Home | Metamath
Proof Explorer Theorem List (p. 206 of 460) | < Previous Next > |
Bad symbols? Try the
GIF version. |
||
Mirrors > Metamath Home Page > MPE Home Page > Theorem List Contents > Recent Proofs This page: Page List |
Color key: | Metamath Proof Explorer
(1-28855) |
Hilbert Space Explorer
(28856-30378) |
Users' Mathboxes
(30379-45991) |
Type | Label | Description |
---|---|---|
Statement | ||
Theorem | iscss 20501 | The predicate "is a closed subspace" (of a pre-Hilbert space). (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑋 → (𝑆 ∈ 𝐶 ↔ 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆)))) | ||
Theorem | cssi 20502 | Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑆 ∈ 𝐶 → 𝑆 = ( ⊥ ‘( ⊥ ‘𝑆))) | ||
Theorem | cssss 20503 | A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑉) | ||
Theorem | iscss2 20504 | 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.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝑆 ∈ 𝐶 ↔ ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ 𝑆)) | ||
Theorem | ocvcss 20505 | The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐶) | ||
Theorem | cssincl 20506 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶) → (𝐴 ∩ 𝐵) ∈ 𝐶) | ||
Theorem | css0 20507 | The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → { 0 } ∈ 𝐶) | ||
Theorem | css1 20508 | The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → 𝑉 ∈ 𝐶) | ||
Theorem | csslss 20509 | A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ 𝐿) | ||
Theorem | lsmcss 20510 | 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.) |
⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ ⊕ = (LSSum‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝑆 ⊆ 𝑉) & ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑆)) ⊆ (𝑆 ⊕ ( ⊥ ‘𝑆))) ⇒ ⊢ (𝜑 → 𝑆 ∈ 𝐶) | ||
Theorem | cssmre 20511 | 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 16965: consider the Hilbert space of sequences ℕ⟶ℝ 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 17030. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) | ||
Theorem | mrccss 20512 | The Moore closure corresponding to the system of closed subspaces is the double orthocomplement operation. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝐹 = (mrCls‘𝐶) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → (𝐹‘𝑆) = ( ⊥ ‘( ⊥ ‘𝑆))) | ||
Theorem | thlval 20513 | Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐾 = (toHL‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝐼 = (toInc‘𝐶) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ (𝑊 ∈ 𝑉 → 𝐾 = (𝐼 sSet 〈(oc‘ndx), ⊥ 〉)) | ||
Theorem | thlbas 20514 | Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐾 = (toHL‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ 𝐶 = (Base‘𝐾) | ||
Theorem | thlle 20515 | Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐾 = (toHL‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) & ⊢ 𝐼 = (toInc‘𝐶) & ⊢ ≤ = (le‘𝐼) ⇒ ⊢ ≤ = (le‘𝐾) | ||
Theorem | thlleval 20516 | 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 20517 | Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐾 = (toHL‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ⊥ = (oc‘𝐾) | ||
Syntax | cpj 20518 | Extend class notation with orthogonal projection function. |
class proj | ||
Syntax | chil 20519 | Extend class notation with class of all Hilbert spaces. |
class Hil | ||
Syntax | cobs 20520 | Extend class notation with the set of orthonormal bases. |
class OBasis | ||
Definition | df-pj 20521* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18882, 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 20522 | 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 20523* | 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 20524* | 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 20525 | 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 20526 | 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 20527* | 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 20528 | 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 20529 | 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 20530 | 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 20531 | 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 20532 | 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 20533 | 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 20534 | 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 20535 | 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 20536 | 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 20537* | 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 20538* | 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 20539 | 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 20540 | A basis element has unit length. (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 20541 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | ||
Theorem | obsss 20542 | 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 20543 | A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 ) | ||
Theorem | obsocv 20544 | 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 20545 | 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 20546 | 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 20547 | 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 20548 | An orthogonal basis is a linear basis iff the span of the basis elements is closed (which is usually not true). (Contributed by Mario Carneiro, 29-Oct-2015.) |
⊢ 𝐽 = (LBasis‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝐵 ∈ (OBasis‘𝑊) → (𝐵 ∈ 𝐽 ↔ (𝑁‘𝐵) ∈ 𝐶)) | ||
According to Wikipedia ("Linear algebra", 03-Mar-2019, https://en.wikipedia.org/wiki/Linear_algebra) "Linear algebra is the branch of mathematics concerning linear equations [...], linear functions [...] and their representations through matrices and vector spaces." Or according to the Merriam-Webster dictionary ("linear algebra", 12-Mar-2019, https://www.merriam-webster.com/dictionary/linear%20algebra) "Definition of linear algebra: a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations." Dealing with modules (over rings) instead of vector spaces (over fields) allows for a more unified approach. Therefore, linear equations, matrices, determinants, are usually regarded as "over a ring" in this part. Unless otherwise stated, the rings of scalars need not be commutative (see df-cring 19421), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring 19420). For readers knowing vector spaces but unfamiliar with modules: the elements of a module are still called "vectors" and they still form a group under addition, with a zero vector as neutral element, like in a vector space. Like in a vector space, vectors can be multiplied by scalars, with the usual rules, the only difference being that the scalars are only required to form a ring, and not necessarily a field or a division ring. Note that any vector space is a (special kind of) module, so any theorem proved below for modules applies to any vector space. | ||
According to Wikipedia ("Direct sum of modules", 28-Mar-2019,
https://en.wikipedia.org/wiki/Direct_sum_of_modules) "Let R be a ring, and
{ Mi: i ∈ I } a family of left R-modules indexed by the set I.
The direct sum of {Mi} is then defined to be the set of all
sequences (αi) where αi ∈ Mi
and αi = 0 for cofinitely many indices i. (The direct product
is analogous but the indices do not need to cofinitely vanish.)". In this
definition, "cofinitely many" means "almost all" or "for all but finitely
many". Furthemore, "This set inherits the module structure via componentwise
addition and scalar multiplication. Explicitly, two such sequences α and
β can be added by writing (α + β)i =
αi + βi for all i (note that this is again
zero for all but finitely many indices), and such a sequence can be multiplied
with an element r from R by defining r(α)i =
(rα)i for all i.".
| ||
Syntax | cdsmm 20549 | Class of module direct sum generator. |
class ⊕m | ||
Definition | df-dsmm 20550* | 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 20551 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
⊢ Rel dom ⊕m | ||
Theorem | dsmmval 20552* | 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 20553* | 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 20554 | 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 20555* | Base set of the direct sum module using the fndmin 6824 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 20556 | 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 20557* | 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 20558 | 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 20559 | 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 20560 | 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 20561 | 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 20562* | 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 20563* | The direct sum of a family of modules is a module. See also the remark in [Lang] p. 128. (Contributed by Stefan O'Rear, 11-Jan-2015.) |
⊢ (𝜑 → 𝐼 ∈ 𝑊) & ⊢ (𝜑 → 𝑆 ∈ Ring) & ⊢ (𝜑 → 𝑅:𝐼⟶LMod) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (Scalar‘(𝑅‘𝑥)) = 𝑆) & ⊢ 𝐶 = (𝑆 ⊕m 𝑅) ⇒ ⊢ (𝜑 → 𝐶 ∈ LMod) | ||
According to Wikipedia ("Free module", 03-Mar-2019, https://en.wikipedia.org/wiki/Free_module) "In mathematics, a free module is a module that has a basis - that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist 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 20565. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20565 (see lmisfree 20660), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20565 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 20564 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 20565* | 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 20550 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 20566 | 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 20567 | 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 20568 | 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 20569 | 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 20570 | 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 20571 | 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 20572 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20569). (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 20573* | 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 20574 | 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 20575 | 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 20576 | 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 20577 | 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 20578 | 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 20579 | 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 20580 | 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 20581 | 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 20582 | 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 20583 | 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 20584 | 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 20585 | 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 20586 | 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 20587* | 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 20588* | 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 20589* | 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 20590* | 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 20591* | 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 20592* | 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 20593* | 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 20594 | 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 20595* | 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 20596* | 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 20597 | 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 20598* | Lemma for frlmphl 20599. (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 20599* | Conditions for a free module to be a pre-Hilbert space. (Contributed by Thierry Arnoux, 21-Jun-2019.) (Proof shortened by AV, 21-Jul-2019.) |
⊢ 𝑌 = (𝑅 freeLMod 𝐼) & ⊢ 𝐵 = (Base‘𝑅) & ⊢ · = (.r‘𝑅) & ⊢ 𝑉 = (Base‘𝑌) & ⊢ , = (·𝑖‘𝑌) & ⊢ 𝑂 = (0g‘𝑌) & ⊢ 0 = (0g‘𝑅) & ⊢ ∗ = (*𝑟‘𝑅) & ⊢ (𝜑 → 𝑅 ∈ Field) & ⊢ ((𝜑 ∧ 𝑔 ∈ 𝑉 ∧ (𝑔 , 𝑔) = 0 ) → 𝑔 = 𝑂) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( ∗ ‘𝑥) = 𝑥) & ⊢ (𝜑 → 𝐼 ∈ 𝑊) ⇒ ⊢ (𝜑 → 𝑌 ∈ PreHil) | ||
According to Wikipedia ("Standard basis", 16-Mar-2019, https://en.wikipedia.org/wiki/Standard_basis) "In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.", and ("Unit vector", 16-Mar-2019, https://en.wikipedia.org/wiki/Unit_vector) "In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.". In the following, the term "unit vector" (or more specific "basic unit vector") is used for the (special) unit vectors forming the standard basis of free modules. However, the length of the unit vectors is not considered here, so it is not required to regard normed spaces. | ||
Syntax | cuvc 20600 | Class of basic unit vectors for an explicit free module. |
class unitVec |
< Previous Next > |
Copyright terms: Public domain | < Previous Next > |