HomeHome Metamath Proof Explorer
Theorem List (p. 232 of 449)
< 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  Metamath Proof Explorer
(1-28623)
  Hilbert Space Explorer  Hilbert Space Explorer
(28624-30146)
  Users' Mathboxes  Users' Mathboxes
(30147-44804)
 

Theorem List for Metamath Proof Explorer - 23101-23200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremblval2 23101 The ball around a point 𝑃, alternative definition. (Contributed by Thierry Arnoux, 7-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑃𝑋𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) = ((𝐷 “ (0[,)𝑅)) “ {𝑃}))
 
Theoremelbl4 23102 Membership in a ball, alternative definition. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Mar-2018.)
(((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑅 ∈ ℝ+) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(𝐷 “ (0[,)𝑅))𝐴))
 
Theoremmetuel 23103* Elementhood in the uniform structure generated by a metric 𝐷 (Contributed by Thierry Arnoux, 8-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉 ∈ (metUnif‘𝐷) ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑤 ∈ ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))𝑤𝑉)))
 
Theoremmetuel2 23104* Elementhood in the uniform structure generated by a metric 𝐷 (Contributed by Thierry Arnoux, 24-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝑈 = (metUnif‘𝐷)       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝑉𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐷𝑦) < 𝑑𝑥𝑉𝑦))))
 
Theoremmetustbl 23105* The "section" image of an entourage at a point 𝑃 always contains a ball (centered on this point). (Contributed by Thierry Arnoux, 8-Dec-2017.)
((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑉 ∈ (metUnif‘𝐷) ∧ 𝑃𝑋) → ∃𝑎 ∈ ran (ball‘𝐷)(𝑃𝑎𝑎 ⊆ (𝑉 “ {𝑃})))
 
Theorempsmetutop 23106 The topology induced by a uniform structure generated by a metric 𝐷 is generated by that metric's open balls. (Contributed by Thierry Arnoux, 6-Dec-2017.) (Revised by Thierry Arnoux, 11-Mar-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (topGen‘ran (ball‘𝐷)))
 
Theoremxmetutop 23107 The topology induced by a uniform structure generated by an extended metric 𝐷 is that metric's open sets. (Contributed by Thierry Arnoux, 11-Mar-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (∞Met‘𝑋)) → (unifTop‘(metUnif‘𝐷)) = (MetOpen‘𝐷))
 
Theoremxmsusp 23108 If the uniform set of a metric space is the uniform structure generated by its metric, then it is a uniform space. (Contributed by Thierry Arnoux, 14-Dec-2017.)
𝑋 = (Base‘𝐹)    &   𝐷 = ((dist‘𝐹) ↾ (𝑋 × 𝑋))    &   𝑈 = (UnifSt‘𝐹)       ((𝑋 ≠ ∅ ∧ 𝐹 ∈ ∞MetSp ∧ 𝑈 = (metUnif‘𝐷)) → 𝐹 ∈ UnifSp)
 
Theoremrestmetu 23109 The uniform structure generated by the restriction of a metric is its trace. (Contributed by Thierry Arnoux, 18-Dec-2017.)
((𝐴 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴))))
 
Theoremmetucn 23110* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 23082. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝑈 = (metUnif‘𝐶)    &   𝑉 = (metUnif‘𝐷)    &   (𝜑𝑋 ≠ ∅)    &   (𝜑𝑌 ≠ ∅)    &   (𝜑𝐶 ∈ (PsMet‘𝑋))    &   (𝜑𝐷 ∈ (PsMet‘𝑌))       (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑐 ∈ ℝ+𝑥𝑋𝑦𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹𝑥)𝐷(𝐹𝑦)) < 𝑑))))
 
12.4.7  Examples of metric spaces
 
Theoremdscmet 23111* The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))       (𝑋𝑉𝐷 ∈ (Met‘𝑋))
 
Theoremdscopn 23112* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
𝐷 = (𝑥𝑋, 𝑦𝑋 ↦ if(𝑥 = 𝑦, 0, 1))       (𝑋𝑉 → (MetOpen‘𝐷) = 𝒫 𝑋)
 
Theoremnrmmetd 23113* Show that a group norm generates a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝐹:𝑋⟶ℝ)    &   ((𝜑𝑥𝑋) → ((𝐹𝑥) = 0 ↔ 𝑥 = 0 ))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝐹‘(𝑥 𝑦)) ≤ ((𝐹𝑥) + (𝐹𝑦)))       (𝜑 → (𝐹 ) ∈ (Met‘𝑋))
 
Theoremabvmet 23114 An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥𝑦), analogously to cnmet 23309. (In fact, the ring structure is not needed at all; the group properties abveq0 19528 and abvtri 19532, abvneg 19536 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝐴 = (AbsVal‘𝑅)    &    = (-g𝑅)       (𝐹𝐴 → (𝐹 ) ∈ (Met‘𝑋))
 
12.4.8  Normed algebraic structures

In the following, the norm of a normed algebraic structure (group, left module, vector space) is defined by the (given) distance function (the norm 𝑁 of an element is its distance 𝐷 from the zero element, see nmval 23128: (𝑁𝐴) = (𝐴𝐷 0 )). By this definition, the norm function 𝑁 is actually a norm (satisfying the properties of being a function into the reals, subadditivity/triangle inequality (𝑛‘(𝑥 + 𝑦)) ≤ ((𝑛𝑥) + (𝑛𝑦)), absolute homogeneity ( n(sx) = |s| n(x) ) [Remark: for group norms, some authors (e.g. Juris Steprans in "A characterization of free abelian groups") demand that n(sx) = |s| n(x) for all s in ZZ, whereas other authors (e.g. N. H. Bingham and A. J. Ostaszewski in "Normed versus topological groups: Dichotomy and duality") only require inversion symmetry, i.e. n(-x) = n(x). The definition df-ngp 23122 of a group norm follows the second aproach, see nminv 23159.] and positive definiteness/point-separating ( n(x) = 0 <-> x = 0 ) if the structure is a metric space with a right-translation-invariant metric (see nmf 23153, nmtri 23164, nmvs 23214 and nmeq0 23156). An alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space is given by theorem tngngp3 23194. For a structure being a group, the (arbitrary) distance function can be restricted to the elements of the group without affecting the norm, see nmfval2 23129.

Usually, however, the norm of a normed structure is given, and the corresponding metric ("induced metric") is achieved by defining a distance function based on the norm (the distance 𝐷 between two elements is the norm 𝑁 of their difference, see ngpds 23142: (𝐴𝐷𝐵) = (𝑁‘(𝐴 𝐵))). The operation toNrmGrp does exactly this, i.e. it adds a distance function (and a topology) to a structure (which should be at least a group) corresponding to a given norm in the just shown way: (dist‘𝑇) = (𝑁 ), see also tngds 23186. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 23190) resp. if the norm is in fact a norm (see tngngpd 23191). If the norm is derived from a given metric, as done with df-nm 23121, the induced metric is the original metric restricted to the base set: (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)), see nrmtngdist 23195, and the norm remains the same: (norm‘𝑇) = (norm‘𝐺), see nrmtngnrm 23196.

 
Syntaxcnm 23115 Norm of a normed ring.
class norm
 
Syntaxcngp 23116 The class of all normed groups.
class NrmGrp
 
Syntaxctng 23117 Make a normed group from a norm and a group.
class toNrmGrp
 
Syntaxcnrg 23118 Normed ring.
class NrmRing
 
Syntaxcnlm 23119 Normed module.
class NrmMod
 
Syntaxcnvc 23120 Normed vector space.
class NrmVec
 
Definitiondf-nm 23121* Define the norm on a group or ring (when it makes sense) in terms of the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
norm = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤) ↦ (𝑥(dist‘𝑤)(0g𝑤))))
 
Definitiondf-ngp 23122 Define a normed group, which is a group with a right-translation-invariant metric. This is not a standard notion, but is helpful as the most general context in which a metric-like norm makes sense. (Contributed by Mario Carneiro, 2-Oct-2015.)
NrmGrp = {𝑔 ∈ (Grp ∩ MetSp) ∣ ((norm‘𝑔) ∘ (-g𝑔)) ⊆ (dist‘𝑔)}
 
Definitiondf-tng 23123* 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 23124 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 23125* 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 23126 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 23127* 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 23128 The value of the norm function. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))
 
Theoremnmfval2 23129* 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 23130 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 23131 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 23132 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 23133* 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 23134 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 23135 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 23136* 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 23137 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
 
Theoremngpms 23138 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
 
Theoremngpxms 23139 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
 
Theoremngptps 23140 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
 
Theoremngpmet 23141 The (induced) metric of a normed group is a metric. Part of Definition 2.2-1 of [Kreyszig] p. 58. (Contributed by NM, 4-Dec-2006.) (Revised by AV, 14-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))       (𝐺 ∈ NrmGrp → 𝐷 ∈ (Met‘𝑋))
 
Theoremngpds 23142 Value of the distance function in terms of the norm of a normed group. Equation 1 of [Kreyszig] p. 59. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 𝐵)))
 
Theoremngpdsr 23143 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 23144 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 23145 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 23146 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 23147 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 23148 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 23149 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 23150* 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 23151 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 23152 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 23153 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 23154 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 23155 The norm of a normed group is nonnegative. Second part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 28-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → 0 ≤ (𝑁𝐴))
 
Theoremnmeq0 23156 The identity is the only element of the group with zero norm. First part of Problem 2 of [Kreyszig] p. 64. (Contributed by NM, 24-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → ((𝑁𝐴) = 0 ↔ 𝐴 = 0 ))
 
Theoremnmne0 23157 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 23158 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 23159 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 23160 The triangle inequality for the norm of a subtraction. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
 
Theoremnmsub 23161 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 23162 Reverse triangle inequality for the norm of a subtraction. Problem 3 of [Kreyszig] p. 64. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (abs‘((𝑁𝐴) − (𝑁𝐵))) ≤ (𝑁‘(𝐴 𝐵)))
 
Theoremnm2dif 23163 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 23164 The triangle inequality for the norm of a sum. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    + = (+g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 + 𝐵)) ≤ ((𝑁𝐴) + (𝑁𝐵)))
 
Theoremnmtri2 23165 Triangle inequality for the norm of two subtractions. (Contributed by NM, 24-Feb-2008.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    = (-g𝐺)       ((𝐺 ∈ NrmGrp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → (𝑁‘(𝐴 𝐶)) ≤ ((𝑁‘(𝐴 𝐵)) + (𝑁‘(𝐵 𝐶))))
 
Theoremngpi 23166* The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    = (-g𝑊)    &    0 = (0g𝑊)       (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥𝑉 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑉 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))))
 
Theoremnm0 23167 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ NrmGrp → (𝑁0 ) = 0)
 
Theoremnmgt0 23168 The norm of a nonzero element is a positive real. (Contributed by NM, 20-Nov-2007.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Base‘𝐺)    &   𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋) → (𝐴0 ↔ 0 < (𝑁𝐴)))
 
Theoremsgrim 23169 The induced metric on a subgroup is the induced metric on the parent group equipped with a norm. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑇s 𝑈)    &   𝐷 = (dist‘𝑇)    &   𝐸 = (dist‘𝑋)       (𝑈𝑆𝐸 = 𝐷)
 
Theoremsgrimval 23170 The induced metric on a subgroup in terms of the induced metric on the parent normed group. (Contributed by NM, 1-Feb-2008.) (Revised by AV, 19-Oct-2021.)
𝑋 = (𝑇s 𝑈)    &   𝐷 = (dist‘𝑇)    &   𝐸 = (dist‘𝑋)    &   𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑁 = (norm‘𝐺)    &   𝑆 = (SubGrp‘𝑇)       (((𝐺 ∈ NrmGrp ∧ 𝑈𝑆) ∧ (𝐴𝑈𝐵𝑈)) → (𝐴𝐸𝐵) = (𝐴𝐷𝐵))
 
Theoremsubgnm 23171 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 23172 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 23173 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 23174 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 23175* 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 23176 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp
 
Theoremtngval 23177 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 23178 Lemma for tngbas 23179 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 23179 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 23180 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 23181 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 23182 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 23183 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 23184 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 23185 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 23186 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 23187 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 23188 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 23189 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 23190 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 23191* 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 23192* 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))))))
 
Theoremtnggrpr 23193 If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)       ((𝑁𝑉𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)
 
Theoremtngngp3 23194* Alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))
 
Theoremnrmtngdist 23195 The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)))
 
Theoremnrmtngnrm 23196 The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))       (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺)))
 
Theoremtngngpim 23197 The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ)
 
Theoremisnrg 23198 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 23199 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 23200 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
    < Previous  Next >

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 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44804
  Copyright terms: Public domain < Previous  Next >