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Theorem List for Metamath Proof Explorer - 19701-19800   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremphlsrng 19701 The scalar ring of a pre-Hilbert space is a star ring. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ PreHil → 𝐹 ∈ *-Ring)

Theoremphllmhm 19702* The inner product of a pre-Hilbert space is linear in its left argument. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐺 = (𝑥𝑉 ↦ (𝑥 , 𝐴))       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → 𝐺 ∈ (𝑊 LMHom (ringLMod‘𝐹)))

Theoremipcl 19703 Closure of the inner product operation in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ 𝐾)

Theoremipcj 19704 Conjugate of an inner product in a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ( ‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremiporthcom 19705 Orthogonality (meaning inner product is 0) is commutative. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 𝑍 ↔ (𝐵 , 𝐴) = 𝑍))

Theoremip0l 19706 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 𝑍)

Theoremip0r 19707 Inner product with a zero second argument. (Contributed by NM, 5-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 𝑍)

Theoremipeq0 19708 The inner product of a vector with itself is zero iff the vector is zero. Part of Definition 3.1-1 of [Kreyszig] p. 129. (Contributed by NM, 24-Jan-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝑍 = (0g𝐹)    &    0 = (0g𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 𝑍𝐴 = 0 ))

Theoremipdir 19709 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) (𝐵 , 𝐶)))

Theoremipdi 19710 Distributive law for inner product (left-distributivity). (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) (𝐴 , 𝐶)))

Theoremip2di 19711 Distributive law for inner product. (Contributed by NM, 17-Apr-2008.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) (𝐵 , 𝐷)) ((𝐴 , 𝐷) (𝐵 , 𝐶))))

Theoremipsubdir 19712 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶)𝑆(𝐵 , 𝐶)))

Theoremipsubdi 19713 Distributive law for inner product subtraction. (Contributed by NM, 20-Nov-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵)𝑆(𝐴 , 𝐶)))

Theoremip2subdi 19714 Distributive law for inner product subtraction. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   𝑆 = (-g𝐹)    &    + = (+g𝐹)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 𝐵) , (𝐶 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷))𝑆((𝐴 , 𝐷) + (𝐵 , 𝐶))))

Theoremipass 19715 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 × (𝐵 , 𝐶)))

Theoremipassr 19716 "Associative" law for second argument of inner product (compare ipass 19715). (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → (𝐴 , (𝐶 · 𝐵)) = ((𝐴 , 𝐵) × ( 𝐶)))

Theoremipassr2 19717 "Associative" law for inner product. Conjugate version of ipassr 19716. (Contributed by NM, 25-Aug-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    , = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &    × = (.r𝐹)    &    = (*𝑟𝐹)       ((𝑊 ∈ PreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝐾)) → ((𝐴 , 𝐵) × 𝐶) = (𝐴 , (( 𝐶) · 𝐵)))

Theoremipffval 19718* The inner product operation as a function. (Contributed by Mario Carneiro, 12-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)        · = (𝑥𝑉, 𝑦𝑉 ↦ (𝑥 , 𝑦))

Theoremipfval 19719 The inner product operation as a function. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ((𝑋𝑉𝑌𝑉) → (𝑋 · 𝑌) = (𝑋 , 𝑌))

Theoremipfeq 19720 If the inner product operation is already a function, the functionalization of it is equal to the original operation. (Contributed by Mario Carneiro, 14-Aug-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &    · = (·if𝑊)       ( , Fn (𝑉 × 𝑉) → · = , )

Theoremipffn 19721 The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)
𝑉 = (Base‘𝑊)    &    , = (·if𝑊)        , Fn (𝑉 × 𝑉)

Theoremphlipf 19722 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 19723* 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 19724* 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 19725* 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))

10.12.2  Orthocomplements and closed subspaces

Syntaxcocv 19726 Extend class notation with orthocomplement of a subspace.
class ocv

Syntaxccss 19727 Extend class notation with set of closed subspaces.
class CSubSp

Syntaxcthl 19728 Extend class notation with the Hilbert lattice.
class toHL

Definitiondf-ocv 19729* 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 19730* Define set of closed subspaces. (Contributed by NM, 7-Oct-2011.)
CSubSp = ( ∈ V ↦ {𝑠𝑠 = ((ocv‘)‘((ocv‘)‘𝑠))})

Definitiondf-thl 19731 Define the Hilbert lattice of closed subspaces. (Contributed by Mario Carneiro, 25-Oct-2015.)
toHL = ( ∈ V ↦ ((toInc‘(CSubSp‘)) sSet ⟨(oc‘ndx), (ocv‘)⟩))

Theoremocvfval 19732* 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 19733* 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 19734* 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 19735 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 19736 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 19737 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 19738 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 19739 Orthocomplements reverse subset inclusion. (Contributed by Mario Carneiro, 13-Oct-2015.)
= (ocv‘𝑊)       (𝑇𝑆 → ( 𝑆) ⊆ ( 𝑇))

Theoremocvin 19740 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 19741 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 19742 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 19743 The orthocomplement of the empty set. (Contributed by Mario Carneiro, 23-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (ocv‘𝑊)       ( ‘∅) = 𝑉

Theoremocvz 19744 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 19745 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 19746 The orthocomplement of a union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)       ( ‘(𝐴𝐵)) = (( 𝐴) ∩ ( 𝐵))

Theoremiunocv 19747* The orthocomplement of an indexed union. (Contributed by Mario Carneiro, 23-Oct-2015.)
= (ocv‘𝑊)    &   𝑉 = (Base‘𝑊)       ( 𝑥𝐴 𝐵) = (𝑉 𝑥𝐴 ( 𝐵))

Theoremcssval 19748* 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 19749 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 19750 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 19751 A closed subspace is a subset of the base. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑆𝐶𝑆𝑉)

Theoremiscss2 19752 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 19753 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 19754 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)       ((𝑊 ∈ PreHil ∧ 𝐴𝐶𝐵𝐶) → (𝐴𝐵) ∈ 𝐶)

Theoremcss0 19755 The zero subspace is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ PreHil → { 0 } ∈ 𝐶)

Theoremcss1 19756 The whole space is a closed subspace. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝑉𝐶)

Theoremcsslss 19757 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 19758 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 19759 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 15964: 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 16030. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐶 = (CSubSp‘𝑊)       (𝑊 ∈ PreHil → 𝐶 ∈ (Moore‘𝑉))

Theoremmrccss 19760 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 19761 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 19762 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 19763 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 19764 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 19765 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 19766 Extend class notation with orthogonal projection function.
class proj

Syntaxchs 19767 Extend class notation with class of all Hilbert spaces.
class Hil

Syntaxcobs 19768 Extend class notation with the set of orthonormal bases.
class OBasis

Definitiondf-pj 19769* Define orthogonal projection onto a subspace. This is just a wrapping of df-pj1 17783, 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 19770 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 19771* 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 19772* 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 19773 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 19774 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 19775* 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 19776 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 19777 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 19778 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 19779 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 19780 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 19781 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 19782 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 19783 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 19784 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 19785* 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 19786* 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 19787 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 19788 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 19789 Reverse closure for an orthonormal basis. (Contributed by Mario Carneiro, 23-Oct-2015.)
(𝐵 ∈ (OBasis‘𝑊) → 𝑊 ∈ PreHil)

Theoremobsss 19790 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 19791 A basis element is nonzero. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)       ((𝐵 ∈ (OBasis‘𝑊) ∧ 𝐴𝐵) → 𝐴0 )

Theoremobsocv 19792 An orthonormal basis has trivial orthocomplement. (Contributed by Mario Carneiro, 23-Oct-2015.)
0 = (0g𝑊)    &    = (ocv‘𝑊)       (𝐵 ∈ (OBasis‘𝑊) → ( 𝐵) = { 0 })

Theoremobs2ocv 19793 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 19794 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 19795 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 19796 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 19813) 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 19904), 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 18280), but the existence of a multiplicative neutral element is always presumed (the ring is a unital ring, see also df-ring 18279). 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 component-wise 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 19798.

Syntaxcdsmm 19797 Class of module direct sum generator.
class m

Definitiondf-dsmm 19798* 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 19799 The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)
Rel dom ⊕m

Theoremdsmmval 19800* 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))

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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 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