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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | ocvval 20601* | Value of the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐹) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ (𝑆 ⊆ 𝑉 → ( ⊥ ‘𝑆) = {𝑥 ∈ 𝑉 ∣ ∀𝑦 ∈ 𝑆 (𝑥 , 𝑦) = 0 }) | ||
Theorem | elocv 20602* | Elementhood in the orthocomplement of a subset (normally a subspace) of a pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐹) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ (𝐴 ∈ ( ⊥ ‘𝑆) ↔ (𝑆 ⊆ 𝑉 ∧ 𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝑆 (𝐴 , 𝑥) = 0 )) | ||
Theorem | ocvi 20603 | Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 0 = (0g‘𝐹) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ((𝐴 ∈ ( ⊥ ‘𝑆) ∧ 𝐵 ∈ 𝑆) → (𝐴 , 𝐵) = 0 ) | ||
Theorem | ocvss 20604 | The orthocomplement of a subset is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ( ⊥ ‘𝑆) ⊆ 𝑉 | ||
Theorem | ocvocv 20605 | A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → 𝑆 ⊆ ( ⊥ ‘( ⊥ ‘𝑆))) | ||
Theorem | ocvlss 20606 | The orthocomplement of a subset is a linear subspace of the pre-Hilbert space. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝐿 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘𝑆) ∈ 𝐿) | ||
Theorem | ocv2ss 20607 | Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ (𝑇 ⊆ 𝑆 → ( ⊥ ‘𝑆) ⊆ ( ⊥ ‘𝑇)) | ||
Theorem | ocvin 20608 | An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝐿 = (LSubSp‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) | ||
Theorem | ocvsscon 20609 | Two ways to say that 𝑆 and 𝑇 are orthogonal subspaces. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉 ∧ 𝑇 ⊆ 𝑉) → (𝑆 ⊆ ( ⊥ ‘𝑇) ↔ 𝑇 ⊆ ( ⊥ ‘𝑆))) | ||
Theorem | ocvlsp 20610 | The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝑁 = (LSpan‘𝑊) ⇒ ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ 𝑉) → ( ⊥ ‘(𝑁‘𝑆)) = ( ⊥ ‘𝑆)) | ||
Theorem | ocv0 20611 | The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ( ⊥ ‘∅) = 𝑉 | ||
Theorem | ocvz 20612 | The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘{ 0 }) = 𝑉) | ||
Theorem | ocv1 20613 | The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘𝑉) = { 0 }) | ||
Theorem | unocv 20614 | The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ( ⊥ ‘(𝐴 ∪ 𝐵)) = (( ⊥ ‘𝐴) ∩ ( ⊥ ‘𝐵)) | ||
Theorem | iunocv 20615* | The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ ⊥ = (ocv‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ( ⊥ ‘∪ 𝑥 ∈ 𝐴 𝐵) = (𝑉 ∩ ∩ 𝑥 ∈ 𝐴 ( ⊥ ‘𝐵)) | ||
Theorem | cssval 20616* | The set of closed subspaces 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 → 𝐶 = {𝑠 ∣ 𝑠 = ( ⊥ ‘( ⊥ ‘𝑠))}) | ||
Theorem | iscss 20617 | 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 20618 | 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 20619 | 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 20620 | 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 20621 | 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 20622 | 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 20623 | 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 20624 | 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 20625 | 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 20626 | 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 20627 | 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 17064: 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 17129. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐶 = (ClSubSp‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉)) | ||
Theorem | mrccss 20628 | 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 20629 | 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 20630 | 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 20631 | 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 20632 | 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 20633 | Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.) |
⊢ 𝐾 = (toHL‘𝑊) & ⊢ ⊥ = (ocv‘𝑊) ⇒ ⊢ ⊥ = (oc‘𝐾) | ||
Syntax | cpj 20634 | Extend class notation with orthogonal projection function. |
class proj | ||
Syntax | chil 20635 | Extend class notation with class of all Hilbert spaces. |
class Hil | ||
Syntax | cobs 20636 | Extend class notation with the set of orthonormal bases. |
class OBasis | ||
Definition | df-pj 20637* | Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18998, 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 20638 | 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 20639* | 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 20640* | 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 20641 | 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 20642 | 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 20643* | 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 20644 | 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 20645 | 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 20646 | 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 20647 | 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 20648 | 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 20649 | 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 20650 | 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 20651 | 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 20652 | 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 20653* | 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 20654* | 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 20655 | 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 20656 | 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 20657 | Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ (𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil) | ||
Theorem | obsss 20658 | 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 20659 | A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.) |
⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴 ∈ 𝐵) → 𝐴 ≠ 0 ) | ||
Theorem | obsocv 20660 | 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 20661 | 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 20662 | 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 20663 | 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 20664 | 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 19537), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring 19536). 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 20665 | Class of module direct sum generator. |
class ⊕m | ||
Definition | df-dsmm 20666* | 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 20667 | The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.) |
⊢ Rel dom ⊕m | ||
Theorem | dsmmval 20668* | 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 20669* | 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 20670 | 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 20671* | Base set of the direct sum module using the fndmin 6854 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 20672 | 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 20673* | 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 20674 | 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 20675 | 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 20676 | 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 20677 | 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 20678* | 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 20679* | 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 20681. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20681 (see lmisfree 20776), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20681 are also taken as a motivation to introduce free modules by [Lang] p. 135. | ||
Syntax | cfrlm 20680 | Class of free module generator. |
class freeLMod | ||
Definition | df-frlm 20681* | 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 20666 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 20682 | 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 20683 | 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 20684 | 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 20685 | 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 20686 | 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 20687 | 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 20688 | Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20685). (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 20689* | 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 20690 | 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 20691 | 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 20692 | 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 20693 | 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 20694 | 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 20695 | 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 20696 | 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 20697 | 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 20698 | 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 20699 | 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 20700 | 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)) |
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