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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | cphnmcl 23801 | 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) & ⊢ , = (·𝑖‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ 𝐾 = (Base‘𝐹) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝑁‘𝐴) ∈ 𝐾) | ||
Theorem | reipcl 23802 | 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 23803 | The inner product in a subcomplex 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 23804 | Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj 20323. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴)) | ||
Theorem | cphipipcj 23805 | 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 23806 | Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 20324. (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 23807 | Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 20325. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → ( 0 , 𝐴) = 0) | ||
Theorem | cphip0r 23808 | Inner product with a zero second argument. Complex version of ip0r 20326. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ , = (·𝑖‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝐴 ∈ 𝑉) → (𝐴 , 0 ) = 0) | ||
Theorem | cphipeq0 23809 | 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 20327. (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 23810 | Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 20328. (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 23811 | Distributive law for inner product (left-distributivity). Complex version of ipdi 20329. (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 23812 | Distributive law for inner product. Complex version of ip2di 20330. (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 23813 | Distributive law for inner product subtraction. Complex version of ipsubdir 20331. (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 23814 | Distributive law for inner product subtraction. Complex version of ipsubdi 20332. (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 23815 | Distributive law for inner product subtraction. Complex version of ip2subdi 20333. (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 23816 | Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 20334, his5 28869. (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 23817 | "Associative" law for second argument of inner product (compare cphass 23816). See ipassr 20335, 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 23818 | Move scalar multiplication to outside of inner product. See his35 28871. (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 23819 | 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 23820 | "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 | tcphex 23821* | Lemma for tcphbas 23823 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V | ||
Theorem | tcphval 23822* | Define a function to augment a subcomplex pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ 𝐺 = (𝑊 toNrmGrp (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) | ||
Theorem | tcphbas 23823 | The base set of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) ⇒ ⊢ 𝑉 = (Base‘𝐺) | ||
Theorem | tchplusg 23824 | The addition operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ + = (+g‘𝑊) ⇒ ⊢ + = (+g‘𝐺) | ||
Theorem | tcphsub 23825 | The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ − = (-g‘𝑊) ⇒ ⊢ − = (-g‘𝐺) | ||
Theorem | tcphmulr 23826 | The ring operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ · = (.r‘𝑊) ⇒ ⊢ · = (.r‘𝐺) | ||
Theorem | tcphsca 23827 | The scalar field of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) ⇒ ⊢ 𝐹 = (Scalar‘𝐺) | ||
Theorem | tcphvsca 23828 | The scalar multiplication of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ · = ( ·𝑠 ‘𝑊) ⇒ ⊢ · = ( ·𝑠 ‘𝐺) | ||
Theorem | tcphip 23829 | The inner product of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ · = (·𝑖‘𝑊) ⇒ ⊢ · = (·𝑖‘𝐺) | ||
Theorem | tcphtopn 23830 | The topology of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝐽 = (TopOpen‘𝐺) ⇒ ⊢ (𝑊 ∈ 𝑉 → 𝐽 = (MetOpen‘𝐷)) | ||
Theorem | tcphphl 23831 | 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 = (toℂPreHil‘𝑊) ⇒ ⊢ (𝑊 ∈ PreHil ↔ 𝐺 ∈ PreHil) | ||
Theorem | tchnmfval 23832* | The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑉 ↦ (√‘(𝑥 , 𝑥)))) | ||
Theorem | tcphnmval 23833 | The norm of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ , = (·𝑖‘𝑊) ⇒ ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉) → (𝑁‘𝑋) = (√‘(𝑋 , 𝑋))) | ||
Theorem | cphtcphnm 23834 | 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 = (toℂPreHil‘𝑊) & ⊢ 𝑁 = (norm‘𝑊) ⇒ ⊢ (𝑊 ∈ ℂPreHil → 𝑁 = (norm‘𝐺)) | ||
Theorem | tcphds 23835 | The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝑊) ⇒ ⊢ (𝑊 ∈ Grp → (𝑁 ∘ − ) = (dist‘𝐺)) | ||
Theorem | phclm 23836 | A pre-Hilbert space whose field of scalars is a restriction of the field of complex numbers is a subcomplex module. TODO: redundant hypotheses. (Contributed by Mario Carneiro, 16-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘𝑊) & ⊢ 𝑉 = (Base‘𝑊) & ⊢ 𝐹 = (Scalar‘𝑊) & ⊢ (𝜑 → 𝑊 ∈ PreHil) & ⊢ (𝜑 → 𝐹 = (ℂfld ↾s 𝐾)) ⇒ ⊢ (𝜑 → 𝑊 ∈ ℂMod) | ||
Theorem | tcphcphlem3 23837 | Lemma for tcphcph 23841: real closure of an inner product of a vector with itself. (Contributed by Mario Carneiro, 10-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘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 23838* | The Cauchy-Schwarz inequality for a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘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 | tcphcphlem1 23839* | Lemma for tcphcph 23841: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘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 | tcphcphlem2 23840* | Lemma for tcphcph 23841: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘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 | tcphcph 23841* | The standard definition of a norm turns any pre-Hilbert space over a subfield of ℂfld closed under square roots of nonnegative reals into a subcomplex pre-Hilbert space (which allows access to a norm, metric, and topology). (Contributed by Mario Carneiro, 11-Oct-2015.) |
⊢ 𝐺 = (toℂPreHil‘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 23842 | The Cauchy-Schwarz inequality for a subcomplex 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 23843 | Lemma for nmpar 23844. (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 23844 | A subcomplex 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 23845 | 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 23846 | Four times the inner product value cphipval2 23845. (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 23847* | 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 23848 | The inner product operation of a subcomplex 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 23849* | The inner product operation of a subcomplex 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 23850 | The inner product operation of a subcomplex 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 23851* | Continuity of inner product; analogue of cnmpt12f 22271 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 23852* | Continuity of inner product; analogue of cnmpt22f 22280 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 23853 | 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 = (ClSubSp‘𝑊) & ⊢ 𝐽 = (TopOpen‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑆 ∈ 𝐶) → 𝑆 ∈ (Clsd‘𝐽)) | ||
Theorem | clsocv 23854 | 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)) | ||
Theorem | cphsscph 23855 | A subspace of a subcomplex pre-Hilbert space is a subcomplex pre-Hilbert space. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 25-Sep-2022.) |
⊢ 𝑋 = (𝑊 ↾s 𝑈) & ⊢ 𝑆 = (LSubSp‘𝑊) ⇒ ⊢ ((𝑊 ∈ ℂPreHil ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ ℂPreHil) | ||
Syntax | ccfil 23856 | Extend class notation with the class of Cauchy filters. |
class CauFil | ||
Syntax | ccau 23857 | Extend class notation with the class of Cauchy sequences. |
class Cau | ||
Syntax | ccmet 23858 | Extend class notation with the class of complete metrics. |
class CMet | ||
Definition | df-cfil 23859* | Define the set of Cauchy filters on a given extended 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 23860* | Define the set of Cauchy sequences on a given extended metric 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 23861* | Define the set of complete metrics on a given set. (Contributed by Mario Carneiro, 1-May-2014.) |
⊢ CMet = (𝑥 ∈ V ↦ {𝑑 ∈ (Met‘𝑥) ∣ ∀𝑓 ∈ (CauFil‘𝑑)((MetOpen‘𝑑) fLim 𝑓) ≠ ∅}) | ||
Theorem | lmmbr 23862* | 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 21834. (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 23863* | 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 21834. (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 23864* | 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 23865* | 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 23866* | 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 23863 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 23867* | 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 23868* | 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 23869* | 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 23870* | 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 23871 | A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋)) | ||
Theorem | cfili 23872* | 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 23873* | 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 23874 | 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 23875* | 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 23876* | 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 23877* | 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 23878 | Similar to ultrafilters (uffclsflim 22636), 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 23879* | 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 23880* | 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 21834. (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 23881* | 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 23882* | 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 23883* | 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 23884* | 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 23885 | 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 23886 | 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 23887 | 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 23888* | 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 23889 | 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 23890 | 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 23891 | A complete metric space is a metric space. (Contributed by NM, 26-Oct-2007.) |
⊢ 𝐷 ∈ (CMet‘𝑋) ⇒ ⊢ 𝐷 ∈ (Met‘𝑋) | ||
Theorem | cmetcaulem 23892* | Lemma for cmetcau 23893. (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 23893 | 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 23894* | Lemma for iscmet3 23897. (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 23895* | Lemma for iscmet3 23897. (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 23896* | Lemma for iscmet3 23897. (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 23897* | 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 23898 | 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))) | ||
Theorem | cfilresi 23899 | 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)) | ||
Theorem | cfilres 23900 | Cauchy filter on a metric subspace. (Contributed by Mario Carneiro, 15-Oct-2015.) |
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝐹) → (𝐹 ∈ (CauFil‘𝐷) ↔ (𝐹 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))) |
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