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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-tng 22101* Define a function that fills in the topology and metric components of a structure given a group and a norm on it. (Contributed by Mario Carneiro, 2-Oct-2015.)
toNrmGrp = (𝑔 ∈ V, 𝑓 ∈ V ↦ ((𝑔 sSet ⟨(dist‘ndx), (𝑓 ∘ (-g𝑔))⟩) sSet ⟨(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g𝑔)))⟩))
 
Definitiondf-nrg 22102 A normed ring is a ring with an induced topology and metric such that the metric is translation-invariant and the norm (distance from 0) is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmRing = {𝑤 ∈ NrmGrp ∣ (norm‘𝑤) ∈ (AbsVal‘𝑤)}
 
Definitiondf-nlm 22103* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmMod = {𝑤 ∈ (NrmGrp ∩ LMod) ∣ [(Scalar‘𝑤) / 𝑓](𝑓 ∈ NrmRing ∧ ∀𝑥 ∈ (Base‘𝑓)∀𝑦 ∈ (Base‘𝑤)((norm‘𝑤)‘(𝑥( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}
 
Definitiondf-nvc 22104 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec = (NrmMod ∩ LVec)
 
Theoremnmfval 22105* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))
 
Theoremnmval 22106 The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
 
Theoremnmfval2 22107* The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))
 
Theoremnmval2 22108 The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))
 
Theoremnmf2 22109 The norm is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       ((𝑊 ∈ Grp ∧ 𝐸 ∈ (Met‘𝑋)) → 𝑁:𝑋⟶ℝ)
 
Theoremnmpropd 22110 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (+g𝐾) = (+g𝐿))    &   (𝜑 → (dist‘𝐾) = (dist‘𝐿))       (𝜑 → (norm‘𝐾) = (norm‘𝐿))
 
Theoremnmpropd2 22111* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐾 ∈ Grp)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))       (𝜑 → (norm‘𝐾) = (norm‘𝐿))
 
Theoremisngp 22112 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))
 
Theoremisngp2 22113 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)    &   𝑋 = (Base‘𝐺)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) = 𝐸))
 
Theoremisngp3 22114* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐷𝑦) = (𝑁‘(𝑥 𝑦))))
 
Theoremngpgrp 22115 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
 
Theoremngpms 22116 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
 
Theoremngpxms 22117 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
 
Theoremngptps 22118 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
 
Theoremngpds 22119 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 𝐵)))
 
Theoremngpdsr 22120 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐵 𝐴)))
 
Theoremngpds2 22121 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ((𝐴 𝐵)𝐷 0 ))
 
Theoremngpds2r 22122 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ((𝐵 𝐴)𝐷 0 ))
 
Theoremngpds3 22123 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐴 𝐵)))
 
Theoremngpds3r 22124 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐵 𝐴)))
 
Theoremngprcan 22125 Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴 + 𝐶)𝐷(𝐵 + 𝐶)) = (𝐴𝐷𝐵))
 
Theoremngplcan 22126 Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶 + 𝐴)𝐷(𝐶 + 𝐵)) = (𝐴𝐷𝐵))
 
Theoremisngp4 22127* Express the property of being a normed group purely in terms of right-translation invariance of the metric instead of using the definition of norm (which itself uses the metric). (Contributed by Mario Carneiro, 29-Oct-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥 + 𝑧)𝐷(𝑦 + 𝑧)) = (𝑥𝐷𝑦)))
 
Theoremngpinvds 22128 Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝐼 = (invg𝐺)    &   𝐷 = (dist‘𝐺)       (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐼𝐴)𝐷(𝐼𝐵)) = (𝐴𝐷𝐵))
 
Theoremngpsubcan 22129 Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴 𝐶)𝐷(𝐵 𝐶)) = (𝐴𝐷𝐵))
 
Theoremnmf 22130 The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)       (𝐺 ∈ NrmGrp → 𝑁:𝑋⟶ℝ)
 
Theoremnmcl 22131 The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → (𝑁𝐴) ∈ ℝ)
 
Theoremnmge0 22132 The norm of a normed group is nonnegative. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → 0 ≤ (𝑁𝐴))
 
Theoremnmeq0 22133 The identity is the only element of the group with zero norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → ((𝑁𝐴) = 0 ↔ 𝐴 = 0 ))
 
Theoremnmne0 22134 The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐴0 ) → (𝑁𝐴) ≠ 0)
 
Theoremnmrpcl 22135 The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐴0 ) → (𝑁𝐴) ∈ ℝ+)
 
Theoremnminv 22136 The norm of a negated element is the same as the norm of the original element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &   𝐼 = (invg𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → (𝑁‘(𝐼𝐴)) = (𝑁𝐴))
 
Theoremnmmtri 22137 The triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
 
Theoremnmsub 22138 The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 𝐵)) = (𝑁‘(𝐵 𝐴)))
 
Theoremnmrtri 22139 Reverse triangle inequality for the norm of a subtraction. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (abs‘((𝑁𝐴) − (𝑁𝐵))) ≤ (𝑁‘(𝐴 𝐵)))
 
Theoremnm2dif 22140 Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → ((𝑁𝐴) − (𝑁𝐵)) ≤ (𝑁‘(𝐴 𝐵)))
 
Theoremnmtri 22141 The triangle inequality for the norm of a sum. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 + 𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
 
Theoremnm0 22142 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ NrmGrp → (𝑁0 ) = 0)
 
Theoremsubgnm 22143 The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)    &   𝑁 = (norm‘𝐺)    &   𝑀 = (norm‘𝐻)       (𝐴 ∈ (SubGrp‘𝐺) → 𝑀 = (𝑁𝐴))
 
Theoremsubgnm2 22144 A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)    &   𝑁 = (norm‘𝐺)    &   𝑀 = (norm‘𝐻)       ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋𝐴) → (𝑀𝑋) = (𝑁𝑋))
 
Theoremsubgngp 22145 A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)       ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ NrmGrp)
 
Theoremngptgp 22146 A normed abelian group is a topological group (with the topology induced by the metric induced by the norm). (Contributed by Mario Carneiro, 4-Oct-2015.)
((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) → 𝐺 ∈ TopGrp)
 
Theoremngppropd 22147* Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))    &   (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿))       (𝜑 → (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp))
 
Theoremreldmtng 22148 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp
 
Theoremtngval 22149 Value of the function which augments a given structure 𝐺 with a norm 𝑁. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    = (-g𝐺)    &   𝐷 = (𝑁 )    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝑇 = ((𝐺 sSet ⟨(dist‘ndx), 𝐷⟩) sSet ⟨(TopSet‘ndx), 𝐽⟩))
 
Theoremtnglem 22150 Lemma for tngbas 22151 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐸 = Slot 𝐾    &   𝐾 ∈ ℕ    &   𝐾 < 9       (𝑁𝑉 → (𝐸𝐺) = (𝐸𝑇))
 
Theoremtngbas 22151 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐵 = (Base‘𝐺)       (𝑁𝑉𝐵 = (Base‘𝑇))
 
Theoremtngplusg 22152 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    + = (+g𝐺)       (𝑁𝑉+ = (+g𝑇))
 
Theoremtng0 22153 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    0 = (0g𝐺)       (𝑁𝑉0 = (0g𝑇))
 
Theoremtngmulr 22154 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = (.r𝐺)       (𝑁𝑉· = (.r𝑇))
 
Theoremtngsca 22155 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐹 = (Scalar‘𝐺)       (𝑁𝑉𝐹 = (Scalar‘𝑇))
 
Theoremtngvsca 22156 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    · = ( ·𝑠𝐺)       (𝑁𝑉· = ( ·𝑠𝑇))
 
Theoremtngip 22157 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    , = (·𝑖𝐺)       (𝑁𝑉, = (·𝑖𝑇))
 
Theoremtngds 22158 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    = (-g𝐺)       (𝑁𝑉 → (𝑁 ) = (dist‘𝑇))
 
Theoremtngtset 22159 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐷 = (dist‘𝑇)    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝐽 = (TopSet‘𝑇))
 
Theoremtngtopn 22160 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐷 = (dist‘𝑇)    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝐽 = (TopOpen‘𝑇))
 
Theoremtngnm 22161 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐴 ∈ V       ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))
 
Theoremtngngp2 22162 A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))
 
Theoremtngngpd 22163* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑁:𝑋⟶ℝ)    &   ((𝜑𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))       (𝜑𝑇 ∈ NrmGrp)
 
Theoremtngngp 22164* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
 
Theoremisnrg 22165 A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))
 
Theoremnrgabv 22166 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing → 𝑁𝐴)
 
Theoremnrgngp 22167 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
 
Theoremnrgring 22168 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
 
Theoremnmmul 22169 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))
 
Theoremnrgdsdi 22170 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝑁𝐴) · (𝐵𝐷𝐶)) = ((𝐴 · 𝐵)𝐷(𝐴 · 𝐶)))
 
Theoremnrgdsdir 22171 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵) · (𝑁𝐶)) = ((𝐴 · 𝐶)𝐷(𝐵 · 𝐶)))
 
Theoremnm1 22172 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁1 ) = 1)
 
Theoremunitnmn0 22173 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴𝑈) → (𝑁𝐴) ≠ 0)
 
Theoremnminvr 22174 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴𝑈) → (𝑁‘(𝐼𝐴)) = (1 / (𝑁𝐴)))
 
Theoremnmdvr 22175 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)       (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴𝑋𝐵𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁𝐴) / (𝑁𝐵)))
 
Theoremnrgdomn 22176 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing))
 
Theoremnrgtgp 22177 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
 
Theoremsubrgnrg 22178 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)       ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing)
 
Theoremtngnrg 22179 Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝑅 toNrmGrp 𝐹)    &   𝐴 = (AbsVal‘𝑅)       (𝐹𝐴𝑇 ∈ NrmRing)
 
Theoremisnlm 22180* A normed (left) module is a module which is also a normed group over a normed ring, such that the norm distributes over scalar multiplication. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))
 
Theoremnmvs 22181 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))
 
Theoremnlmngp 22182 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
 
Theoremnlmlmod 22183 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
 
Theoremnlmnrg 22184 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)
 
Theoremnlmngp2 22185 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)
 
Theoremnlmdsdi 22186 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝑉𝑍𝑉)) → ((𝐴𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)))
 
Theoremnlmdsdir 22187 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐸 = (dist‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝐾𝑍𝑉)) → ((𝑋𝐸𝑌) · (𝑁𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)))
 
Theoremnlmmul0or 22188 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ NrmMod ∧ 𝐴𝐾𝐵𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂𝐵 = 0 )))
 
Theoremsranlm 22189 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)
 
Theoremnlmvscnlem2 22190 Lemma for nlmvscn 22192. Compare this proof with the similar elementary proof mulcn2 14040 for continuity of multiplication on . (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (dist‘𝐹)    &   𝑁 = (norm‘𝑊)    &   𝐴 = (norm‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝐶𝐾)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝐵𝐸𝐶) < 𝑈)    &   (𝜑 → (𝑋𝐷𝑌) < 𝑇)       (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)
 
Theoremnlmvscnlem1 22191* Lemma for nlmvscn 22192. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (dist‘𝐹)    &   𝑁 = (norm‘𝑊)    &   𝐴 = (norm‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑟 ∈ ℝ+𝑥𝐾𝑦𝑉 (((𝐵𝐸𝑥) < 𝑟 ∧ (𝑋𝐷𝑦) < 𝑟) → ((𝐵 · 𝑋)𝐷(𝑥 · 𝑦)) < 𝑅))
 
Theoremnlmvscn 22192 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 22194 and nlmtlm 22198. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·sf𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘𝐹)       (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
 
Theoremrlmnlm 22193 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
 
Theoremnrgtrg 22194 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
 
Theoremnrginvrcnlem 22195* Lemma for nrginvrcn 22196. Compare this proof with reccn2 14041, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝑁 = (norm‘𝑅)    &   𝐷 = (dist‘𝑅)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑇 = (if(1 ≤ ((𝑁𝐴) · 𝐵), 1, ((𝑁𝐴) · 𝐵)) · ((𝑁𝐴) / 2))       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝑈 ((𝐴𝐷𝑦) < 𝑥 → ((𝐼𝐴)𝐷(𝐼𝑦)) < 𝐵))
 
Theoremnrginvrcn 22196 The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝐽 = (TopOpen‘𝑅)       (𝑅 ∈ NrmRing → 𝐼 ∈ ((𝐽t 𝑈) Cn (𝐽t 𝑈)))
 
Theoremnrgtdrg 22197 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)
 
Theoremnlmtlm 22198 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)
 
Theoremisnvc 22199 A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))
 
Theoremnvcnlm 22200 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
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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 42401-42426
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