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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-ocv 20001* Define orthocomplement of a subspace. (Contributed by NM, 7-Oct-2011.)
ocv = ( ∈ V ↦ (𝑠 ∈ 𝒫 (Base‘) ↦ {𝑥 ∈ (Base‘) ∣ ∀𝑦𝑠 (𝑥(·𝑖)𝑦) = (0g‘(Scalar‘))}))

Definitiondf-css 20002* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
CSubSp = ( ∈ V ↦ {𝑠𝑠 = ((ocv‘)‘((ocv‘)‘𝑠))})

Definitiondf-thl 20003 Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))

Theoremocvfval 20004* 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 20005* 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 20006* 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 20007 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 20008 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 20009 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 20010 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 20011 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)       (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))

Theoremocvin 20012 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 20013 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 20014 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 20015 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ( ‘∅) = 𝑉

Theoremocvz 20016 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 20017 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 20018 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)       ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Theoremiunocv 20019* The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       ( 𝑥𝐴 𝐵) = (𝑉 𝑥𝐴 ( 𝐵))

Theoremcssval 20020* 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 = (CSubSp‘𝑊)       (𝑊𝑋𝐶 = {𝑠𝑠 = ( ‘( 𝑠))})

Theoremiscss 20021 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 = (CSubSp‘𝑊)       (𝑊𝑋 → (𝑆𝐶𝑆 = ( ‘( 𝑆))))

Theoremcssi 20022 Property of a closed subspace (of a pre-Hilbert space). (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑆𝐶𝑆 = ( ‘( 𝑆)))

Theoremcssss 20023 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑆𝐶𝑆𝑉)

Theoremiscss2 20024 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 = (CSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝑆𝐶 ↔ ( ‘( 𝑆)) ⊆ 𝑆))

Theoremocvcss 20025 The orthocomplement of any set is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &    = (ocv‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → ( 𝑆) ∈ 𝐶)

Theoremcssincl 20026 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremcss0 20027 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)

Theoremcss1 20028 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝑉𝐶)

Theoremcsslss 20029 A closed subspace of a pre-Hilbert space is a linear subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &   𝐿 = (LSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝑆𝐶) → 𝑆𝐿)

Theoremlsmcss 20030 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 = (CSubSp‘𝑊)    &   𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)    &    = (LSSum‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝑆𝑉)    &   (𝜑 → ( ‘( 𝑆)) ⊆ (𝑆 ( 𝑆)))       (𝜑𝑆𝐶)

Theoremcssmre 20031 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 16243: 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 16309. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))

Theoremmrccss 20032 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 = (CSubSp‘𝑊)    &   𝐹 = (mrCls‘𝐶)       ((𝑊 ∈ PreHil ∧ 𝑆𝑉) → (𝐹𝑆) = ( ‘( 𝑆)))

Theoremthlval 20033 Value of the Hilbert lattice. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (ocv‘𝑊)       (𝑊𝑉𝐾 = (𝐼 sSet ⟨(oc‘ndx), ⟩))

Theoremthlbas 20034 Base set of the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       𝐶 = (Base‘𝐾)

Theoremthlle 20035 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &   𝐼 = (toInc‘𝐶)    &    = (le‘𝐼)        = (le‘𝐾)

Theoremthlleval 20036 Ordering on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &   𝐶 = (CSubSp‘𝑊)    &    = (le‘𝐾)       ((𝑆𝐶𝑇𝐶) → (𝑆 𝑇𝑆𝑇))

Theoremthloc 20037 Orthocomplement on the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
𝐾 = (toHL‘𝑊)    &    = (ocv‘𝑊)        = (oc‘𝐾)

10.12.3  Orthogonal projection and orthonormal bases

Syntaxcpj 20038 Extend class notation with orthogonal projection function.
class proj

Syntaxchs 20039 Extend class notation with class of all Hilbert spaces.
class Hil

Syntaxcobs 20040 Extend class notation with the set of orthonormal bases.
class OBasis

Definitiondf-pj 20041* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 18046, 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 20042 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‘) = (CSubSp‘)}

Definitiondf-obs 20043* 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 20044* 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 20045 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 20046 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 20047* 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 20048 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 20049 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 20050 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 20051 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 20052 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 20053 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 20054 A projection subspace is an (algebraically) closed subspace. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐾 = (proj‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → dom 𝐾𝐶)

Theoremocvpj 20055 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 20056 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 = (CSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ dom 𝐾 = 𝐶))

Theoremishil2 20057* 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 = (CSubSp‘𝐻)       (𝐻 ∈ Hil ↔ (𝐻 ∈ PreHil ∧ ∀𝑠𝐶 (𝑠 ( 𝑠)) = 𝑉))

Theoremisobs 20058* 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 20059 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 20060 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 20061 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
(𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)

Theoremobsss 20062 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 20063 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → 𝐴0 )

Theoremobsocv 20064 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)    &    = (ocv‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })

Theoremobs2ocv 20065 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 20066 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 20067 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 20068 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 = (CSubSp‘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." However, dealing with modules (over rings) instead of vector spaces (over fields) allows for a more general approach. Therefore, "vectors" are regarded as members (elements of the base set) of a (free) module over a ring (see df-frlm 20085) in the following. By this, the number of entries in a vector is determined by the size of the index set of the direct sum building the free module the vector is belonging to. Since every vector space is isomorphic to a free module (see lvecisfrlm 20176), the theorems stated for free modules are also valid for vector spaces.

Until not explicitly stated, the underlying ring needs not to be commutative (see df-cring 18544), but the existence of a multiplicative neutral element is always presumed (the ring is a unital ring, see also df-ring 18543). In this sense, linear equations, matrices and determinants are usually regarded as "over a ring" in this part.

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 the definition df-dsmm 20070.

Syntaxcdsmm 20069 Class of module direct sum generator.
class m

Definitiondf-dsmm 20070* 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 20071 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m

Theoremdsmmval 20072* 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 20073* 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 20074 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 20075* Base set of the direct sum module using the fndmin 6322 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 20076 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 20077* 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 20078 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 20079 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 20080 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 20081 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 20082* 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 20083* 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, however, a free module is defined as direct sum of a family consisting of the same ring regarded as a (left) module over itself, see df-frlm 20085. Since a module has a basis if and only if it is isomorphic to a free module as defined by df-frlm 20085 (see lmisfree 20175), the two definitions are essentially equivalent. The free modules as defined by df-frlm 20085 are also taken for the motivation of free modules by [Lang] p. 135.

Syntaxcfrlm 20084 Class of free module generator.
class freeLMod

Definitiondf-frlm 20085* The 𝑖-dimensional free module over a ring 𝑟 is the product of 𝑖-many copies of the ring with componentwise addition and multiplication. If 𝑖 is infinite, the allowed vectors are restricted to those with finitely many nonzero coordinates; this ensures that the resulting module is actually spanned by its unit vectors. (Contributed by Stefan O'Rear, 1-Feb-2015.)
freeLMod = (𝑟 ∈ V, 𝑖 ∈ V ↦ (𝑟m (𝑖 × {(ringLMod‘𝑟)})))

Theoremfrlmval 20086 Value of the free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝐹 = (𝑅m (𝐼 × {(ringLMod‘𝑅)})))

Theoremfrlmlmod 20087 The free module is a module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → 𝐹 ∈ LMod)

Theoremfrlmpws 20088 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))

Theoremfrlmlss 20089 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 ∧ 𝐼𝑊) → 𝐵𝑈)

Theoremfrlmpwsfi 20090 The finite free module is a power of the ring module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼 ∈ Fin) → 𝐹 = ((ringLMod‘𝑅) ↑s 𝐼))

Theoremfrlmsca 20091 The ring of scalars of a free module. (Contributed by Stefan O'Rear, 1-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)       ((𝑅𝑉𝐼𝑊) → 𝑅 = (Scalar‘𝐹))

Theoremfrlm0 20092 Zero in a free module (ring constraint is stronger than necessary, but allows use of frlmlss 20089). (Contributed by Stefan O'Rear, 4-Feb-2015.)
𝐹 = (𝑅 freeLMod 𝐼)    &    0 = (0g𝑅)       ((𝑅 ∈ Ring ∧ 𝐼𝑊) → (𝐼 × { 0 }) = (0g𝐹))

Theoremfrlmbas 20093* 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 = (0g𝑅)    &   𝐵 = {𝑘 ∈ (𝑁𝑚 𝐼) ∣ 𝑘 finSupp 0 }       ((𝑅𝑉𝐼𝑊) → 𝐵 = (Base‘𝐹))

Theoremfrlmelbas 20094 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 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝑅𝑉𝐼𝑊) → (𝑋𝐵 ↔ (𝑋 ∈ (𝑁𝑚 𝐼) ∧ 𝑋 finSupp 0 )))

Theoremfrlmrcl 20095 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)       (𝑋𝐵𝑅 ∈ V)

Theoremfrlmbasfsupp 20096 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 = (0g𝑅)    &   𝐵 = (Base‘𝐹)       ((𝐼𝑊𝑋𝐵) → 𝑋 finSupp 0 )

Theoremfrlmbasmap 20097 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)       ((𝐼𝑊𝑋𝐵) → 𝑋 ∈ (𝑁𝑚 𝐼))

Theoremfrlmbasf 20098 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)       ((𝐼𝑊𝑋𝐵) → 𝑋:𝐼𝑁)

Theoremfrlmfibas 20099 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)       ((𝑅𝑉𝐼 ∈ Fin) → (𝑁𝑚 𝐼) = (Base‘𝐹))

Theoremelfrlmbasn0 20100 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)       ((𝐼𝑉𝐼 ≠ ∅) → (𝑋𝐵𝑋 ≠ ∅))

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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 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42322
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