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Type | Label | Description |
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Statement | ||
Theorem | cphnm 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) = (√‘(𝐴 , 𝐴))) | ||
Theorem | nmsq 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝑁‘𝐴)↑2) = (𝐴 , 𝐴)) | ||
Theorem | cphnmf 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (𝑊 ∈ ℂPreHil → 𝑁:𝑉⟶𝐾) | ||
Theorem | cphnmcl 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ 𝐾) | ||
Theorem | reipcl 22805 | An inner product of an element with itself is real. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 𝐴) ∈ ℝ) | ||
Theorem | ipge0 22806 | The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → 0 ≤ (𝐴 , 𝐴)) | ||
Theorem | cphipcj 22807 | Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 19798. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) | ||
Theorem | cphipipcj 22808 | An inner product times its conjugate. (Contributed by NM, 23-Nov-2007.) (Revised by AV, 19-Oct-2021.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) · (𝐵 , 𝐴)) = ((abs‘(𝐴 , 𝐵))↑2)) | ||
Theorem | cphorthcom 22809 | Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 19799. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0)) | ||
Theorem | cphip0l 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.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 0) | ||
Theorem | cphip0r 22811 | Inner product with a zero second argument. Complex version of ip0r 19801. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 0 ) = 0) | ||
Theorem | cphipeq0 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.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 )) | ||
Theorem | cphdir 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.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) + (𝐵 , 𝐶))) | ||
Theorem | cphdi 22814 | Distributive law for inner product (left-distributivity). Complex version of ipdi 19804. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) + (𝐴 , 𝐶))) | ||
Theorem | cph2di 22815 | Distributive law for inner product. Complex version of ip2di 19805. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 + 𝐵) , (𝐶 + 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) + ((𝐴 , 𝐷) + (𝐵 , 𝐶)))) | ||
Theorem | cphsubdir 22816 | Distributive law for inner product subtraction. Complex version of ipsubdir 19806. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 − 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶))) | ||
Theorem | cphsubdi 22817 | Distributive law for inner product subtraction. Complex version of ipsubdi 19807. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐴 , (𝐵 − 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶))) | ||
Theorem | cph2subdi 22818 | Distributive law for inner product subtraction. Complex version of ip2subdi 19808. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝐶 ∈ 𝑉) & ⊢ (𝜑 → 𝐷 ∈ 𝑉) ⇒ ⊢ (𝜑 → ((𝐴 − 𝐵) , (𝐶 − 𝐷)) = (((𝐴 , 𝐶) + (𝐵 , 𝐷)) − ((𝐴 , 𝐷) + (𝐵 , 𝐶)))) | ||
Theorem | cphass 22819 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 19809, his5 27327. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐴 · 𝐵) , 𝐶) = (𝐴 · (𝐵 , 𝐶))) | ||
Theorem | cphassr 22820 | "Associative" law for second argument of inner product (compare cphass 22819). See ipassr 19810, his52 . (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐵 , (𝐴 · 𝐶)) = ((∗‘𝐴) · (𝐵 , 𝐶))) | ||
Theorem | cph2ass 22821 | Move scalar multiplication to outside of inner product. See his35 27329. (Contributed by Mario Carneiro, 17-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ (𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝐾) ∧ (𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉)) → ((𝐴 · 𝐶) , (𝐵 · 𝐷)) = ((𝐴 · (∗‘𝐵)) · (𝐶 , 𝐷))) | ||
Theorem | cphassi 22822 | Associative law for the first argument of an inner product with scalar _𝑖. (Contributed by AV, 17-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((i · 𝐵) , 𝐴) = (i · (𝐵 , 𝐴))) | ||
Theorem | cphassir 22823 | "Associative" law for the second argument of an inner product with scalar _𝑖. (Contributed by AV, 17-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , (i · 𝐵)) = (-i · (𝐴 , 𝐵))) | ||
Theorem | tchex 22824* | Lemma for tchbas 22826 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V | ||
Theorem | tchval 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) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) | ||
Theorem | tchbas 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) | ||
Theorem | tchplusg 22827 | The addition operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ + = (+g‘𝐺) | ||
Theorem | tchsub 22828 | The subtraction operation of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ − = (-g‘𝑊) ⇒ ⊢ − = (-g‘𝐺) | ||
Theorem | tchmulr 22829 | The ring operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ · = (.r‘𝑊) ⇒ ⊢ · = (.r‘𝐺) | ||
Theorem | tchsca 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) | ||
Theorem | tchvsca 22831 | The scalar multiplication of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ · = ( ·𝑠 ‘𝐺) | ||
Theorem | tchip 22832 | The inner product of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ · = (·𝑖‘𝑊) ⇒ ⊢ · = (·𝑖‘𝐺) | ||
Theorem | tchtopn 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) ⇒ ⊢ (𝑊 ∈ 𝑉 → 𝐽 = (MetOpen‘𝐷)) | ||
Theorem | tchphl 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) | ||
Theorem | tchnmfval 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) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) | ||
Theorem | tchnmval 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) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) = (√‘(𝑋 , 𝑋))) | ||
Theorem | cphtchnm 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)) | ||
Theorem | tchds 22838 | The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝑊) ⇒ ⊢ (𝑊 ∈ Grp → (𝑁 ∘ − ) = (dist‘𝐺)) | ||
Theorem | tchclm 22839 | Lemma for tchcph 22844. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) ⇒ ⊢ (𝜑 → 𝑊 ∈ ℂMod) | ||
Theorem | tchcphlem3 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) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → (𝑋 , 𝑋) ∈ ℝ) | ||
Theorem | ipcau2 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) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) & ⊢ , = (·𝑖‘𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝐶 = ((𝑌 , 𝑋) / (𝑌 , 𝑌)) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌))) | ||
Theorem | tchcphlem1 22842* | Lemma for tchcph 22844: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) & ⊢ , = (·𝑖‘𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ − = (-g‘𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (√‘((𝑋 − 𝑌) , (𝑋 − 𝑌))) ≤ ((√‘(𝑋 , 𝑋)) + (√‘(𝑌 , 𝑌)))) | ||
Theorem | tchcphlem2 22843* | Lemma for tchcph 22844: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) & ⊢ , = (·𝑖‘𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ (𝜑 → 𝑋 ∈ 𝐾) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) ⇒ ⊢ (𝜑 → (√‘((𝑋 · 𝑌) , (𝑋 · 𝑌))) = ((abs‘𝑋) · (√‘(𝑌 , 𝑌)))) | ||
Theorem | tchcph 22844* | The standard definition of a norm turns any pre-Hilbert space over a quadratically closed subfield of ℂ 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) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) & ⊢ , = (·𝑖‘𝑊) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐾 ∧ 𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → 0 ≤ (𝑥 , 𝑥)) ⇒ ⊢ (𝜑 → 𝐺 ∈ ℂPreHil) | ||
Theorem | ipcau 22845 | The Cauchy-Schwarz inequality for a complex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (abs‘(𝑋 , 𝑌)) ≤ ((𝑁‘𝑋) · (𝑁‘𝑌))) | ||
Theorem | nmparlem 22846 | Lemma for nmpar 22847. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) ⇒ ⊢ (𝜑 → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
Theorem | nmpar 22847 | A complex pre-Hilbert space satisfies the parallelogram law. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (((𝑁‘(𝐴 + 𝐵))↑2) + ((𝑁‘(𝐴 − 𝐵))↑2)) = (2 · (((𝑁‘𝐴)↑2) + ((𝑁‘𝐵)↑2)))) | ||
Theorem | cphipval2 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) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2)))) / 4)) | ||
Theorem | 4cphipval2 22849 | Four times the inner product value cphipval2 22848. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 18-Oct-2021.) |
⊢ 𝑋 = (Base‘𝑊) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ − = (-g‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (4 · (𝐴 , 𝐵)) = ((((𝑁‘(𝐴 + 𝐵))↑2) − ((𝑁‘(𝐴 − 𝐵))↑2)) + (i · (((𝑁‘(𝐴 + (i · 𝐵)))↑2) − ((𝑁‘(𝐴 − (i · 𝐵)))↑2))))) | ||
Theorem | cphipval 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) & ⊢ + = (+g‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ (((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾) ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴 , 𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴 + ((i↑𝑘) · 𝐵)))↑2)) / 4)) | ||
Theorem | ipcnlem2 22851 | The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝑇 = ((𝑅 / 2) / ((𝑁‘𝐴) + 1)) & ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇)) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) & ⊢ (𝜑 → 𝑋 ∈ 𝑉) & ⊢ (𝜑 → 𝑌 ∈ 𝑉) & ⊢ (𝜑 → (𝐴𝐷𝑋) < 𝑈) & ⊢ (𝜑 → (𝐵𝐷𝑌) < 𝑇) ⇒ ⊢ (𝜑 → (abs‘((𝐴 , 𝐵) − (𝑋 , 𝑌))) < 𝑅) | ||
Theorem | ipcnlem1 22852* | The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝑇 = ((𝑅 / 2) / ((𝑁‘𝐴) + 1)) & ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝐵) + 𝑇)) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐴 ∈ 𝑉) & ⊢ (𝜑 → 𝐵 ∈ 𝑉) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ 𝑉 ∀𝑦 ∈ 𝑉 (((𝐴𝐷𝑥) < 𝑟 ∧ (𝐵𝐷𝑦) < 𝑟) → (abs‘((𝐴 , 𝐵) − (𝑥 , 𝑦))) < 𝑅)) | ||
Theorem | ipcn 22853 | The inner product operation of a complex pre-Hilbert space is continuous. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ , = (·if‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) & ⊢ 𝐾 = (TopOpen‘ℂfld) ⇒ ⊢ (𝑊 ∈ ℂPreHil → , ∈ ((𝐽 ×t 𝐽) Cn 𝐾)) | ||
Theorem | cnmpt1ip 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ (𝐾 Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ 𝐵) ∈ (𝐾 Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝐴 , 𝐵)) ∈ (𝐾 Cn 𝐶)) | ||
Theorem | cnmpt2ip 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ ℂPreHil) & ⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝑋)) & ⊢ (𝜑 → 𝐿 ∈ (TopOn‘𝑌)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐴) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) & ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ 𝐵) ∈ ((𝐾 ×t 𝐿) Cn 𝐽)) ⇒ ⊢ (𝜑 → (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑌 ↦ (𝐴 , 𝐵)) ∈ ((𝐾 ×t 𝐿) Cn 𝐶)) | ||
Theorem | csscld 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 ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ (Clsd‘𝐽)) | ||
Theorem | clsocv 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 ∧ 𝑆 ⊆ 𝑉) → (𝑂‘((cls‘𝐽)‘𝑆)) = (𝑂‘𝑆)) | ||
Syntax | ccfil 22858 | Extend class notation with the set of Cauchy filters. |
class CauFil | ||
Syntax | cca 22859 | Extend class notation with a function on metric spaces whose value is the set of all Cauchy sequences of the space. |
class Cau | ||
Syntax | cms 22860 | Extend class notation with class of complete metric spaces. |
class CMet | ||
Definition | df-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+ ∃𝑦 ∈ 𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) | ||
Definition | df-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 𝑑 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑗)):(ℤ≥‘𝑗)⟶((𝑓‘𝑗)(ball‘𝑑)𝑥)}) | ||
Definition | df-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) ≠ ∅}) | ||
Theorem | lmmbr 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ ran ℤ≥(𝐹 ↾ 𝑦):𝑦⟶(𝑃(ball‘𝐷)𝑥)))) | ||
Theorem | lmmbr2 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑃) < 𝑥)))) | ||
Theorem | lmmbr3 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ 𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷𝑃) < 𝑥)))) | ||
Theorem | lmmcvg 22867* | Convergence property of a converging sequence. (Contributed by NM, 1-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) & ⊢ (𝜑 → 𝑅 ∈ ℝ+) ⇒ ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝑃) < 𝑅)) | ||
Theorem | lmmbrf 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) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) ⇒ ⊢ (𝜑 → (𝐹(⇝𝑡‘𝐽)𝑃 ↔ (𝑃 ∈ 𝑋 ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐴𝐷𝑃) < 𝑥))) | ||
Theorem | lmnn 22869* | A condition that implies convergence. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 1-May-2014.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝐹:ℕ⟶𝑋) & ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝐹‘𝑘)𝐷𝑃) < (1 / 𝑘)) ⇒ ⊢ (𝜑 → 𝐹(⇝𝑡‘𝐽)𝑃) | ||
Theorem | cfilfval 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+ ∃𝑦 ∈ 𝑓 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)}) | ||
Theorem | iscfil 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+ ∃𝑦 ∈ 𝐹 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)))) | ||
Theorem | iscfil2 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+ ∃𝑦 ∈ 𝐹 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥))) | ||
Theorem | cfilfil 22873 | A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋)) | ||
Theorem | cfili 22874* | Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥 ∈ 𝐹 ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦𝐷𝑧) < 𝑅) | ||
Theorem | cfil3i 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+) → ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝐹) | ||
Theorem | cfilss 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) ∧ 𝐹 ⊆ 𝐺)) → 𝐺 ∈ (CauFil‘𝐷)) | ||
Theorem | fgcfil 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+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 (𝑧𝐷𝑤) < 𝑥)) | ||
Theorem | fmcfil 22878* | The Cauchy filter condition for a filter map. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐵 ∈ (fBas‘𝑌) ∧ 𝐹:𝑌⟶𝑋) → (((𝑋 FilMap 𝐹)‘𝐵) ∈ (CauFil‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝑦 ∀𝑤 ∈ 𝑦 ((𝐹‘𝑧)𝐷(𝐹‘𝑤)) < 𝑥)) | ||
Theorem | iscfil3 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+ ∃𝑥 ∈ 𝑋 (𝑥(ball‘𝐷)𝑟) ∈ 𝐹))) | ||
Theorem | cfilfcls 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)) | ||
Theorem | caufval 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 ∈ (𝑋 ↑pm ℂ) ∣ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝑓 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝑓‘𝑘)(ball‘𝐷)𝑥)}) | ||
Theorem | iscau 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 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑘 ∈ ℤ (𝐹 ↾ (ℤ≥‘𝑘)):(ℤ≥‘𝑘)⟶((𝐹‘𝑘)(ball‘𝐷)𝑥)))) | ||
Theorem | iscau2 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 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ((𝐹‘𝑘)𝐷(𝐹‘𝑗)) < 𝑥)))) | ||
Theorem | iscau3 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) ⇒ ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ (𝐹‘𝑘) ∈ 𝑋 ∧ ∀𝑚 ∈ (ℤ≥‘𝑘)((𝐹‘𝑘)𝐷(𝐹‘𝑚)) < 𝑥)))) | ||
Theorem | iscau4 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵) ⇒ ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ (𝐹 ∈ (𝑋 ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝑘 ∈ dom 𝐹 ∧ 𝐴 ∈ 𝑋 ∧ (𝐴𝐷𝐵) < 𝑥)))) | ||
Theorem | iscauf 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 = (ℤ≥‘𝑀) & ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑋)) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) & ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗) = 𝐵) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) ⇒ ⊢ (𝜑 → (𝐹 ∈ (Cau‘𝐷) ↔ ∀𝑥 ∈ ℝ+ ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)(𝐵𝐷𝐴) < 𝑥)) | ||
Theorem | caun0 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 ≠ ∅) | ||
Theorem | caufpm 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 ∈ (𝑋 ↑pm ℂ)) | ||
Theorem | caucfil 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 = (ℤ≥‘𝑀) & ⊢ 𝐿 = ((𝑋 FilMap 𝐹)‘(ℤ≥ “ 𝑍)) ⇒ ⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑀 ∈ ℤ ∧ 𝐹:𝑍⟶𝑋) → (𝐹 ∈ (Cau‘𝐷) ↔ 𝐿 ∈ (CauFil‘𝐷))) | ||
Theorem | iscmet 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) ≠ ∅)) | ||
Theorem | cmetcvg 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) ≠ ∅) | ||
Theorem | cmetmet 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)) | ||
Theorem | cmetmeti 22893 | A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.) |
⊢ 𝐷 ∈ (CMet‘𝑋) ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
Theorem | cmetcaulem 22894* | Lemma for cmetcau 22895. (Contributed by Mario Carneiro, 14-Oct-2015.) |
⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝐷 ∈ (CMet‘𝑋)) & ⊢ (𝜑 → 𝑃 ∈ 𝑋) & ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) & ⊢ 𝐺 = (𝑥 ∈ ℕ ↦ if(𝑥 ∈ dom 𝐹, (𝐹‘𝑥), 𝑃)) ⇒ ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) | ||
Theorem | cmetcau 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 (⇝𝑡‘𝐽)) | ||
Theorem | iscmet3lem3 22896* | Lemma for iscmet3 22899. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) ⇒ ⊢ ((𝑀 ∈ ℤ ∧ 𝑅 ∈ ℝ+) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ≥‘𝑗)((1 / 2)↑𝑘) < 𝑅) | ||
Theorem | iscmet3lem1 22897* | Lemma for iscmet3 22899. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) & ⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) & ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛)) ⇒ ⊢ (𝜑 → 𝐹 ∈ (Cau‘𝐷)) | ||
Theorem | iscmet3lem2 22898* | Lemma for iscmet3 22899. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ 𝑍 = (ℤ≥‘𝑀) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) & ⊢ (𝜑 → 𝐹:𝑍⟶𝑋) & ⊢ (𝜑 → ∀𝑘 ∈ ℤ ∀𝑢 ∈ (𝑆‘𝑘)∀𝑣 ∈ (𝑆‘𝑘)(𝑢𝐷𝑣) < ((1 / 2)↑𝑘)) & ⊢ (𝜑 → ∀𝑘 ∈ 𝑍 ∀𝑛 ∈ (𝑀...𝑘)(𝐹‘𝑘) ∈ (𝑆‘𝑛)) & ⊢ (𝜑 → 𝐺 ∈ (Fil‘𝑋)) & ⊢ (𝜑 → 𝑆:ℤ⟶𝐺) & ⊢ (𝜑 → 𝐹 ∈ dom (⇝𝑡‘𝐽)) ⇒ ⊢ (𝜑 → (𝐽 fLim 𝐺) ≠ ∅) | ||
Theorem | iscmet3 22899* | The property "𝐷 is a complete metric" expressed in terms of functions on ℕ (or any other upper integer set). Thus, we only have to look at functions on ℕ, 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 = (ℤ≥‘𝑀) & ⊢ 𝐽 = (MetOpen‘𝐷) & ⊢ (𝜑 → 𝑀 ∈ ℤ) & ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑋)) ⇒ ⊢ (𝜑 → (𝐷 ∈ (CMet‘𝑋) ↔ ∀𝑓 ∈ (Cau‘𝐷)(𝑓:𝑍⟶𝑋 → 𝑓 ∈ dom (⇝𝑡‘𝐽)))) | ||
Theorem | iscmet2 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 (⇝𝑡‘𝐽))) |
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