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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremncvsdif 23001 The norm of the difference between two vectors. (Contributed by NM, 1-Dec-2006.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑉𝐵𝑉) → (𝑁‘(𝐴 + (-1 · 𝐵))) = (𝑁‘(𝐵 + (-1 · 𝐴))))

Theoremncvspi 23002 The norm of a vector plus the imaginary scalar product of another. (Contributed by NM, 2-Feb-2007.) (Revised by AV, 8-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &    + = (+g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑉𝐵𝑉) ∧ i ∈ 𝐾) → (𝑁‘(𝐴 + (i · 𝐵))) = (𝑁‘(𝐵 + (-i · 𝐴))))

Theoremncvs1 23003 From any nonzero vector, construct a vector whose norm is one. (Contributed by NM, 6-Dec-2007.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)    &    · = ( ·𝑠𝐺)    &   𝐹 = (Scalar‘𝐺)    &   𝐾 = (Base‘𝐹)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ (𝐴𝑋𝐴0 ) ∧ (1 / (𝑁𝐴)) ∈ 𝐾) → (𝑁‘((1 / (𝑁𝐴)) · 𝐴)) = 1)

Theoremcnrnvc 23004 The set of complex numbers (as a subring of itself) is a normed vector space over itself. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ NrmVec

Theoremcnncvs 23005 The set of complex numbers (as a subring of itself) is a normed subcomplex vector space. The vector operation is +, and the scalar product is ·, and the norm function is abs. (Contributed by Steve Rodriguez, 3-Dec-2006.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       𝐶 ∈ (NrmVec ∩ ℂVec)

Theoremcnnm 23006 The norm operation of the normed subcomplex vector space of complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.)
𝐶 = (ringLMod‘ℂfld)       (norm‘𝐶) = abs

Theoremncvspds 23007 Value of the distance function in terms of the norm of a normed subcomplex vector space. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by AV, 13-Oct-2021.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)    &    · = ( ·𝑠𝐺)       ((𝐺 ∈ (NrmVec ∩ ℂVec) ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 + (-1 · 𝐵))))

Theoremcnindmet 23008 The metric induced on the complex numbers. cnmet 22622 proves that it is a metric. The induced metric is identical with the original metric on the complex numbers, see cnfldds 19804 and also cnmet 22622. (Contributed by Steve Rodriguez, 5-Dec-2006.) (Revised by AV, 17-Oct-2021.)
𝑇 = (ℂfld toNrmGrp abs)       (dist‘𝑇) = (abs ∘ − )

Theoremcnncvsaddassdemo 23009 Derive the associative law for complex number addition addass 10061 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by NM, 12-Jan-2008.) (Revised by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 + 𝐵) + 𝐶) = (𝐴 + (𝐵 + 𝐶)))

Theoremcnncvsmulassdemo 23010 Derive the associative law for complex number multiplication mulass 10062 interpreted as scalar multiplication to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → ((𝐴 · 𝐵) · 𝐶) = (𝐴 · (𝐵 · 𝐶)))

Theoremcnncvsabsnegdemo 23011 Derive the absolute value of a negative complex number absneg 14061 to demonstrate the use of the properties of a normed subcomplex vector space for the complex numbers. (Contributed by AV, 9-Oct-2021.) (Proof modification is discouraged.)
(𝐴 ∈ ℂ → (abs‘-𝐴) = (abs‘𝐴))

12.5.4  Subcomplex pre-Hilbert space

Syntaxccph 23012 Extend class notation with a class of subcomplex pre-Hilbert spaces.
class ℂPreHil

Syntaxctch 23013 Function to put a norm on a Hilbert space.
class toℂHil

Definitiondf-cph 23014* Define the class of subcomplex pre-Hilbert spaces. By restricting the scalar field to a quadratically closed subfield of fld, we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}

Definitiondf-tch 23015* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))

Theoremiscph 23016* A subcomplex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the field of complex numbers, with a norm defined. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))

Theoremcphphl 23017 A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Theoremcphnlm 23018 A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Theoremcphngp 23019 A subcomplex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)

Theoremcphlmod 23020 A subcomplex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)

Theoremcphlvec 23021 A subcomplex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Theoremcphnvc 23022 A subcomplex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)

Theoremcphsubrglem 23023 Lemma for cphsubrg 23026. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       (𝜑 → (𝐹 = (ℂflds 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld)))

Theoremcphreccllem 23024 Lemma for cphreccl 23027. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       ((𝜑𝑋𝐾𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)

Theoremcphsca 23025 A subcomplex pre-Hilbert space is a vector space over a subfield of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))

Theoremcphsubrg 23026 The scalar field of a subcomplex pre-Hilbert space is a subring of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))

Theoremcphreccl 23027 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾)

Theoremcphdivcl 23028 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcphcjcl 23029 The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (∗‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl 23030 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)

Theoremcphabscl 23031 The scalar field of a subcomplex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (abs‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl2 23032 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl3 23033 If the scalar field contains i, it is completely closed under square roots (i.e. it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾𝐴𝐾) → (√‘𝐴) ∈ 𝐾)

Theoremcphqss 23034 The scalar field of a subcomplex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾)

Theoremcphclm 23035 A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)

Theoremcphnmvs 23036 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((abs‘𝑋) · (𝑁𝑌)))

Theoremcphipcl 23037 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ ℂ)

Theoremcphnmfval 23038* The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremcphnm 23039 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))

Theoremnmsq 23040 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝑁𝐴)↑2) = (𝐴 , 𝐴))

Theoremcphnmf 23041 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝑁:𝑉𝐾)

Theoremcphnmcl 23042 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) ∈ 𝐾)

Theoremreipcl 23043 An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝐴 , 𝐴) ∈ ℝ)

Theoremipge0 23044 The inner product in a subcomplex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → 0 ≤ (𝐴 , 𝐴))

Theoremcphipcj 23045 Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj 20027. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremcphipipcj 23046 An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (Revised by AV, 19-Oct-2021.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) · (𝐵 , 𝐴)) = ((abs‘(𝐴 , 𝐵))↑2))

Theoremcphorthcom 23047 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 20028. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))

Theoremcphip0l 23048 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 20029. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 0)

Theoremcphip0r 23049 Inner product with a zero second argument. Complex version of ip0r 20030. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 0)

Theoremcphipeq0 23050 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. Complex version of ipeq0 20031. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 ))

Theoremcphdir 23051 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 20032. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) + (𝐵 , 𝐶)))

Theoremcphdi 23052 Distributive law for inner product (left-distributivity). Complex version of ipdi 20033. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) + (𝐴 , 𝐶)))

Theoremcph2di 23053 Distributive law for inner product. Complex version of ip2di 20034. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) + ((𝐴 , 𝐷) + (𝐵 , 𝐶))))

Theoremcphsubdir 23054 Distributive law for inner product subtraction. Complex version of ipsubdir 20035. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶)))

Theoremcphsubdi 23055 Distributive law for inner product subtraction. Complex version of ipsubdi 20036. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶)))

Theoremcph2subdi 23056 Distributive law for inner product subtraction. Complex version of ip2subdi 20037. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝐶𝑉)    &   (𝜑𝐷𝑉)       (𝜑 → ((𝐴 𝐵) , (𝐶 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶))))

Theoremcphass 23057 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 20038, his5 28071. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 · (𝐵 , 𝐶)))

Theoremcphassr 23058 "Associative" law for second argument of inner product (compare cphass 23057). See ipassr 20039, his52 . (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝑉𝐶𝑉)) → (𝐵 , (𝐴 · 𝐶)) = ((∗‘𝐴) · (𝐵 , 𝐶)))

Theoremcph2ass 23059 Move scalar multiplication to outside of inner product. See his35 28073. (Contributed by Mario Carneiro, 17-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝐾) ∧ (𝐶𝑉𝐷𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷)))

Theoremcphassi 23060 Associative law for the first argument of an inner product with scalar 𝑖. (Contributed by AV, 17-Oct-2021.)
𝑋 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴𝑋𝐵𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴)))

Theoremcphassir 23061 "Associative" law for the second argument of an inner product with scalar 𝑖. (Contributed by AV, 17-Oct-2021.)
𝑋 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴𝑋𝐵𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵)))

Theoremtchex 23062* Lemma for tchbas 23064 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V

Theoremtchval 23063* Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremtchbas 23064 The base set of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)       𝑉 = (Base‘𝐺)

Theoremtchplusg 23065 The addition operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    + = (+g𝑊)        + = (+g𝐺)

Theoremtchsub 23066 The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂHil‘𝑊)    &    = (-g𝑊)        = (-g𝐺)

Theoremtchmulr 23067 The ring operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (.r𝑊)        · = (.r𝐺)

Theoremtchsca 23068 The scalar field of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝐹 = (Scalar‘𝑊)       𝐹 = (Scalar‘𝐺)

Theoremtchvsca 23069 The scalar multiplication of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = ( ·𝑠𝑊)        · = ( ·𝑠𝐺)

Theoremtchip 23070 The inner product of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (·𝑖𝑊)        · = (·𝑖𝐺)

Theoremtchtopn 23071 The topology of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝐷 = (dist‘𝐺)    &   𝐽 = (TopOpen‘𝐺)       (𝑊𝑉𝐽 = (MetOpen‘𝐷))

Theoremtchphl 23072 Augmentation of a subcomplex pre-Hilbert space with a norm does not affect whether it is still a pre-Hilbert space because all the original components are the same. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)       (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil)

Theoremtchnmfval 23073* The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       (𝑊 ∈ Grp → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremtchnmval 23074 The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ Grp ∧ 𝑋𝑉) → (𝑁𝑋) = (√‘(𝑋 , 𝑋)))

Theoremcphtchnm 23075 The norm of a norm-augmented subcomplex pre-Hilbert space is the same as the original norm on it. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (norm‘𝐺))

Theoremtchds 23076 The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &    = (-g𝑊)       (𝑊 ∈ Grp → (𝑁 ) = (dist‘𝐺))

Theoremtchclm 23077 Lemma for tchcph 23082. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))       (𝜑𝑊 ∈ ℂMod)

Theoremtchcphlem3 23078 Lemma for tchcph 23082: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)       ((𝜑𝑋𝑉) → (𝑋 , 𝑋) ∈ ℝ)

Theoremipcau2 23079* The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &   𝑁 = (norm‘𝐺)    &   𝐶 = ((𝑌 , 𝑋) / (𝑌 , 𝑌))    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁𝑋) · (𝑁𝑌)))

Theoremtchcphlem1 23080* Lemma for tchcph 23082: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)       (𝜑 → (√‘((𝑋 𝑌) , (𝑋 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌))))

Theoremtchcphlem2 23081* Lemma for tchcph 23082: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))    &   𝐾 = (Base‘𝐹)    &    · = ( ·𝑠𝑊)    &   (𝜑𝑋𝐾)    &   (𝜑𝑌𝑉)       (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌))))

Theoremtchcph 23082* The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of fld into a subcomplex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))       (𝜑𝐺 ∈ ℂPreHil)

Theoremipcau 23083 The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝑉𝑌𝑉) → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁𝑋) · (𝑁𝑌)))

Theoremnmparlem 23084 Lemma for nmpar 23085. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝑁 = (norm‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)       (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremnmpar 23085 A subcomplex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    + = (+g𝑊)    &    = (-g𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 𝐵))↑2)) = (2 · (((𝑁𝐴)↑2) + ((𝑁𝐵)↑2))))

Theoremcphipval2 23086 Value of the inner product expressed by the norm defined by it. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (norm‘𝑊)    &    , = (·𝑖𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴𝑋𝐵𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 (i · 𝐵)))↑2)))) / 4))

Theorem4cphipval2 23087 Four times the inner product value cphipval2 23086. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (norm‘𝑊)    &    , = (·𝑖𝑊)    &    = (-g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴𝑋𝐵𝑋) → (4 · (𝐴 , 𝐵)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 (i · 𝐵)))↑2)))))

Theoremcphipval 23088* Value of the inner product expressed by a sum of terms with the norm defined by the inner product. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-2007.) (Revised by AV, 18-Oct-2021.)
𝑋 = (Base‘𝑊)    &    + = (+g𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (norm‘𝑊)    &    , = (·𝑖𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴𝑋𝐵𝑋) → (𝐴 , 𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴 + ((i↑𝑘) · 𝐵)))↑2)) / 4))

Theoremipcnlem2 23089 The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝑁𝐴) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝐵) + 𝑇))    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝐴𝐷𝑋) < 𝑈)    &   (𝜑 → (𝐵𝐷𝑌) < 𝑇)       (𝜑 → (abs‘((𝐴 , 𝐵) − (𝑋 , 𝑌))) < 𝑅)

Theoremipcnlem1 23090* The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝑁𝐴) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝐵) + 𝑇))    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑉)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∃𝑟 ∈ ℝ+𝑥𝑉𝑦𝑉 (((𝐴𝐷𝑥) < 𝑟 ∧ (𝐵𝐷𝑦) < 𝑟) → (abs‘((𝐴 , 𝐵) − (𝑥 , 𝑦))) < 𝑅))

Theoremipcn 23091 The inner product operation of a subcomplex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.)
, = (·if𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘ℂfld)       (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×t 𝐽) Cn 𝐾))

Theoremcnmpt1ip 23092* Continuity of inner product; analogue of cnmpt12f 21517 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐶 = (TopOpen‘ℂfld)    &    , = (·𝑖𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑 → (𝑥𝑋𝐴) ∈ (𝐾 Cn 𝐽))    &   (𝜑 → (𝑥𝑋𝐵) ∈ (𝐾 Cn 𝐽))       (𝜑 → (𝑥𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶))

Theoremcnmpt2ip 23093* Continuity of inner product; analogue of cnmpt22f 21526 which cannot be used directly because ·𝑖 is not a function. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐽 = (TopOpen‘𝑊)    &   𝐶 = (TopOpen‘ℂfld)    &    , = (·𝑖𝑊)    &   (𝜑𝑊 ∈ ℂPreHil)    &   (𝜑𝐾 ∈ (TopOn‘𝑋))    &   (𝜑𝐿 ∈ (TopOn‘𝑌))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))    &   (𝜑 → (𝑥𝑋, 𝑦𝑌𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽))       (𝜑 → (𝑥𝑋, 𝑦𝑌 ↦ (𝐴 , 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐶))

Theoremcsscld 23094 A "closed subspace" in a subcomplex pre-Hilbert space is actually closed in the topology induced by the norm, thus justifying the terminology "closed subspace". (Contributed by Mario Carneiro, 13-Oct-2015.)
𝐶 = (CSubSp‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑆𝐶) → 𝑆 ∈ (Clsd‘𝐽))

Theoremclsocv 23095 The orthogonal complement of the closure of a subset is the same as the orthogonal complement of the subset itself. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑂 = (ocv‘𝑊)    &   𝐽 = (TopOpen‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑆𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂𝑆))

12.5.5  Convergence and completeness

Syntaxccfil 23096 Extend class notation with the set of Cauchy filters.
class CauFil

Syntaxcca 23097 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
class Cau

Syntaxcms 23098 Extend class notation with class of complete metric spaces.
class CMet

Definitiondf-cfil 23099* Define the set of Cauchy filters on a metric space. A Cauchy filter is a filter on the set such that for every 0 < 𝑥 there is an element of the filter whose metric diameter is less than 𝑥. (Contributed by Mario Carneiro, 13-Oct-2015.)
CauFil = (𝑑 ran ∞Met ↦ {𝑓 ∈ (Fil‘dom dom 𝑑) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Definitiondf-cau 23100* Define a function on metric spaces whose value is the set of Cauchy sequences of the space. (Contributed by NM, 8-Sep-2006.)
Cau = (𝑑 ran ∞Met ↦ {𝑓 ∈ (dom dom 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑓𝑗)(ball‘𝑑)𝑥)})

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