ( ⊥ ‘𝑇) ∈ dom 𝐾)
Theorem	ishil 21124	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.)
⊢ 𝐾 = (proj‘𝐻)    &   ⊢ 𝐶 = (ClSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))
Theorem	ishil2 21125*	The predicate "is a Hilbert space" (over a *-division ring). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 22-Jun-2014.)
⊢ 𝑉 = (Base‘𝐻)    &   ⊢ ⊕ = (LSSum‘𝐻)    &   ⊢ ⊥ = (ocv‘𝐻)    &   ⊢ 𝐶 = (ClSubSp‘𝐻)    ⇒   ⊢ (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠 ∈ 𝐶 (𝑠 ⊕ ( ⊥ ‘𝑠)) = 𝑉))
Theorem	isobs 21126*	The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    &   ⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑌 = (0g‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵 ⊆ 𝑉 ∧ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( ⊥ ‘𝐵) = {𝑌})))
Theorem	obsip 21127	The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    &   ⊢ 0 = (0g‘𝐹)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃 ∈ 𝐵 ∧ 𝑄 ∈ 𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))
Theorem	obsipid 21128	A basis element has length one. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ , = (·𝑖‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 1 = (1r‘𝐹)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → (𝐴 , 𝐴) = 1 )
Theorem	obsrcl 21129	Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)
Theorem	obsss 21130	An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝐵 ⊆ 𝑉)
Theorem	obsne0 21131	A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 )
Theorem	obsocv 21132	An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ 0 = (0g‘𝑊)    &   ⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘𝐵) = { 0 })
Theorem	obs2ocv 21133	The double orthocomplement (closure) of an orthonormal basis is the whole space. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    &   ⊢ 𝑉 = (Base‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → ( ⊥ ‘( ⊥ ‘𝐵)) = 𝑉)
Theorem	obselocv 21134	A basis element is in the orthocomplement of a subset of the basis iff it is not in the subset. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ⊥ = (ocv‘𝑊)    ⇒   ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵 ∧ 𝐴 ∈ 𝐵) → (𝐴 ∈ ( ⊥ ‘𝐶) ↔ ¬ 𝐴 ∈ 𝐶))
Theorem	obs2ss 21135	A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶 ⊆ 𝐵) → 𝐶 = 𝐵)
Theorem	obslbs 21136	An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.)
⊢ 𝐽 = (LBasis‘𝑊)    &   ⊢ 𝑁 = (LSpan‘𝑊)    &   ⊢ 𝐶 = (ClSubSp‘𝑊)    ⇒   ⊢ (𝐵 ∈ (OBasis‘𝑊) → (𝐵 ∈ 𝐽 ↔ (𝑁‘𝐵) ∈ 𝐶))
PART 11  BASIC LINEAR ALGEBRA 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 19967), but the existence of a unity element is always assumed (our rings are unital, see df-ring 19966). 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.
11.1  Vectors and free modules
11.1.1  Direct sum of left modules According to Wikipedia ("Direct sum of modules", 28-Mar-2019, https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and { Mi: i ∈ I } a family of left R-modules indexed by the set I. The direct sum of {Mi} is then defined to be the set of all sequences (αi) where αi ∈ Mi and αi = 0 for cofinitely many indices i. (The direct product is analogous but the indices do not need to cofinitely vanish.)". In this definition, "cofinitely many" means "almost all" or "for all but finitely many". Furthemore, "This set inherits the module structure via componentwise addition and scalar multiplication. Explicitly, two such sequences α and β can be added by writing (α + β)i = αi + βi for all i (note that this is again zero for all but finitely many indices), and such a sequence can be multiplied with an element r from R by defining r(α)i = (rα)i for all i.". In [Lang] p. 128, the definition of the direct sum of left modules is based on direct sums of abelian groups ("We define on [the direct sum of abelian groups Mi] M a structure of A-module: If (xi)i ∈ I is an element of M, i.e. a familiy of elements xi ∈ Mi such that xi = 0 for almost all i, and if a ∈ A, then we define a(xi)i ∈ I = (axi)i ∈ I, that is we define multiplication by a componentwise.") which itself is based on the direct product of abelian groups ([Lang] p. 36: "Let {Ai}i ∈ I be a family of abelian groups. We define their direct sum A ... to be the subset of the direct product ... consisting of all families (xi)i ∈ I with xi ∈ Ai such that xi = 0 for all but a finite number of indices i"). In short, the direct sum of a familiy of (left) modules {Mi}i ∈ I is the restriction of the direct product of {Mi}i ∈ I to the elements with index function having finite support, as formalized by Definition df-dsmm 21138.
Syntax	cdsmm 21137	Class of module direct sum generator.
class ⊕m
Definition	df-dsmm 21138*	The direct sum of a family of Abelian groups or left modules is the induced group structure on finite linear combinations of elements, here represented as functions with finite support. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ ⊕m = (𝑠 ∈ V, 𝑟 ∈ V ↦ ((𝑠Xs𝑟) ↾s {𝑓 ∈ X𝑥 ∈ dom 𝑟(Base‘(𝑟‘𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑟‘𝑥))} ∈ Fin}))
Theorem	reldmdsmm 21139	The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ Rel dom ⊕m
Theorem	dsmmval 21140*	Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}    ⇒   ⊢ (𝑅 ∈ 𝑉 → (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵))
Theorem	dsmmbase 21141*	Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
⊢ 𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓‘𝑥) ≠ (0g‘(𝑅‘𝑥))} ∈ Fin}    ⇒   ⊢ (𝑅 ∈ 𝑉 → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))
Theorem	dsmmval2 21142	Self-referential definition of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝐵 = (Base‘(𝑆 ⊕m 𝑅))    ⇒   ⊢ (𝑆 ⊕m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)
Theorem	dsmmbas2 21143*	Base set of the direct sum module using the fndmin 6995 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g ∘ 𝑅)) ∈ Fin}    ⇒   ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ 𝑉) → 𝐵 = (Base‘(𝑆 ⊕m 𝑅)))
Theorem	dsmmfi 21144	For finite products, the direct sum is just the module product. See also the observation in [Lang] p. 129. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → (𝑆 ⊕m 𝑅) = (𝑆Xs𝑅))
Theorem	dsmmelbas 21145*	Membership in the finitely supported hull of a structure product in terms of the index set. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐶 = (𝑆 ⊕m 𝑅)    &   ⊢ 𝐵 = (Base‘𝑃)    &   ⊢ 𝐻 = (Base‘𝐶)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅 Fn 𝐼)    ⇒   ⊢ (𝜑 → (𝑋 ∈ 𝐻 ↔ (𝑋 ∈ 𝐵 ∧ {𝑎 ∈ 𝐼 ∣ (𝑋‘𝑎) ≠ (0g‘(𝑅‘𝑎))} ∈ Fin)))
Theorem	dsmm0cl 21146	The all-zero vector is contained in the finite hull, since its support is empty and therefore finite. This theorem along with the next one effectively proves that the finite hull is a "submonoid", although that does not exist as a defined concept yet. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ 0 = (0g‘𝑃)    ⇒   ⊢ (𝜑 → 0 ∈ 𝐻)
Theorem	dsmmacl 21147	The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Mnd)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐻)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐻)    &   ⊢ + = (+g‘𝑃)    ⇒   ⊢ (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)
Theorem	prdsinvgd2 21148	Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑌 = (𝑆Xs𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝑁 = (invg‘𝑌)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑁‘𝑋)‘𝐽) = ((invg‘(𝑅‘𝐽))‘(𝑋‘𝐽)))
Theorem	dsmmsubg 21149	The finite hull of a product of groups is additionally closed under negation and thus is a subgroup of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ 𝑉)    &   ⊢ (𝜑 → 𝑅:𝐼⟶Grp)    ⇒   ⊢ (𝜑 → 𝐻 ∈ (SubGrp‘𝑃))
Theorem	dsmmlss 21150*	The finite hull of a product of modules is additionally closed under scalar multiplication and thus is a linear subspace of the product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → 𝑅:𝐼⟶LMod)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆)    &   ⊢ 𝑃 = (𝑆Xs𝑅)    &   ⊢ 𝑈 = (LSubSp‘𝑃)    &   ⊢ 𝐻 = (Base‘(𝑆 ⊕m 𝑅))    ⇒   ⊢ (𝜑 → 𝐻 ∈ 𝑈)
Theorem	dsmmlmod 21151*	The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.)
⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑆 ∈ Ring)    &   ⊢ (𝜑 → 𝑅:𝐼⟶LMod)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆)    &   ⊢ 𝐶 = (𝑆 ⊕m 𝑅)    ⇒   ⊢ (𝜑 → 𝐶 ∈ LMod)
11.1.2  Free modules According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist 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 21153. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 21153 (see lmisfree 21248), the two definitions are essentially equivalent. The free modules as defined by df-frlm 21153 are also taken as a motivation to introduce free modules by [Lang] p. 135.
Syntax	cfrlm 21152	Class of free module generator.
class freeLMod
Definition	df-frlm 21153*	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 21138 that an element of a direct sum has finitely many nonzero coordinates. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟 ⊕m (𝑖 × {(ringLMod‘𝑟)})))
Theorem	frlmval 21154	Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (𝑅 ⊕m (𝐼 × {(ringLMod‘𝑅)})))
Theorem	frlmlmod 21155	The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LMod)
Theorem	frlmpws 21156	The free module as a restriction of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))
Theorem	frlmlss 21157	The base set of the free module is a subspace of the power module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝑈 = (LSubSp‘((ringLMod‘𝑅) ↑s 𝐼))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝐵 ∈ 𝑈)
Theorem	frlmpwsfi 21158	The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼))
Theorem	frlmsca 21159	The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑅 = (Scalar‘𝐹))
Theorem	frlm0 21160	Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 21157). (Contributed by Stefan O'Rear, 4-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → (𝐼 × { 0 }) = (0g‘𝐹))
Theorem	frlmbas 21161*	Base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = {𝑘 ∈ (𝑁 ↑m 𝐼) ∣ 𝑘 finSupp 0 }    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝐵 = (Base‘𝐹))
Theorem	frlmelbas 21162	Membership in the base set of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → (𝑋 ∈ 𝐵 ↔ (𝑋 ∈ (𝑁 ↑m 𝐼) ∧ 𝑋 finSupp 0 )))
Theorem	frlmrcl 21163	If a free module is inhabited, this is sufficient to conclude that the ring expression defines a set. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ (𝑋 ∈ 𝐵 → 𝑅 ∈ V)
Theorem	frlmbasfsupp 21164	Elements of the free module are finitely supported. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Revised by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 20-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 finSupp 0 )
Theorem	frlmbasmap 21165	Elements of the free module are set functions. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 ∈ (𝑁 ↑m 𝐼))
Theorem	frlmbasf 21166	Elements of the free module are functions. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋:𝐼⟶𝑁)
Theorem	frlmlvec 21167	The free module over a division ring is a left vector space. (Contributed by Steven Nguyen, 29-Apr-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    ⇒   ⊢ ((𝑅 ∈ DivRing ∧ 𝐼 ∈ 𝑊) → 𝐹 ∈ LVec)
Theorem	frlmfibas 21168	The base set of the finite free module as a set exponential. (Contributed by AV, 6-Dec-2018.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ Fin) → (𝑁 ↑m 𝐼) = (Base‘𝐹))
Theorem	elfrlmbasn0 21169	If the dimension of a free module over a ring is not 0, every element of its base set is not empty. (Contributed by AV, 10-Feb-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑁 = (Base‘𝑅)    &   ⊢ 𝐵 = (Base‘𝐹)    ⇒   ⊢ ((𝐼 ∈ 𝑉 ∧ 𝐼 ≠ ∅) → (𝑋 ∈ 𝐵 → 𝑋 ≠ ∅))
Theorem	frlmplusgval 21170	Addition in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹 ✚ 𝐺) = (𝐹 ∘f + 𝐺))
Theorem	frlmsubgval 21171	Subtraction in a free module. (Contributed by Thierry Arnoux, 30-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐹 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐺 ∈ 𝐵)    &   ⊢ − = (-g‘𝑅)    &   ⊢ 𝑀 = (-g‘𝑌)    ⇒   ⊢ (𝜑 → (𝐹𝑀𝐺) = (𝐹 ∘f − 𝐺))
Theorem	frlmvscafval 21172	Scalar multiplication in a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Stefan O'Rear, 6-May-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ ∙ = ( ·𝑠 ‘𝑌)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → (𝐴 ∙ 𝑋) = ((𝐼 × {𝐴}) ∘f · 𝑋))
Theorem	frlmvplusgvalc 21173	Coordinates of a sum with respect to a basis in a free module. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    ⇒   ⊢ (𝜑 → ((𝑋 ✚ 𝑌)‘𝐽) = ((𝑋‘𝐽) + (𝑌‘𝐽)))
Theorem	frlmvscaval 21174	Coordinates of a scalar multiple with respect to a basis in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ ∙ = ( ·𝑠 ‘𝑌)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝐴 ∙ 𝑋)‘𝐽) = (𝐴 · (𝑋‘𝐽)))
Theorem	frlmplusgvalb 21175*	Addition in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    ⇒   ⊢ (𝜑 → (𝑍 = (𝑋 ✚ 𝑌) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝑋‘𝑖) + (𝑌‘𝑖))))
Theorem	frlmvscavalb 21176*	Scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ (𝜑 → (𝑍 = (𝐴 ∙ 𝑋) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = (𝐴 · (𝑋‘𝑖))))
Theorem	frlmvplusgscavalb 21177*	Addition combined with scalar multiplication in a free module at the coordinates. (Contributed by AV, 16-Jan-2023.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑍 ∈ 𝐵)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐾)    &   ⊢ ∙ = ( ·𝑠 ‘𝐹)    &   ⊢ · = (.r‘𝑅)    &   ⊢ (𝜑 → 𝑌 ∈ 𝐵)    &   ⊢ + = (+g‘𝑅)    &   ⊢ ✚ = (+g‘𝐹)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑍 = ((𝐴 ∙ 𝑋) ✚ (𝐶 ∙ 𝑌)) ↔ ∀𝑖 ∈ 𝐼 (𝑍‘𝑖) = ((𝐴 · (𝑋‘𝑖)) + (𝐶 · (𝑌‘𝑖)))))
Theorem	frlmgsum 21178*	Finite commutative sums in a free module are taken componentwise. (Contributed by Stefan O'Rear, 1-Feb-2015.) (Revised by Mario Carneiro, 5-Jul-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑌)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ∈ 𝑊)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐽) → (𝑥 ∈ 𝐼 ↦ 𝑈) ∈ 𝐵)    &   ⊢ (𝜑 → (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈)) finSupp 0 )    ⇒   ⊢ (𝜑 → (𝑌 Σg (𝑦 ∈ 𝐽 ↦ (𝑥 ∈ 𝐼 ↦ 𝑈))) = (𝑥 ∈ 𝐼 ↦ (𝑅 Σg (𝑦 ∈ 𝐽 ↦ 𝑈))))
Theorem	frlmsplit2 21179*	Restriction is homomorphic on free modules. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝑈)    &   ⊢ 𝑍 = (𝑅 freeLMod 𝑉)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 𝐶 = (Base‘𝑍)    &   ⊢ 𝐹 = (𝑥 ∈ 𝐵 ↦ (𝑥 ↾ 𝑉))    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝑈 ∈ 𝑋 ∧ 𝑉 ⊆ 𝑈) → 𝐹 ∈ (𝑌 LMHom 𝑍))
Theorem	frlmsslss 21180*	A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of forbidden unit vectors. (Contributed by Stefan O'Rear, 4-Feb-2015.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (LSubSp‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 ↾ 𝐽) = (𝐽 × { 0 })}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)
Theorem	frlmsslss2 21181*	A subset of a free module obtained by restricting the support set is a submodule. 𝐽 is the set of permitted unit vectors. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 23-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (LSubSp‘𝑌)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑉 ∧ 𝐽 ⊆ 𝐼) → 𝐶 ∈ 𝑈)
Theorem	frlmbas3 21182	An element of the base set of a finite free module with a Cartesian product as index set as operation value. (Contributed by AV, 14-Feb-2019.)
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ 𝑉 = (Base‘𝐹)    ⇒   ⊢ (((𝑅 ∈ 𝑊 ∧ 𝑋 ∈ 𝑉) ∧ (𝑁 ∈ Fin ∧ 𝑀 ∈ Fin) ∧ (𝐼 ∈ 𝑁 ∧ 𝐽 ∈ 𝑀)) → (𝐼𝑋𝐽) ∈ 𝐵)
Theorem	mpofrlmd 21183*	Elements of the free module are mappings with two arguments defined by their operation values. (Contributed by AV, 20-Feb-2019.) (Proof shortened by AV, 3-Jul-2022.)
⊢ 𝐹 = (𝑅 freeLMod (𝑁 × 𝑀))    &   ⊢ 𝑉 = (Base‘𝐹)    &   ⊢ ((𝑖 = 𝑎 ∧ 𝑗 = 𝑏) → 𝐴 = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑁 ∧ 𝑗 ∈ 𝑀) → 𝐴 ∈ 𝑋)    &   ⊢ ((𝜑 ∧ 𝑎 ∈ 𝑁 ∧ 𝑏 ∈ 𝑀) → 𝐵 ∈ 𝑌)    &   ⊢ (𝜑 → (𝑁 ∈ 𝑈 ∧ 𝑀 ∈ 𝑊 ∧ 𝑍 ∈ 𝑉))    ⇒   ⊢ (𝜑 → (𝑍 = (𝑎 ∈ 𝑁, 𝑏 ∈ 𝑀 ↦ 𝐵) ↔ ∀𝑖 ∈ 𝑁 ∀𝑗 ∈ 𝑀 (𝑖𝑍𝑗) = 𝐴))
Theorem	frlmip 21184*	The inner product of a free module. (Contributed by Thierry Arnoux, 20-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑉) → (𝑓 ∈ (𝐵 ↑m 𝐼), 𝑔 ∈ (𝐵 ↑m 𝐼) ↦ (𝑅 Σg (𝑥 ∈ 𝐼 ↦ ((𝑓‘𝑥) · (𝑔‘𝑥))))) = (·𝑖‘𝑌))
Theorem	frlmipval 21185	The inner product of a free module. (Contributed by Thierry Arnoux, 21-Jun-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    ⇒   ⊢ (((𝐼 ∈ 𝑊 ∧ 𝑅 ∈ 𝑋) ∧ (𝐹 ∈ 𝑉 ∧ 𝐺 ∈ 𝑉)) → (𝐹 , 𝐺) = (𝑅 Σg (𝐹 ∘f · 𝐺)))
Theorem	frlmphllem 21186*	Lemma for frlmphl 21187. (Contributed by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    &   ⊢ 𝑂 = (0g‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ ℎ ∈ 𝑉) → (𝑥 ∈ 𝐼 ↦ ((𝑔‘𝑥) · (ℎ‘𝑥))) finSupp 0 )
Theorem	frlmphl 21187*	Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.)
⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝑉 = (Base‘𝑌)    &   ⊢ , = (·𝑖‘𝑌)    &   ⊢ 𝑂 = (0g‘𝑌)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ ∗ = (*𝑟‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Field)    &   ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂)    &   ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    ⇒   ⊢ (𝜑 → 𝑌 ∈ PreHil)
11.1.3  Standard basis (unit vectors) According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces.
Syntax	cuvc 21188	Class of basic unit vectors for an explicit free module.
class unitVec
Definition	df-uvc 21189*	((𝑅 unitVec 𝐼)‘𝑗) is the unit vector in (𝑅 freeLMod 𝐼) along the 𝑗 axis. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ unitVec = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑗 ∈ 𝑖 ↦ (𝑘 ∈ 𝑖 ↦ if(𝑘 = 𝑗, (1r‘𝑟), (0g‘𝑟)))))
Theorem	uvcfval 21190*	Value of the unit-vector generator for a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊) → 𝑈 = (𝑗 ∈ 𝐼 ↦ (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝑗, 1 , 0 ))))
Theorem	uvcval 21191*	Value of a single unit vector in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ ((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) → (𝑈‘𝐽) = (𝑘 ∈ 𝐼 ↦ if(𝑘 = 𝐽, 1 , 0 )))
Theorem	uvcvval 21192	Value of a unit vector coordinate in a free module. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) = if(𝐾 = 𝐽, 1 , 0 ))
Theorem	uvcvvcl 21193	A coordinate of a unit vector is either 0 or 1. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 1 = (1r‘𝑅)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (((𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ∧ 𝐽 ∈ 𝐼) ∧ 𝐾 ∈ 𝐼) → ((𝑈‘𝐽)‘𝐾) ∈ { 0 , 1 })
Theorem	uvcvvcl2 21194	A unit vector coordinate is a ring element. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) ∈ 𝐵)
Theorem	uvcvv1 21195	The unit vector is one at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐽) = 1 )
Theorem	uvcvv0 21196	The unit vector is zero at its designated coordinate. (Contributed by Stefan O'Rear, 3-Feb-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ (𝜑 → 𝑅 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑊)    &   ⊢ (𝜑 → 𝐽 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐾 ∈ 𝐼)    &   ⊢ (𝜑 → 𝐽 ≠ 𝐾)    &   ⊢ 0 = (0g‘𝑅)    ⇒   ⊢ (𝜑 → ((𝑈‘𝐽)‘𝐾) = 0 )
Theorem	uvcff 21197	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.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼⟶𝐵)
Theorem	uvcf1 21198	In a nonzero ring, each unit vector is different. (Contributed by Stefan O'Rear, 7-Feb-2015.) (Revised by Mario Carneiro, 14-Jun-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    ⇒   ⊢ ((𝑅 ∈ NzRing ∧ 𝐼 ∈ 𝑊) → 𝑈:𝐼–1-1→𝐵)
Theorem	uvcresum 21199	Any element of a free module can be expressed as a finite linear combination of unit vectors. (Contributed by Stefan O'Rear, 3-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Jul-2015.)
⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝑌 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝐵 = (Base‘𝑌)    &   ⊢ · = ( ·𝑠 ‘𝑌)    ⇒   ⊢ ((𝑅 ∈ Ring ∧ 𝐼 ∈ 𝑊 ∧ 𝑋 ∈ 𝐵) → 𝑋 = (𝑌 Σg (𝑋 ∘f · 𝑈)))
Theorem	frlmssuvc1 21200*	A scalar multiple of a unit vector included in a support-restriction subspace is included in the subspace. (Contributed by Stefan O'Rear, 5-Feb-2015.) (Revised by AV, 24-Jun-2019.)
⊢ 𝐹 = (𝑅 freeLMod 𝐼)    &   ⊢ 𝑈 = (𝑅 unitVec 𝐼)    &   ⊢ 𝐵 = (Base‘𝐹)    &   ⊢ 𝐾 = (Base‘𝑅)    &   ⊢ · = ( ·𝑠 ‘𝐹)    &   ⊢ 0 = (0g‘𝑅)    &   ⊢ 𝐶 = {𝑥 ∈ 𝐵 ∣ (𝑥 supp 0 ) ⊆ 𝐽}    &   ⊢ (𝜑 → 𝑅 ∈ Ring)    &   ⊢ (𝜑 → 𝐼 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐽 ⊆ 𝐼)    &   ⊢ (𝜑 → 𝐿 ∈ 𝐽)    &   ⊢ (𝜑 → 𝑋 ∈ 𝐾)    ⇒   ⊢ (𝜑 → (𝑋 · (𝑈‘𝐿)) ∈ 𝐶)