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Theorem List for Metamath Proof Explorer - 22801-22900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcphnm 22801 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))

Theoremnmsq 22802 The square of the norm is the norm of an inner product in a normed 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 22803 The norm of a vector is a member of the scalar field in a complex 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 22804 The norm of a vector is a member of the scalar field in a complex 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 22805 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 22806 The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → 0 ≤ (𝐴 , 𝐴))

Theoremcphipcj 22807 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 19798. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremcphipipcj 22808 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 22809 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 19799. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))

Theoremcphip0l 22810 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 19800. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 0)

Theoremcphip0r 22811 Inner product with a zero second argument. Complex version of ip0r 19801. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 0)

Theoremcphipeq0 22812 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 19802. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 ))

Theoremcphdir 22813 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 19803. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) + (𝐵 , 𝐶)))

Theoremcphdi 22814 Distributive law for inner product (left-distributivity). Complex version of ipdi 19804. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) + (𝐴 , 𝐶)))

Theoremcph2di 22815 Distributive law for inner product. Complex version of ip2di 19805. (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 22816 Distributive law for inner product subtraction. Complex version of ipsubdir 19806. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶)))

Theoremcphsubdi 22817 Distributive law for inner product subtraction. Complex version of ipsubdi 19807. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶)))

Theoremcph2subdi 22818 Distributive law for inner product subtraction. Complex version of ip2subdi 19808. (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 22819 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 19809, his5 27327. (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 22820 "Associative" law for second argument of inner product (compare cphass 22819). See ipassr 19810, 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 22821 Move scalar multiplication to outside of inner product. See his35 27329. (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 22822 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 22823 "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 22824* Lemma for tchbas 22826 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V

Theoremtchval 22825* Define a function to augment a 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 22826 The base set of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)       𝑉 = (Base‘𝐺)

Theoremtchplusg 22827 The addition operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    + = (+g𝑊)        + = (+g𝐺)

Theoremtchsub 22828 The subtraction operation of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂHil‘𝑊)    &    = (-g𝑊)        = (-g𝐺)

Theoremtchmulr 22829 The ring operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (.r𝑊)        · = (.r𝐺)

Theoremtchsca 22830 The scalar field of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝐹 = (Scalar‘𝑊)       𝐹 = (Scalar‘𝐺)

Theoremtchvsca 22831 The scalar multiplication of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = ( ·𝑠𝑊)        · = ( ·𝑠𝐺)

Theoremtchip 22832 The inner product of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (·𝑖𝑊)        · = (·𝑖𝐺)

Theoremtchtopn 22833 The topology of a 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 22834 Augmentation of a 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 22835* The norm of a 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 22836 The norm of a 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 22837 The norm of a norm-augmented complex 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 22838 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 22839 Lemma for tchcph 22844. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))       (𝜑𝑊 ∈ ℂMod)

Theoremtchcphlem3 22840 Lemma for tchcph 22844: 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 22841* The Cauchy-Schwarz inequality for a complex 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 22842* Lemma for tchcph 22844: 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 22843* Lemma for tchcph 22844: 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 22844* The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of into a complex 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 22845 The Cauchy-Schwarz inequality for a complex 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 22846 Lemma for nmpar 22847. (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 22847 A complex 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 22848 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 22849 Four times the inner product value cphipval2 22848. (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 22850* 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 22851 The inner product operation of a complex 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 22852* The inner product operation of a complex 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 22853 The inner product operation of a complex 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 22854* Continuity of inner product; analogue of cnmpt12f 21279 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 22855* Continuity of inner product; analogue of cnmpt22f 21288 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 22856 A "closed subspace" in a complex 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 22857 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 22858 Extend class notation with the set of Cauchy filters.
class CauFil

Syntaxcca 22859 Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space.
class Cau

Syntaxcms 22860 Extend class notation with class of complete metric spaces.
class CMet

Definitiondf-cfil 22861* 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 22862* 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 22863* 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 22864* 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 20843. (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 22865* 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 20843. (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 22866* 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 22867* 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 22868* 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 22865 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 22869* 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 22870* 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 22871* 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 22872* 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 22873 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))

Theoremcfili 22874* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝐹𝑦𝑥𝑧𝑥 (𝑦𝐷𝑧) < 𝑅)

Theoremcfil3i 22875* 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 22876 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 22877* 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 22878* 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 22879* 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 22880 Similar to ultrafilters (uffclsflim 21645), 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 22881* 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 22882* 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 20843. (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 22883* 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 22884* 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 22885* 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 22886* 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 22887 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 22888 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 22889 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 22890* 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 22891 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 22892 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 22893 A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.)
𝐷 ∈ (CMet‘𝑋)       𝐷 ∈ (Met‘𝑋)

Theoremcmetcaulem 22894* Lemma for cmetcau 22895. (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 22895 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 22896* Lemma for iscmet3 22899. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑍 = (ℤ𝑀)       ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗𝑍𝑘 ∈ (ℤ𝑗)((1 / 2)↑𝑘) < 𝑅)

Theoremiscmet3lem1 22897* Lemma for iscmet3 22899. (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 22898* Lemma for iscmet3 22899. (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 22899* 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 22900 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)))

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