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Theorem List for Metamath Proof Explorer - 20401-20500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremipffn 20401 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)        , Fn (𝑉 × 𝑉)

Theoremphlipf 20402 The inner product operation is a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)    &   𝑆 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝑆)       (𝑊 ∈ PreHil → , :(𝑉 × 𝑉)⟶𝐾)

Theoremip2eq 20403* Two vectors are equal iff their inner products with all other vectors are equal. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 = 𝐵 ↔ ∀𝑥𝑉 (𝑥 , 𝐴) = (𝑥 , 𝐵)))

Theoremisphld 20404* Properties that determine a pre-Hilbert (inner product) space. (Contributed by Mario Carneiro, 18-Nov-2013.) (Revised by Mario Carneiro, 7-Oct-2015.)
(𝜑𝑉 = (Base‘𝑊))    &   (𝜑+ = (+g𝑊))    &   (𝜑· = ( ·𝑠𝑊))    &   (𝜑𝐼 = (·𝑖𝑊))    &   (𝜑0 = (0g𝑊))    &   (𝜑𝐹 = (Scalar‘𝑊))    &   (𝜑𝐾 = (Base‘𝐹))    &   (𝜑 = (+g𝐹))    &   (𝜑× = (.r𝐹))    &   (𝜑 = (*𝑟𝐹))    &   (𝜑𝑂 = (0g𝐹))    &   (𝜑𝑊 ∈ LVec)    &   (𝜑𝐹 ∈ *-Ring)    &   ((𝜑𝑥𝑉𝑦𝑉) → (𝑥𝐼𝑦) ∈ 𝐾)    &   ((𝜑𝑞𝐾 ∧ (𝑥𝑉𝑦𝑉𝑧𝑉)) → (((𝑞 · 𝑥) + 𝑦)𝐼𝑧) = ((𝑞 × (𝑥𝐼𝑧)) (𝑦𝐼𝑧)))    &   ((𝜑𝑥𝑉 ∧ (𝑥𝐼𝑥) = 𝑂) → 𝑥 = 0 )    &   ((𝜑𝑥𝑉𝑦𝑉) → ( ‘(𝑥𝐼𝑦)) = (𝑦𝐼𝑥))       (𝜑𝑊 ∈ PreHil)

Theoremphlpropd 20405* If two structures have the same components (properties), one is a pre-Hilbert space iff the other one is. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑𝐹 = (Scalar‘𝐾))    &   (𝜑𝐹 = (Scalar‘𝐿))    &   𝑃 = (Base‘𝐹)    &   ((𝜑 ∧ (𝑥𝑃𝑦𝐵)) → (𝑥( ·𝑠𝐾)𝑦) = (𝑥( ·𝑠𝐿)𝑦))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(·𝑖𝐾)𝑦) = (𝑥(·𝑖𝐿)𝑦))       (𝜑 → (𝐾 ∈ PreHil ↔ 𝐿 ∈ PreHil))

Theoremssipeq 20406 The inner product on a subspace equals the inner product on the parent space. (Contributed by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)       (𝑈𝑆𝑃 = , )

Theoremphssipval 20407 The inner product on a subspace in terms of the inner product on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &    , = (·𝑖𝑊)    &   𝑃 = (·𝑖𝑋)    &   𝑆 = (LSubSp‘𝑊)       (((𝑊 ∈ PreHil ∧ 𝑈𝑆) ∧ (𝐴𝑈𝐵𝑈)) → (𝐴𝑃𝐵) = (𝐴 , 𝐵))

Theoremphssip 20408 The inner product (as a function) on a subspace is a restriction of the inner product (as a function) on the parent space. (Contributed by NM, 28-Jan-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)    &    · = (·if𝑊)    &   𝑃 = (·if𝑋)       ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑃 = ( · ↾ (𝑈 × 𝑈)))

Theoremphlssphl 20409 A subspace of an inner product space (pre-Hilbert space) is an inner product space. (Contributed by AV, 25-Sep-2022.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑈𝑆) → 𝑋 ∈ PreHil)

10.9.2  Orthocomplements and closed subspaces

Syntaxcocv 20410 Extend class notation with orthocomplement of a subset.
class ocv

Syntaxccss 20411 Extend class notation with set of closed subspaces.
class ClSubSp

Syntaxcthl 20412 Extend class notation with the Hilbert lattice.
class toHL

Definitiondf-ocv 20413* Define the orthocomplement function in a given set (which usually is a pre-Hilbert space): it associates with a subset its orthogonal subset (which in the case of a closed linear subspace is its orthocomplement). (Contributed by NM, 7-Oct-2011.)
ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))

Definitiondf-css 20414* Define the set of closed (linear) subspaces of a given pre-Hilbert space. (Contributed by NM, 7-Oct-2011.)
ClSubSp = ( ∈ V ↦ {𝑠𝑠 = ((ocv‘)‘((ocv‘)‘𝑠))})

Definitiondf-thl 20415 Define the Hilbert lattice of closed subspaces of a given pre-Hilbert space. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL = ( ∈ V ↦ ((toInc‘(ClSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))

Theoremocvfval 20416* The orthocomplement operation. (Contributed by NM, 7-Oct-2011.) (Revised by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝑊𝑋 = (𝑠 ∈ 𝒫 𝑉 ↦ {𝑥𝑉 ∣ ∀𝑦𝑠 (𝑥 , 𝑦) = 0 }))

Theoremocvval 20417* 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)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝑆𝑉 → ( 𝑆) = {𝑥𝑉 ∣ ∀𝑦𝑆 (𝑥 , 𝑦) = 0 })

Theoremelocv 20418* 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)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       (𝐴 ∈ ( 𝑆) ↔ (𝑆𝑉𝐴𝑉 ∧ ∀𝑥𝑆 (𝐴 , 𝑥) = 0 ))

Theoremocvi 20419 Property of a member of the orthocomplement of a subset. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)       ((𝐴 ∈ ( 𝑆) ∧ 𝐵𝑆) → (𝐴 , 𝐵) = 0 )

Theoremocvss 20420 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

Theoremocvocv 20421 A set is contained in its double orthocomplement. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → 𝑆 ⊆ ( ‘( 𝑆)))

Theoremocvlss 20422 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 ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐿)

Theoremocv2ss 20423 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)       (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))

Theoremocvin 20424 An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝐿) → (𝑆 ∩ ( 𝑆)) = { 0 })

Theoremocvsscon 20425 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 ∧ 𝑆𝑉𝑇𝑉) → (𝑆 ⊆ ( 𝑇) ↔ 𝑇 ⊆ ( 𝑆)))

Theoremocvlsp 20426 The orthocomplement of a linear span. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &   𝑁 = (LSpan‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( ‘(𝑁𝑆)) = ( 𝑆))

Theoremocv0 20427 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ( ‘∅) = 𝑉

Theoremocvz 20428 The orthocomplement of the zero subspace. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → ( ‘{ 0 }) = 𝑉)

Theoremocv1 20429 The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → ( 𝑉) = { 0 })

Theoremunocv 20430 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)       ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Theoremiunocv 20431* The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       ( 𝑥𝐴 𝐵) = (𝑉 𝑥𝐴 ( 𝐵))

Theoremcssval 20432* 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)       (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})

Theoremiscss 20433 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)       (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))

Theoremcssi 20434 Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑆𝐶𝑆 = ( ‘( 𝑆)))

Theoremcssss 20435 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑆𝐶𝑆𝑉)

Theoremiscss2 20436 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 ∧ 𝑆𝑉) → (𝑆𝐶 ↔ ( ‘( 𝑆)) ⊆ 𝑆))

Theoremocvcss 20437 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 ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐶)

Theoremcssincl 20438 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (ClSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremcss0 20439 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (ClSubSp‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)

Theoremcss1 20440 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝑉𝐶)

Theoremcsslss 20441 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 ∧ 𝑆𝐶) → 𝑆𝐿)

Theoremlsmcss 20442 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)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝑆𝑉)    &   (𝜑 → ( ‘( 𝑆)) ⊆ (𝑆 ( 𝑆)))       (𝜑𝑆𝐶)

Theoremcssmre 20443 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 16903: 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 16968. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))

Theoremmrccss 20444 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 ∧ 𝑆𝑉) → (𝐹𝑆) = ( ‘( 𝑆)))

Theoremthlval 20445 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (ocv‘𝑊)       (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))

Theoremthlbas 20446 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)

Theoremthlle 20447 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)

Theoremthlleval 20448 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)    &    = (le‘𝐾)       ((𝑆𝐶𝑇𝐶) → (𝑆 𝑇𝑆𝑇))

Theoremthloc 20449 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

Syntaxcpj 20450 Extend class notation with orthogonal projection function.
class proj

Syntaxchil 20451 Extend class notation with class of all Hilbert spaces.
class Hil

Syntaxcobs 20452 Extend class notation with the set of orthonormal bases.
class OBasis

Definitiondf-pj 20453* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18814, but we restrict the domain of this function to only total projection functions. (Contributed by Mario Carneiro, 16-Oct-2015.)
proj = ( ∈ V ↦ ((𝑥 ∈ (LSubSp‘) ↦ (𝑥(proj1)((ocv‘)‘𝑥))) ∩ (V × ((Base‘) ↑m (Base‘)))))

Definitiondf-hil 20454 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 = { ∈ PreHil ∣ dom (proj‘) = (ClSubSp‘)}

Definitiondf-obs 20455* 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 = ( ∈ PreHil ↦ {𝑏 ∈ 𝒫 (Base‘) ∣ (∀𝑥𝑏𝑦𝑏 (𝑥(·𝑖)𝑦) = if(𝑥 = 𝑦, (1r‘(Scalar‘)), (0g‘(Scalar‘))) ∧ ((ocv‘)‘𝑏) = {(0g)})})

Theorempjfval 20456* The value of the projection function. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐿 = (LSubSp‘𝑊)    &    = (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = ((𝑥𝐿 ↦ (𝑥𝑃( 𝑥))) ∩ (V × (𝑉m 𝑉)))

Theorempjdm 20457 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 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 ↔ (𝑇𝐿 ∧ (𝑇𝑃( 𝑇)):𝑉𝑉))

Theorempjpm 20458 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 ∈ ((𝑉m 𝑉) ↑pm 𝐿)

Theorempjfval2 20459* Value of the projection map with implicit domain. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       𝐾 = (𝑥 ∈ dom 𝐾 ↦ (𝑥𝑃( 𝑥)))

Theorempjval 20460 Value of the projection map. (Contributed by Mario Carneiro, 16-Oct-2015.)
= (ocv‘𝑊)    &   𝑃 = (proj1𝑊)    &   𝐾 = (proj‘𝑊)       (𝑇 ∈ dom 𝐾 → (𝐾𝑇) = (𝑇𝑃( 𝑇)))

Theorempjdm2 20461 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 ↔ (𝑇𝐿 ∧ (𝑇 ( 𝑇)) = 𝑉)))

Theorempjff 20462 A projection is a linear operator. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)       (𝑊 ∈ PreHil → 𝐾:dom 𝐾⟶(𝑊 LMHom 𝑊))

Theorempjf 20463 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 → (𝐾𝑇):𝑉𝑉)

Theorempjf2 20464 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) → (𝐾𝑇):𝑉𝑇)

Theorempjfo 20465 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) → (𝐾𝑇):𝑉onto𝑇)

Theorempjcss 20466 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝐶 = (ClSubSp‘𝑊)       (𝑊 ∈ PreHil → dom 𝐾𝐶)

Theoremocvpj 20467 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)

Theoremishil 20468 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))

Theoremishil2 20469* 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 ∧ ∀𝑠𝐶 (𝑠 ( 𝑠)) = 𝑉))

Theoremisobs 20470* The predicate "is an orthonormal basis" (over a pre-Hilbert space). (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)    &    = (ocv‘𝑊)    &   𝑌 = (0g𝑊)       (𝐵 ∈ (OBasis‘𝑊) ↔ (𝑊 ∈ PreHil ∧ 𝐵𝑉 ∧ (∀𝑥𝐵𝑦𝐵 (𝑥 , 𝑦) = if(𝑥 = 𝑦, 1 , 0 ) ∧ ( 𝐵) = {𝑌})))

Theoremobsip 20471 The inner product of two elements of an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)    &    0 = (0g𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝑃𝐵𝑄𝐵) → (𝑃 , 𝑄) = if(𝑃 = 𝑄, 1 , 0 ))

Theoremobsipid 20472 A basis element has unit length. (Contributed by Mario Carneiro, 23-Oct-2015.)
, = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &    1 = (1r𝐹)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → (𝐴 , 𝐴) = 1 )

Theoremobsrcl 20473 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
(𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)

Theoremobsss 20474 An orthonormal basis is a subset of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → 𝐵𝑉)

Theoremobsne0 20475 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → 𝐴0 )

Theoremobsocv 20476 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)    &    = (ocv‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })

Theoremobs2ocv 20477 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)

Theoremobselocv 20478 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) ∧ 𝐶𝐵𝐴𝐵) → (𝐴 ∈ ( 𝐶) ↔ ¬ 𝐴𝐶))

Theoremobs2ss 20479 A basis has no proper subsets that are also bases. (Contributed by Mario Carneiro, 23-Oct-2015.)
((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐶 ∈ (OBasis‘𝑊) ∧ 𝐶𝐵) → 𝐶 = 𝐵)

Theoremobslbs 20480 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) → (𝐵𝐽 ↔ (𝑁𝐵) ∈ 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 19353), but the existence of a multiplicative neutral element is always assumed (our rings are unital, see df-ring 19352).

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 20482.

Syntaxcdsmm 20481 Class of module direct sum generator.
class m

Definitiondf-dsmm 20482* 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 {𝑓X𝑥 ∈ dom 𝑟(Base‘(𝑟𝑥)) ∣ {𝑥 ∈ dom 𝑟 ∣ (𝑓𝑥) ≠ (0g‘(𝑟𝑥))} ∈ Fin}))

Theoremreldmdsmm 20483 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m

Theoremdsmmval 20484* Value of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉 → (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵))

Theoremdsmmbase 20485* Base set of the module direct sum. (Contributed by Stefan O'Rear, 7-Jan-2015.)
𝐵 = {𝑓 ∈ (Base‘(𝑆Xs𝑅)) ∣ {𝑥 ∈ dom 𝑅 ∣ (𝑓𝑥) ≠ (0g‘(𝑅𝑥))} ∈ Fin}       (𝑅𝑉𝐵 = (Base‘(𝑆m 𝑅)))

Theoremdsmmval2 20486 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‘(𝑆m 𝑅))       (𝑆m 𝑅) = ((𝑆Xs𝑅) ↾s 𝐵)

Theoremdsmmbas2 20487* Base set of the direct sum module using the fndmin 6799 abbreviation. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐵 = {𝑓 ∈ (Base‘𝑃) ∣ dom (𝑓 ∖ (0g𝑅)) ∈ Fin}       ((𝑅 Fn 𝐼𝐼𝑉) → 𝐵 = (Base‘(𝑆m 𝑅)))

Theoremdsmmfi 20488 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 𝐼𝐼 ∈ Fin) → (𝑆m 𝑅) = (𝑆Xs𝑅))

Theoremdsmmelbas 20489* 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 = (𝑆m 𝑅)    &   𝐵 = (Base‘𝑃)    &   𝐻 = (Base‘𝐶)    &   (𝜑𝐼𝑉)    &   (𝜑𝑅 Fn 𝐼)       (𝜑 → (𝑋𝐻 ↔ (𝑋𝐵 ∧ {𝑎𝐼 ∣ (𝑋𝑎) ≠ (0g‘(𝑅𝑎))} ∈ Fin)))

Theoremdsmm0cl 20490 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‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &    0 = (0g𝑃)       (𝜑0𝐻)

Theoremdsmmacl 20491 The finite hull is closed under addition. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑃 = (𝑆Xs𝑅)    &   𝐻 = (Base‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Mnd)    &   (𝜑𝐽𝐻)    &   (𝜑𝐾𝐻)    &    + = (+g𝑃)       (𝜑 → (𝐽 + 𝐾) ∈ 𝐻)

Theoremprdsinvgd2 20492 Negation of a single coordinate in a structure product. (Contributed by Stefan O'Rear, 11-Jan-2015.)
𝑌 = (𝑆Xs𝑅)    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)    &   𝐵 = (Base‘𝑌)    &   𝑁 = (invg𝑌)    &   (𝜑𝑋𝐵)    &   (𝜑𝐽𝐼)       (𝜑 → ((𝑁𝑋)‘𝐽) = ((invg‘(𝑅𝐽))‘(𝑋𝐽)))

Theoremdsmmsubg 20493 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‘(𝑆m 𝑅))    &   (𝜑𝐼𝑊)    &   (𝜑𝑆𝑉)    &   (𝜑𝑅:𝐼⟶Grp)       (𝜑𝐻 ∈ (SubGrp‘𝑃))

Theoremdsmmlss 20494* 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.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝑃 = (𝑆Xs𝑅)    &   𝑈 = (LSubSp‘𝑃)    &   𝐻 = (Base‘(𝑆m 𝑅))       (𝜑𝐻𝑈)

Theoremdsmmlmod 20495* 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.)
(𝜑𝐼𝑊)    &   (𝜑𝑆 ∈ Ring)    &   (𝜑𝑅:𝐼⟶LMod)    &   ((𝜑𝑥𝐼) → (Scalar‘(𝑅𝑥)) = 𝑆)    &   𝐶 = (𝑆m 𝑅)       (𝜑𝐶 ∈ 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 non-free 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 20497. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20497 (see lmisfree 20592), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20497 are also taken as a motivation to introduce free modules by [Lang] p. 135.

Syntaxcfrlm 20496 Class of free module generator.
class freeLMod

Definitiondf-frlm 20497* 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 20482 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 ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))

Theoremfrlmval 20498 Value of the "free module" function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))

Theoremfrlmlmod 20499 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ LMod)

Theoremfrlmpws 20500 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)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (((ringLMod‘𝑅) ↑s 𝐼) ↾s 𝐵))

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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45683
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