 Home Metamath Proof ExplorerTheorem List (p. 236 of 444) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28490) Hilbert Space Explorer (28491-30015) Users' Mathboxes (30016-44303)

Theorem List for Metamath Proof Explorer - 23501-23600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremcphreccllem 23501 Lemma for cphreccl 23504. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       ((𝜑𝑋𝐾𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)

Theoremcphsca 23502 A subcomplex pre-Hilbert space is a vector space over a subfield of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))

Theoremcphsubrg 23503 The scalar field of a subcomplex pre-Hilbert space is a subring of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))

Theoremcphreccl 23504 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾𝐴 ≠ 0) → (1 / 𝐴) ∈ 𝐾)

Theoremcphdivcl 23505 The scalar field of a subcomplex pre-Hilbert space is closed under reciprocal. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)

Theoremcphcjcl 23506 The scalar field of a subcomplex pre-Hilbert space is closed under conjugation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (∗‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl 23507 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of nonnegative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)

Theoremcphabscl 23508 The scalar field of a subcomplex pre-Hilbert space is closed under the absolute value operation. (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾) → (abs‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl2 23509 The scalar field of a subcomplex pre-Hilbert space is closed under square roots of all numbers except possibly the negative reals. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝐾 ∧ ¬ -𝐴 ∈ ℝ+) → (√‘𝐴) ∈ 𝐾)

Theoremcphsqrtcl3 23510 If the scalar field of a subcomplex pre-Hilbert space contains the imaginary unit i, then it is closed under square roots (i.e., it is quadratically closed). (Contributed by Mario Carneiro, 11-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ i ∈ 𝐾𝐴𝐾) → (√‘𝐴) ∈ 𝐾)

Theoremcphqss 23511 The scalar field of a subcomplex pre-Hilbert space contains the rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾)

Theoremcphclm 23512 A subcomplex pre-Hilbert space is a subcomplex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)

Theoremcphnmvs 23513 Norm of a scalar product. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((abs‘𝑋) · (𝑁𝑌)))

Theoremcphipcl 23514 An inner product is a member of the complex numbers. (Contributed by Mario Carneiro, 13-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (𝐴 , 𝐵) ∈ ℂ)

Theoremcphnmfval 23515* The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremcphnm 23516 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))

Theoremnmsq 23517 The square of the norm is the norm of an inner product in a subcomplex pre-Hilbert space. Equation I4 of [Ponnusamy] p. 362. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝑁𝐴)↑2) = (𝐴 , 𝐴))

Theoremcphnmf 23518 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝑁:𝑉𝐾)

Theoremcphnmcl 23519 The norm of a vector is a member of the scalar field in a subcomplex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) ∈ 𝐾)

Theoremreipcl 23520 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 23521 The inner product in a subcomplex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → 0 ≤ (𝐴 , 𝐴))

Theoremcphipcj 23522 Conjugate of an inner product in a subcomplex pre-Hilbert space. Complex version of ipcj 20496. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))

Theoremcphipipcj 23523 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 23524 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 20497. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))

Theoremcphip0l 23525 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 20498. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 0)

Theoremcphip0r 23526 Inner product with a zero second argument. Complex version of ip0r 20499. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 0)

Theoremcphipeq0 23527 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 20500. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 ))

Theoremcphdir 23528 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 20501. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) + (𝐵 , 𝐶)))

Theoremcphdi 23529 Distributive law for inner product (left-distributivity). Complex version of ipdi 20502. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) + (𝐴 , 𝐶)))

Theoremcph2di 23530 Distributive law for inner product. Complex version of ip2di 20503. (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 23531 Distributive law for inner product subtraction. Complex version of ipsubdir 20504. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶)))

Theoremcphsubdi 23532 Distributive law for inner product subtraction. Complex version of ipsubdi 20505. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶)))

Theoremcph2subdi 23533 Distributive law for inner product subtraction. Complex version of ip2subdi 20506. (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 23534 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 20507, his5 28658. (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 23535 "Associative" law for second argument of inner product (compare cphass 23534). See ipassr 20508, 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 23536 Move scalar multiplication to outside of inner product. See his35 28660. (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 23537 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 23538 "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)))

Theoremtcphex 23539* Lemma for tcphbas 23541 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V

Theoremtcphval 23540* 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)    &    , = (·𝑖𝑊)       𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremtcphbas 23541 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)

Theoremtchplusg 23542 The addition operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘𝑊)    &    + = (+g𝑊)        + = (+g𝐺)

Theoremtcphsub 23543 The subtraction operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂPreHil‘𝑊)    &    = (-g𝑊)        = (-g𝐺)

Theoremtcphmulr 23544 The ring operation of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘𝑊)    &    · = (.r𝑊)        · = (.r𝐺)

Theoremtcphsca 23545 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)

Theoremtcphvsca 23546 The scalar multiplication of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘𝑊)    &    · = ( ·𝑠𝑊)        · = ( ·𝑠𝐺)

Theoremtcphip 23547 The inner product of a subcomplex pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘𝑊)    &    · = (·𝑖𝑊)        · = (·𝑖𝐺)

Theoremtcphtopn 23548 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)       (𝑊𝑉𝐽 = (MetOpen‘𝐷))

Theoremtcphphl 23549 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)

Theoremtchnmfval 23550* 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)    &    , = (·𝑖𝑊)       (𝑊 ∈ Grp → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))

Theoremtcphnmval 23551 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)    &    , = (·𝑖𝑊)       ((𝑊 ∈ Grp ∧ 𝑋𝑉) → (𝑁𝑋) = (√‘(𝑋 , 𝑋)))

Theoremcphtcphnm 23552 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))

Theoremtcphds 23553 The distance of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂPreHil‘𝑊)    &   𝑁 = (norm‘𝐺)    &    = (-g𝑊)       (𝑊 ∈ Grp → (𝑁 ) = (dist‘𝐺))

Theoremphclm 23554 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)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))       (𝜑𝑊 ∈ ℂMod)

Theoremtcphcphlem3 23555 Lemma for tcphcph 23559: 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)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)       ((𝜑𝑋𝑉) → (𝑋 , 𝑋) ∈ ℝ)

Theoremipcau2 23556* 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)    &   (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)))

Theoremtcphcphlem1 23557* Lemma for tcphcph 23559: the triangle inequality. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘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))))

Theoremtcphcphlem2 23558* Lemma for tcphcph 23559: homogeneity. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂPreHil‘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))))

Theoremtcphcph 23559* The standard definition of a norm turns any pre-Hilbert space over a subfield of fld 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)    &   (𝜑𝑊 ∈ PreHil)    &   (𝜑𝐹 = (ℂflds 𝐾))    &    , = (·𝑖𝑊)    &   ((𝜑 ∧ (𝑥𝐾𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) → (√‘𝑥) ∈ 𝐾)    &   ((𝜑𝑥𝑉) → 0 ≤ (𝑥 , 𝑥))       (𝜑𝐺 ∈ ℂPreHil)

Theoremipcau 23560 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 23561 Lemma for nmpar 23562. (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 23562 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 23563 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 23564 Four times the inner product value cphipval2 23563. (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 23565* 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 23566 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 23567* 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 23568 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 23569* Continuity of inner product; analogue of cnmpt12f 21994 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 23570* Continuity of inner product; analogue of cnmpt22f 22003 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 23571 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 ∧ 𝑆𝐶) → 𝑆 ∈ (Clsd‘𝐽))

Theoremclsocv 23572 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))

Theoremcphsscph 23573 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 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝑈𝑆) → 𝑋 ∈ ℂPreHil)

12.5.5  Convergence and completeness

Syntaxccfil 23574 Extend class notation with the class of Cauchy filters.
class CauFil

Syntaxccau 23575 Extend class notation with the class of Cauchy sequences.
class Cau

Syntaxccmet 23576 Extend class notation with the class of complete metrics.
class CMet

Definitiondf-cfil 23577* 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+𝑦𝑓 (𝑑 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥)})

Definitiondf-cau 23578* 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 𝑑pm ℂ) ∣ ∀𝑥 ∈ ℝ+𝑗 ∈ ℤ (𝑓 ↾ (ℤ𝑗)):(ℤ𝑗)⟶((𝑓𝑗)(ball‘𝑑)𝑥)})

Definitiondf-cmet 23579* 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) ≠ ∅})

Theoremlmmbr 23580* 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 21557. (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 23581* 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 21557. (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 23582* 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 23583* 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 23584* 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 23581 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 23585* 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 23586* 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 23587* 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 23588* 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 23589 A Cauchy filter is a filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐷 ∈ (∞Met‘𝑋) ∧ 𝐹 ∈ (CauFil‘𝐷)) → 𝐹 ∈ (Fil‘𝑋))

Theoremcfili 23590* Property of a Cauchy filter. (Contributed by Mario Carneiro, 13-Oct-2015.)
((𝐹 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝐹𝑦𝑥𝑧𝑥 (𝑦𝐷𝑧) < 𝑅)

Theoremcfil3i 23591* 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 23592 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 23593* 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 23594* 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 23595* 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 23596 Similar to ultrafilters (uffclsflim 22359), 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 23597* 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 23598* 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 21557. (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 23599* 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 23600* 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))))

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44303
 Copyright terms: Public domain < Previous  Next >