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Theorem List for Metamath Proof Explorer - 23001-23100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcph2di 23001 Distributive law for inner product. Complex version of ip2di 19980. (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 23002 Distributive law for inner product subtraction. Complex version of ipsubdir 19981. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶)))
 
Theoremcphsubdi 23003 Distributive law for inner product subtraction. Complex version of ipsubdi 19982. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶)))
 
Theoremcph2subdi 23004 Distributive law for inner product subtraction. Complex version of ip2subdi 19983. (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 23005 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 19984, his5 27927. (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 23006 "Associative" law for second argument of inner product (compare cphass 23005). See ipassr 19985, 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 23007 Move scalar multiplication to outside of inner product. See his35 27929. (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 23008 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 23009 "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 23010* Lemma for tchbas 23012 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V
 
Theoremtchval 23011* 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 23012 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 23013 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 23014 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 23015 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 23016 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 23017 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 23018 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 23019 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 23020 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 23021* 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 23022 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 23023 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 23024 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 23025 Lemma for tchcph 23030. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))       (𝜑𝑊 ∈ ℂMod)
 
Theoremtchcphlem3 23026 Lemma for tchcph 23030: 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 23027* 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 23028* Lemma for tchcph 23030: 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 23029* Lemma for tchcph 23030: 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 23030* 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 23031 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 23032 Lemma for nmpar 23033. (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 23033 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 23034 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 23035 Four times the inner product value cphipval2 23034. (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 23036* 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 23037 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 23038* 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 23039 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 23040* Continuity of inner product; analogue of cnmpt12f 21463 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 23041* Continuity of inner product; analogue of cnmpt22f 21472 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 23042 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 23043 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 23044 Extend class notation with the set of Cauchy filters.
class CauFil
 
Syntaxcca 23045 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
class Cau
 
Syntaxcms 23046 Extend class notation with class of complete metric spaces.
class CMet
 
Definitiondf-cfil 23047* 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 23048* 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)})
 
Definitiondf-cmet 23049* Define the class of complete metrics. (Contributed by Mario Carneiro, 1-May-2014.)
CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅})
 
Theoremlmmbr 23050* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21027. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ran ℤ(𝐹𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥))))
 
Theoremlmmbr2 23051* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space. Definition 1.4-1 of [Kreyszig] p. 25. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21027. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷𝑃) < 𝑥))))
 
Theoremlmmbr3 23052* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ 𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷𝑃) < 𝑥))))
 
Theoremlmmcvg 23053* Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑅 ∈ ℝ+)       (𝜑 → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐴𝑋 ∧ (𝐴𝐷𝑃) < 𝑅))
 
Theoremlmmbrf 23054* Express the binary relation "sequence 𝐹 converges to point 𝑃 " in a metric space using an arbitrary upper set of integers. This version of lmmbr2 23051 presupposes that 𝐹 is a function. (Contributed by NM, 20-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   𝑍 = (ℤ𝑀)    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   (𝜑𝐹:𝑍𝑋)       (𝜑 → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝑃𝑋 ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐴𝐷𝑃) < 𝑥)))
 
Theoremlmnn 23055* A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝐹:ℕ⟶𝑋)    &   ((𝜑𝑘 ∈ ℕ) → ((𝐹𝑘)𝐷𝑃) < (1 / 𝑘))       (𝜑𝐹(⇝𝑡𝐽)𝑃)
 
Theoremcfilfval 23056* The set of Cauchy filters on a metric space. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (CauFil‘𝐷) = {𝑓 ∈ (Fil‘𝑋) ∣ ∀𝑥 ∈ ℝ+𝑦𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})
 
Theoremiscfil 23057* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremiscfil2 23058* The property of being a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐹𝑧𝑦𝑤𝑦 (𝑧𝐷𝑤) < 𝑥)))
 
Theoremcfilfil 23059 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))
 
Theoremcfili 23060* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝐹𝑦𝑥𝑧𝑥 (𝑦𝐷𝑧) < 𝑅)
 
Theoremcfil3i 23061* A Cauchy filter contains balls of any pre-chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝐹)
 
Theoremcfilss 23062 A filter finer than a Cauchy filter is Cauchy. (Contributed by Mario Carneiro, 13-Oct-2015.)
(((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) ∧ (𝐺 ∈ (Fil‘𝑋) ∧ 𝐹𝐺)) → 𝐺 ∈ (CauFil‘𝐷))
 
Theoremfgcfil 23063* The Cauchy filter condition for a filter base. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑋)) → ((𝑋filGen𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑦𝐵𝑧𝑦𝑤𝑦 (𝑧𝐷𝑤) < 𝑥))
 
Theoremfmcfil 23064* The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑦𝐵𝑧𝑦𝑤𝑦 ((𝐹𝑧)𝐷(𝐹𝑤)) < 𝑥))
 
Theoremiscfil3 23065* A filter is Cauchy iff it contains a ball of any chosen size. (Contributed by Mario Carneiro, 15-Oct-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ∈ (Fil‘𝑋) ∧ ∀𝑟 ∈ ℝ+𝑥𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹)))
 
Theoremcfilfcls 23066 Similar to ultrafilters (uffclsflim 21829), the cluster points and limit points of a Cauchy filter coincide. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   𝑋 = dom dom 𝐷       (𝐹 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝐹) = (𝐽 fLim 𝐹))
 
Theoremcaufval 23067* The set of Cauchy sequences on a metric space. (Contributed by NM, 8-Sep-2006.) (Revised by Mario Carneiro, 5-Sep-2015.)
(𝐷 ∈ (∞Met‘𝑋) → (Cau‘𝐷) = {𝑓 ∈ (𝑋pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝑓 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝑓𝑘)(ball‘𝐷)𝑥)})
 
Theoremiscau 23068* Express the property "𝐹 is a Cauchy sequence of metric 𝐷." Part of Definition 1.4-3 of [Kreyszig] p. 28. The condition 𝐹 ⊆ (ℂ × 𝑋) allows us to use objects more general than sequences when convenient; see the comment in df-lm 21027. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑘 ∈ ℤ (𝐹 ↾ (ℤ𝑘)):(ℤ𝑘)⟶((𝐹𝑘)(ball‘𝐷)𝑥))))
 
Theoremiscau2 23069* Express the property "𝐹 is a Cauchy sequence of metric 𝐷," using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ((𝐹𝑘)𝐷(𝐹𝑗)) < 𝑥))))
 
Theoremiscau3 23070* Express the Cauchy sequence property in the more conventional three-quantifier form. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Nov-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ𝑘)((𝐹𝑘)𝐷(𝐹𝑚)) < 𝑥))))
 
Theoremiscau4 23071* Express the property "𝐹 is a Cauchy sequence of metric 𝐷," using an arbitrary upper set of integers. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝑘 ∈ dom 𝐹𝐴𝑋 ∧ (𝐴𝐷𝐵) < 𝑥))))
 
Theoremiscauf 23072* Express the property "𝐹 is a Cauchy sequence of metric 𝐷 " presupposing 𝐹 is a function. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 23-Dec-2013.)
𝑍 = (ℤ𝑀)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   ((𝜑𝑘𝑍) → (𝐹𝑘) = 𝐴)    &   ((𝜑𝑗𝑍) → (𝐹𝑗) = 𝐵)    &   (𝜑𝐹:𝑍𝑋)       (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+𝑗𝑍𝑘 ∈ (ℤ𝑗)(𝐵𝐷𝐴) < 𝑥))
 
Theoremcaun0 23073 A metric with a Cauchy sequence cannot be empty. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝑋 ≠ ∅)
 
Theoremcaufpm 23074 Inclusion of a Cauchy sequence, under our definition. (Contributed by NM, 7-Dec-2006.) (Revised by Mario Carneiro, 24-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ (𝑋pm ℂ))
 
Theoremcaucfil 23075 A Cauchy sequence predicate can be expressed in terms of the Cauchy filter predicate for a suitably chosen filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ𝑍))       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷)))
 
Theoremiscmet 23076* The property "𝐷 is a complete metric." meaning all Cauchy filters converge to a point in the space. (Contributed by Mario Carneiro, 1-May-2014.) (Revised by Mario Carneiro, 13-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
 
Theoremcmetcvg 23077 The convergence of a Cauchy filter in a complete metric space. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝐹) ≠ ∅)
 
Theoremcmetmet 23078 A complete metric space is a metric space. (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 29-Jan-2014.)
(𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋))
 
Theoremcmetmeti 23079 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
𝐷 ∈ (CMet‘𝑋)       𝐷 ∈ (Met‘𝑋)
 
Theoremcmetcaulem 23080* Lemma for cmetcau 23081. (Contributed by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (CMet‘𝑋))    &   (𝜑𝑃𝑋)    &   (𝜑𝐹 ∈ (Cau‘𝐷))    &   𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹𝑥), 𝑃))       (𝜑𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremcmetcau 23081 The convergence of a Cauchy sequence in a complete metric space. (Contributed by NM, 19-Dec-2006.) (Revised by Mario Carneiro, 14-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (CMet‘𝑋) ∧ 𝐹 ∈ (Cau‘𝐷)) → 𝐹 ∈ dom (⇝𝑡𝐽))
 
Theoremiscmet3lem3 23082* Lemma for iscmet3 23085. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((1 / 2)↑𝑘) < 𝑅)
 
Theoremiscmet3lem1 23083* Lemma for iscmet3 23085. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))       (𝜑𝐹 ∈ (Cau‘𝐷))
 
Theoremiscmet3lem2 23084* Lemma for iscmet3 23085. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝐹:𝑍𝑋)    &   (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆𝑘)∀𝑣 ∈ (𝑆𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘))    &   (𝜑 → ∀𝑘𝑍𝑛 ∈ (𝑀...𝑘)(𝐹𝑘) ∈ (𝑆𝑛))    &   (𝜑𝐺 ∈ (Fil‘𝑋))    &   (𝜑𝑆:ℤ⟶𝐺)    &   (𝜑𝐹 ∈ dom (⇝𝑡𝐽))       (𝜑 → (𝐽 fLim 𝐺) ≠ ∅)
 
Theoremiscmet3 23085* The property "𝐷 is a complete metric" expressed in terms of functions on (or any other upper integer set). Thus, we only have to look at functions on , and not all possible Cauchy filters, to determine completeness. (The proof uses countable choice.) (Contributed by NM, 18-Dec-2006.) (Revised by Mario Carneiro, 5-May-2014.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐷 ∈ (Met‘𝑋))       (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍𝑋𝑓 ∈ dom (⇝𝑡𝐽))))
 
Theoremiscmet2 23086 A metric 𝐷 is complete iff all Cauchy sequences converge to a point in the space. The proof uses countable choice. Part of Definition 1.4-3 of [Kreyszig] p. 28. (Contributed by NM, 7-Sep-2006.) (Revised by Mario Carneiro, 15-Oct-2015.)
𝐽 = (MetOpen‘𝐷)       (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ (Cau‘𝐷) ⊆ dom (⇝𝑡𝐽)))
 
Theoremcfilresi 23087 A Cauchy filter on a metric subspace extends to a Cauchy filter in the larger space. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝐹) ∈ (CauFil‘𝐷))
 
Theoremcfilres 23088 Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))))
 
Theoremcaussi 23089 Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
(𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷))
 
Theoremcauss 23090 Cauchy sequence on a metric subspace. (Contributed by NM, 29-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹:ℕ⟶𝑌) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐹 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌)))))
 
Theoremequivcfil 23091* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy filters are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (CauFil‘𝐷) ⊆ (CauFil‘𝐶))
 
Theoremequivcau 23092* If the metric 𝐷 is "strongly finer" than 𝐶 (meaning that there is a positive real constant 𝑅 such that 𝐶(𝑥, 𝑦) ≤ 𝑅 · 𝐷(𝑥, 𝑦)), all the 𝐷-Cauchy sequences are also 𝐶-Cauchy. (Using this theorem twice in each direction states that if two metrics are strongly equivalent, then they have the same Cauchy sequences.) (Contributed by Mario Carneiro, 14-Sep-2015.)
(𝜑𝐶 ∈ (Met‘𝑋))    &   (𝜑𝐷 ∈ (Met‘𝑋))    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑥𝐶𝑦) ≤ (𝑅 · (𝑥𝐷𝑦)))       (𝜑 → (Cau‘𝐷) ⊆ (Cau‘𝐶))
 
Theoremlmle 23093* If the distance from each member of a converging sequence to a given point is less than or equal to a given amount, so is the convergence value. (Contributed by NM, 23-Dec-2007.) (Proof shortened by Mario Carneiro, 1-May-2014.)
𝑍 = (ℤ𝑀)    &   𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑀 ∈ ℤ)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑄𝑋)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑘𝑍) → (𝑄𝐷(𝐹𝑘)) ≤ 𝑅)       (𝜑 → (𝑄𝐷𝑃) ≤ 𝑅)
 
Theoremnglmle 23094* If the norm of each member of a converging sequence is less than or equal to a given amount, so is the norm of the convergence value. (Contributed by NM, 25-Dec-2007.) (Revised by AV, 16-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))    &   𝐽 = (MetOpen‘𝐷)    &   𝑁 = (norm‘𝐺)    &   (𝜑𝐺 ∈ NrmGrp)    &   (𝜑𝐹:ℕ⟶𝑋)    &   (𝜑𝐹(⇝𝑡𝐽)𝑃)    &   (𝜑𝑅 ∈ ℝ*)    &   ((𝜑𝑘 ∈ ℕ) → (𝑁‘(𝐹𝑘)) ≤ 𝑅)       (𝜑 → (𝑁𝑃) ≤ 𝑅)
 
Theoremlmclim 23095 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑍 ⊆ dom 𝐹) → (𝐹(⇝𝑡𝐽)𝑃 ↔ (𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝐹𝑃)))
 
Theoremlmclimf 23096 Relate a limit on the metric space of complex numbers to our complex number limit notation. (Contributed by NM, 24-Jul-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (TopOpen‘ℂfld)    &   𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶ℂ) → (𝐹(⇝𝑡𝐽)𝑃𝐹𝑃))
 
Theoremmetelcls 23097* A point belongs to the closure of a subset iff there is a sequence in the subset converging to it. Theorem 1.4-6(a) of [Kreyszig] p. 30. This proof uses countable choice ax-cc 9254. The statement can be generalized to first-countable spaces, not just metrizable spaces. (Contributed by NM, 8-Nov-2007.) (Proof shortened by Mario Carneiro, 1-May-2015.)
𝐽 = (MetOpen‘𝐷)    &   (𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝑆𝑋)       (𝜑 → (𝑃 ∈ ((cls‘𝐽)‘𝑆) ↔ ∃𝑓(𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑃)))
 
Theoremmetcld 23098* A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by NM, 11-Nov-2007.) (Revised by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ∀𝑥𝑓((𝑓:ℕ⟶𝑆𝑓(⇝𝑡𝐽)𝑥) → 𝑥𝑆)))
 
Theoremmetcld2 23099 A subset of a metric space is closed iff every convergent sequence on it converges to a point in the subset. Theorem 1.4-6(b) of [Kreyszig] p. 30. (Contributed by Mario Carneiro, 1-May-2014.)
𝐽 = (MetOpen‘𝐷)       ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑆𝑋) → (𝑆 ∈ (Clsd‘𝐽) ↔ ((⇝𝑡𝐽) “ (𝑆𝑚 ℕ)) ⊆ 𝑆))
 
Theoremcaubl 23100* Sufficient condition to ensure a sequence of nested balls is Cauchy. (Contributed by Mario Carneiro, 18-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
(𝜑𝐷 ∈ (∞Met‘𝑋))    &   (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))    &   (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))    &   (𝜑 → ∀𝑟 ∈ ℝ+𝑛 ∈ ℕ (2nd ‘(𝐹𝑛)) < 𝑟)       (𝜑 → (1st𝐹) ∈ (Cau‘𝐷))
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