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Theorem List for Metamath Proof Explorer - 22601-22700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremclmring 22601 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Ring)
 
Theoremclmfgrp 22602 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 𝐹 ∈ Grp)
 
Theoremclm0 22603 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 0 = (0g𝐹))
 
Theoremclm1 22604 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → 1 = (1r𝐹))
 
Theoremclmadd 22605 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → + = (+g𝐹))
 
Theoremclmmul 22606 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → · = (.r𝐹))
 
Theoremclmcj 22607 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ ℂMod → ∗ = (*𝑟𝐹))
 
Theoremisclmi 22608 Reverse direction of isclm 22595. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝐹 = (Scalar‘𝑊)       ((𝑊 ∈ LMod ∧ 𝐹 = (ℂflds 𝐾) ∧ 𝐾 ∈ (SubRing‘ℂfld)) → 𝑊 ∈ ℂMod)
 
Theoremclmzss 22609 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → ℤ ⊆ 𝐾)
 
Theoremclmsscn 22610 The scalar ring of a complex module is a subset of the complex numbers. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂMod → 𝐾 ⊆ ℂ)
 
Theoremclmsub 22611 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾𝐵𝐾) → (𝐴𝐵) = (𝐴(-g𝐹)𝐵))
 
Theoremclmneg 22612 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → -𝐴 = ((invg𝐹)‘𝐴))
 
Theoremclmabs 22613 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝐴𝐾) → (abs‘𝐴) = ((norm‘𝐹)‘𝐴))
 
Theoremclmacl 22614 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 + 𝑌) ∈ 𝐾)
 
Theoremclmmcl 22615 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋 · 𝑌) ∈ 𝐾)
 
Theoremclmsubcl 22616 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ 𝑋𝐾𝑌𝐾) → (𝑋𝑌) ∈ 𝐾)
 
Theoremlmhmclm 22617 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
(𝐹 ∈ (𝑆 LMHom 𝑇) → (𝑆 ∈ ℂMod ↔ 𝑇 ∈ ℂMod))
 
Theoremclmvsass 22618 Associative law for scalar product. (lmodvsass 18618 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 · 𝑅) · 𝑋) = (𝑄 · (𝑅 · 𝑋)))
 
Theoremclmvsdir 22619 Distributive law for scalar product (right-distributivity). (lmodvsdir 18617 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐾 = (Base‘𝐹)    &    + = (+g𝑊)       ((𝑊 ∈ ℂMod ∧ (𝑄𝐾𝑅𝐾𝑋𝑉)) → ((𝑄 + 𝑅) · 𝑋) = ((𝑄 · 𝑋) + (𝑅 · 𝑋)))
 
Theoremclmvs1 22620 Scalar product with ring unit. (lmodvs1 18621 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (1 · 𝑋) = 𝑋)
 
Theoremclm0vs 22621 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 18626 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (0 · 𝑋) = 0 )
 
Theoremclmvneg1 22622 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 18633 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (invg𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝑋𝑉) → (-1 · 𝑋) = (𝑁𝑋))
 
Theoremclmvsneg 22623 Multiplication of a vector by a negated scalar. (lmodvsneg 18634 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝐵 = (Base‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝑁 = (invg𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝑋𝐵)    &   (𝜑𝑅𝐾)       (𝜑 → (𝑁‘(𝑅 · 𝑋)) = (-𝑅 · 𝑋))
 
Theoremclmmulg 22624 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝑉 = (Base‘𝑊)    &    = (.g𝑊)    &    · = ( ·𝑠𝑊)       ((𝑊 ∈ ℂMod ∧ 𝐴 ∈ ℤ ∧ 𝐵𝑉) → (𝐴 𝐵) = (𝐴 · 𝐵))
 
Theoremclmsubdir 22625 Scalar multiplication distributive law for subtraction. (lmodsubdir 18648 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &    = (-g𝑊)    &   (𝜑𝑊 ∈ ℂMod)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ((𝐴𝐵) · 𝑋) = ((𝐴 · 𝑋) (𝐵 · 𝑋)))
 
Theoremzlmclm 22626 The -module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)       (𝐺 ∈ Abel ↔ 𝑊 ∈ ℂMod)
 
Theoremclmzlmvsca 22627 The scalar product of a complex module matches the scalar product of the derived -module, which implies, together with zlmbas 19591 and zlmplusg 19592, that any module over is structure-equivalent to the canonical -module ℤMod‘𝐺. (Contributed by Mario Carneiro, 30-Oct-2015.)
𝑊 = (ℤMod‘𝐺)    &   𝑋 = (Base‘𝐺)       ((𝐺 ∈ ℂMod ∧ (𝐴 ∈ ℤ ∧ 𝐵𝑋)) → (𝐴( ·𝑠𝐺)𝐵) = (𝐴( ·𝑠𝑊)𝐵))
 
Theoremnmoleub2lem 22628* Lemma for nmoleub2a 22631 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   ((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) → 0 ≤ 𝐴)    &   ((((𝜑 ∧ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)) ∧ 𝐴 ∈ ℝ) ∧ (𝑦𝑉𝑦 ≠ (0g𝑆))) → (𝑀‘(𝐹𝑦)) ≤ (𝐴 · (𝐿𝑦)))    &   ((𝜑𝑥𝑉) → (𝜓 → (𝐿𝑥) ≤ 𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 (𝜓 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))
 
Theoremnmoleub2lem3 22629* Lemma for nmoleub2a 22631 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &    · = ( ·𝑠𝑆)    &   (𝜑𝐴 ∈ ℝ)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑𝐵𝑉)    &   (𝜑𝐵 ≠ (0g𝑆))    &   (𝜑 → ((𝑟 · 𝐵) ∈ 𝑉 → ((𝐿‘(𝑟 · 𝐵)) < 𝑅 → ((𝑀‘(𝐹‘(𝑟 · 𝐵))) / 𝑅) ≤ 𝐴)))    &   (𝜑 → ¬ (𝑀‘(𝐹𝐵)) ≤ (𝐴 · (𝐿𝐵)))        ¬ 𝜑
 
Theoremnmoleub2lem2 22630* Lemma for nmoleub2a 22631 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥)𝑂𝑅 → (𝐿𝑥) ≤ 𝑅))    &   (((𝐿𝑥) ∈ ℝ ∧ 𝑅 ∈ ℝ) → ((𝐿𝑥) < 𝑅 → (𝐿𝑥)𝑂𝑅))       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥)𝑂𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))
 
Theoremnmoleub2a 22631* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) ≤ 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))
 
Theoremnmoleub2b 22632* The operator norm is the supremum of the value of a linear operator in the open unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → ℚ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) < 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))
 
Theoremnmoleub3 22633* The operator norm is the supremum of the value of a linear operator on the closed unit sphere. (Contributed by Mario Carneiro, 19-Oct-2015.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐾 = (Base‘𝐺)    &   (𝜑𝑆 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝑇 ∈ (NrmMod ∩ ℂMod))    &   (𝜑𝐹 ∈ (𝑆 LMHom 𝑇))    &   (𝜑𝐴 ∈ ℝ*)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑 → 0 ≤ 𝐴)    &   (𝜑 → ℝ ⊆ 𝐾)       (𝜑 → ((𝑁𝐹) ≤ 𝐴 ↔ ∀𝑥𝑉 ((𝐿𝑥) = 𝑅 → ((𝑀‘(𝐹𝑥)) / 𝑅) ≤ 𝐴)))
 
Theoremnmhmcn 22634 A linear operator over a normed complex module is bounded iff it is continuous. (Contributed by Mario Carneiro, 22-Oct-2015.)
𝐽 = (TopOpen‘𝑆)    &   𝐾 = (TopOpen‘𝑇)    &   𝐺 = (Scalar‘𝑆)    &   𝐵 = (Base‘𝐺)       ((𝑆 ∈ (NrmMod ∩ ℂMod) ∧ 𝑇 ∈ (NrmMod ∩ ℂMod) ∧ ℚ ⊆ 𝐵) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝐽 Cn 𝐾))))
 
12.5.2  Complex vector spaces
 
Syntaxccvs 22635 Complex vector space.
class ℂVec
 
Definitiondf-cvs 22636 Define a complex vector space, which is just a complex left module and a vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
ℂVec = (ℂMod ∩ LVec)
 
Theoremcvslvec 22637 A complex vector space is a (left) vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ LVec)
 
Theoremcvsclm 22638 A complex vector space is a complex left module. (Contributed by Thierry Arnoux, 22-May-2019.)
(𝜑𝑊 ∈ ℂVec)       (𝜑𝑊 ∈ ℂMod)
 
Theoremcvsunit 22639 Unit group of the scalar ring of a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂVec → (𝐾 ∖ {0}) = (Unit‘𝐹))
 
Theoremcvsdiv 22640 Division of the scalar ring of a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) = (𝐴(/r𝐹)𝐵))
 
Theoremcvsdivcl 22641 The scalar field of a complex vector space is closed under division. (Contributed by Thierry Arnoux, 22-May-2019.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂVec ∧ (𝐴𝐾𝐵𝐾𝐵 ≠ 0)) → (𝐴 / 𝐵) ∈ 𝐾)
 
Theoremcvsmuleqdivd 22642 An equality involving ratios in a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑 → (𝐴 · 𝑋) = (𝐵 · 𝑌))       (𝜑𝑋 = ((𝐵 / 𝐴) · 𝑌))
 
Theoremcvsdiveqd 22643 An equality involving ratios in a complex vector space. (Contributed by Thierry Arnoux, 22-May-2019.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   (𝜑𝑊 ∈ ℂVec)    &   (𝜑𝐴𝐾)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝑌𝑉)    &   (𝜑𝐴 ≠ 0)    &   (𝜑𝐵 ≠ 0)    &   (𝜑𝑋 = ((𝐴 / 𝐵) · 𝑌))       (𝜑 → ((𝐵 / 𝐴) · 𝑋) = 𝑌)
 
12.5.3  Complex pre-Hilbert space
 
Syntaxccph 22644 Extend class notation with a complex pre-Hilbert space.
class ℂPreHil
 
Syntaxctch 22645 Function to put a norm on a Hilbert space.
class toℂHil
 
Definitiondf-cph 22646* Define a complex pre-Hilbert space. By restricting the scalar field to a quadratically closed subfield of , we have enough structure to define a norm, with the associated connection to a metric and topology. (Contributed by Mario Carneiro, 8-Oct-2015.)
ℂPreHil = {𝑤 ∈ (PreHil ∩ NrmMod) ∣ [(Scalar‘𝑤) / 𝑓][(Base‘𝑓) / 𝑘](𝑓 = (ℂflds 𝑘) ∧ (√ “ (𝑘 ∩ (0[,)+∞))) ⊆ 𝑘 ∧ (norm‘𝑤) = (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥))))}
 
Definitiondf-tch 22647* Define a function to augment a (pre-)Hilbert space with a norm. No extra parameters are needed, but some conditions must be satisfied to ensure that this in fact creates a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
toℂHil = (𝑤 ∈ V ↦ (𝑤 toNrmGrp (𝑥 ∈ (Base‘𝑤) ↦ (√‘(𝑥(·𝑖𝑤)𝑥)))))
 
Theoremiscph 22648* A complex pre-Hilbert space is a pre-Hilbert space over a quadratically closed subfield of the complex numbers, with a norm defined. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
 
Theoremcphphl 22649 A complex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
 
Theoremcphnlm 22650 A complex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
 
Theoremcphngp 22651 A complex pre-Hilbert space is a normed group. (Contributed by Mario Carneiro, 13-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmGrp)
 
Theoremcphlmod 22652 A complex pre-Hilbert space is a left module. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LMod)
 
Theoremcphlvec 22653 A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
 
Theoremcphnvc 22654 A complex pre-Hilbert space is a normed vector space. (Contributed by Mario Carneiro, 8-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmVec)
 
Theoremcphsubrglem 22655 Lemma for cphsubrg 22658. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       (𝜑 → (𝐹 = (ℂflds 𝐾) ∧ 𝐾 = (𝐴 ∩ ℂ) ∧ 𝐾 ∈ (SubRing‘ℂfld)))
 
Theoremcphreccllem 22656 Lemma for cphreccl 22659. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐾 = (Base‘𝐹)    &   (𝜑𝐹 = (ℂflds 𝐴))    &   (𝜑𝐹 ∈ DivRing)       ((𝜑𝑋𝐾𝑋 ≠ 0) → (1 / 𝑋) ∈ 𝐾)
 
Theoremcphsca 22657 A complex pre-Hilbert space is a vector space over a subfield of . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
 
Theoremcphsubrg 22658 The scalar field of a complex pre-Hilbert space is a subring of . (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝐾 ∈ (SubRing‘ℂfld))
 
Theoremcphreccl 22659 The scalar field of a complex 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 22660 The scalar field of a complex 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 22661 The scalar field of a complex 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 22662 The scalar field of a complex pre-Hilbert space is closed under square roots of positive reals (i.e. it is quadratically closed relative to ). (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝐾𝐴 ∈ ℝ ∧ 0 ≤ 𝐴)) → (√‘𝐴) ∈ 𝐾)
 
Theoremcphabscl 22663 The scalar field of a complex 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 22664 The scalar field of a complex 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 22665 If the scalar field contains i, it is completely 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 22666 The scalar field of a complex pre-Hilbert space contains all rational numbers. (Contributed by Mario Carneiro, 15-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → ℚ ⊆ 𝐾)
 
Theoremcphclm 22667 A complex pre-Hilbert space is a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
(𝑊 ∈ ℂPreHil → 𝑊 ∈ ℂMod)
 
Theoremcphnmvs 22668 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 22669 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 22670* The value of the norm in a complex 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 22671 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) = (√‘(𝐴 , 𝐴)))
 
Theoremnmsq 22672 The square of the norm is the norm of an inner product in a normed pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝑁𝐴)↑2) = (𝐴 , 𝐴))
 
Theoremcphnmf 22673 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       (𝑊 ∈ ℂPreHil → 𝑁:𝑉𝐾)
 
Theoremcphnmcl 22674 The norm of a vector is a member of the scalar field in a complex pre-Hilbert space. (Contributed by Mario Carneiro, 9-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝑁𝐴) ∈ 𝐾)
 
Theoremreipcl 22675 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 22676 The inner product in a complex pre-Hilbert space is positive definite. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → 0 ≤ (𝐴 , 𝐴))
 
Theoremcphipcj 22677 Conjugate of an inner product in a complex pre-Hilbert space. Complex version of ipcj 19704. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → (∗‘(𝐴 , 𝐵)) = (𝐵 , 𝐴))
 
Theoremcphorthcom 22678 Orthogonality (meaning inner product is 0) is commutative. Complex version of iporthcom 19705. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉𝐵𝑉) → ((𝐴 , 𝐵) = 0 ↔ (𝐵 , 𝐴) = 0))
 
Theoremcphip0l 22679 Inner product with a zero first argument. Part of proof of Theorem 6.44 of [Ponnusamy] p. 361. Complex version of ip0l 19706. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ( 0 , 𝐴) = 0)
 
Theoremcphip0r 22680 Inner product with a zero second argument. Complex version of ip0r 19707. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → (𝐴 , 0 ) = 0)
 
Theoremcphipeq0 22681 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 19708. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    0 = (0g𝑊)       ((𝑊 ∈ ℂPreHil ∧ 𝐴𝑉) → ((𝐴 , 𝐴) = 0 ↔ 𝐴 = 0 ))
 
Theoremcphdir 22682 Distributive law for inner product (right-distributivity). Equation I3 of [Ponnusamy] p. 362. Complex version of ipdir 19709. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 + 𝐵) , 𝐶) = ((𝐴 , 𝐶) + (𝐵 , 𝐶)))
 
Theoremcphdi 22683 Distributive law for inner product (left-distributivity). Complex version of ipdi 19710. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    + = (+g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 + 𝐶)) = ((𝐴 , 𝐵) + (𝐴 , 𝐶)))
 
Theoremcph2di 22684 Distributive law for inner product. Complex version of ip2di 19711. (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 22685 Distributive law for inner product subtraction. Complex version of ipsubdir 19712. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → ((𝐴 𝐵) , 𝐶) = ((𝐴 , 𝐶) − (𝐵 , 𝐶)))
 
Theoremcphsubdi 22686 Distributive law for inner product subtraction. Complex version of ipsubdi 19713. (Contributed by Mario Carneiro, 16-Oct-2015.)
, = (·𝑖𝑊)    &   𝑉 = (Base‘𝑊)    &    = (-g𝑊)       ((𝑊 ∈ ℂPreHil ∧ (𝐴𝑉𝐵𝑉𝐶𝑉)) → (𝐴 , (𝐵 𝐶)) = ((𝐴 , 𝐵) − (𝐴 , 𝐶)))
 
Theoremcph2subdi 22687 Distributive law for inner product subtraction. Complex version of ip2subdi 19714. (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 22688 Associative law for inner product. Equation I2 of [Ponnusamy] p. 363. See ipass 19715, his5 27115. (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 22689 "Associative" law for second argument of inner product (compare cphass 22688). See ipassr 19716, 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 22690 Move scalar multiplication to outside of inner product. See his35 27117. (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)))
 
Theoremtchex 22691* Lemma for tchbas 22693 and similar theorems. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝑉 = (Base‘𝑊)       (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))) ∈ V
 
Theoremtchval 22692* Define a function to augment a pre-Hilbert space with norm. (Contributed by Mario Carneiro, 7-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)    &    , = (·𝑖𝑊)       𝐺 = (𝑊 toNrmGrp (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
 
Theoremtchbas 22693 The base set of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝑉 = (Base‘𝑊)       𝑉 = (Base‘𝐺)
 
Theoremtchplusg 22694 The addition operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    + = (+g𝑊)        + = (+g𝐺)
 
Theoremtchsub 22695 The subtraction operation of a pre-Hilbert space augmented with norm. (Contributed by Thierry Arnoux, 30-Jun-2019.)
𝐺 = (toℂHil‘𝑊)    &    = (-g𝑊)        = (-g𝐺)
 
Theoremtchmulr 22696 The ring operation of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (.r𝑊)        · = (.r𝐺)
 
Theoremtchsca 22697 The scalar field of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝐹 = (Scalar‘𝑊)       𝐹 = (Scalar‘𝐺)
 
Theoremtchvsca 22698 The scalar multiplication of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = ( ·𝑠𝑊)        · = ( ·𝑠𝐺)
 
Theoremtchip 22699 The inner product of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &    · = (·𝑖𝑊)        · = (·𝑖𝐺)
 
Theoremtchtopn 22700 The topology of a pre-Hilbert space augmented with norm. (Contributed by Mario Carneiro, 8-Oct-2015.)
𝐺 = (toℂHil‘𝑊)    &   𝐷 = (dist‘𝐺)    &   𝐽 = (TopOpen‘𝐺)       (𝑊𝑉𝐽 = (MetOpen‘𝐷))
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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 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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 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