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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
12.4.6  The uniform structure generated by a metric
 
Theoremmetuval 22401* Value of the uniform structure generated by metric 𝐷. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
(𝐷 ∈ (PsMet‘𝑋) → (metUnif‘𝐷) = ((𝑋 × 𝑋)filGenran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))))
 
Theoremmetustel 22402* Define a filter base 𝐹 generated by a metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       (𝐷 ∈ (PsMet‘𝑋) → (𝐵𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐵 = (𝐷 “ (0[,)𝑎))))
 
Theoremmetustss 22403* Range of the elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 ⊆ (𝑋 × 𝑋))
 
Theoremmetustrel 22404* Elements of the filter base generated by the metric 𝐷 are relations. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → Rel 𝐴)
 
Theoremmetustto 22405* Any two elements of the filter base generated by the metric 𝐷 can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹𝐵𝐹) → (𝐴𝐵𝐵𝐴))
 
Theoremmetustid 22406* The identity diagonal is included in all elements of the filter base generated by the metric 𝐷. (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → ( I ↾ 𝑋) ⊆ 𝐴)
 
Theoremmetustsym 22407* Elements of the filter base generated by the metric 𝐷 are symmetric. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝐴𝐹) → 𝐴 = 𝐴)
 
Theoremmetustexhalf 22408* For any element 𝐴 of the filter base generated by the metric 𝐷, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
 
Theoremmetustfbas 22409* The filter base generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → 𝐹 ∈ (fBas‘(𝑋 × 𝑋)))
 
Theoremmetust 22410* The uniform structure generated by a metric 𝐷. (Contributed by Thierry Arnoux, 26-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑋 × 𝑋)filGen𝐹) ∈ (UnifOn‘𝑋))
 
Theoremcfilucfil 22411* Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 23109. (Contributed by Thierry Arnoux, 29-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))       ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘((𝑋 × 𝑋)filGen𝐹)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremmetuust 22412 The uniform structure generated by metric 𝐷 is a uniform structure. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (metUnif‘𝐷) ∈ (UnifOn‘𝑋))
 
Theoremcfilucfil2 22413* Given a metric 𝐷 and a uniform structure generated by that metric, Cauchy filter bases on that uniform structure are exactly the filter bases which contain balls of any pre-chosen size. See iscfil 23109. (Contributed by Thierry Arnoux, 1-Dec-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐶 ∈ (CauFilu‘(metUnif‘𝐷)) ↔ (𝐶 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ ℝ+𝑦𝐶 (𝐷 “ (𝑦 × 𝑦)) ⊆ (0[,)𝑥))))
 
Theoremblval2 22414 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 22415 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 22416* 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 22417* 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 22418* 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 22419 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 22420 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 22421 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 22422 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 22423* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 22395. (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 22424* 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 22425* 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 22426* 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 22427 An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥𝑦), analogously to cnmet 22622. (In fact, the ring structure is not needed at all; the group properties abveq0 18874 and abvtri 18878, abvneg 18882 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 22441: (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) <_ n(x)+n(y) ), 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 22435 of a group norm follows the second aproach, see nminv 22472.] 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 22466, nmtri 22477, nmvs 22527 and nmeq0 22469). 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 22507. 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 22442.

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 22455: (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 22499. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 22503) resp. if the norm is in fact a norm (see tngngpd 22504). If the norm is derived from a given metric, as done with df-nm 22434, the induced metric is the original metric restricted to the base set: (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)), see nrmtngdist 22508, and the norm remains the same: (norm‘𝑇) = (norm‘𝐺), see nrmtngnrm 22509.

 
Syntaxcnm 22428 Norm of a normed ring.
class norm
 
Syntaxcngp 22429 The class of all normed groups.
class NrmGrp
 
Syntaxctng 22430 Make a normed group from a norm and a group.
class toNrmGrp
 
Syntaxcnrg 22431 Normed ring.
class NrmRing
 
Syntaxcnlm 22432 Normed module.
class NrmMod
 
Syntaxcnvc 22433 Normed vector space.
class NrmVec
 
Definitiondf-nm 22434* 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 22435 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 22436* 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 22437 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 22438* 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 22439 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 22440* 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 22441 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 22442* 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 22443 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 22444 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 22445 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 22446* 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 22447 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 22448 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 22449* 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 22450 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
 
Theoremngpms 22451 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
 
Theoremngpxms 22452 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
 
Theoremngptps 22453 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
 
Theoremngpmet 22454 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 22455 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 22456 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 22457 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 22458 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 22459 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 22460 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 22461 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 22462 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 22463* 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 22464 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 22465 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 22466 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 22467 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 22468 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 22469 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 22470 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 22471 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 22472 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 22473 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 22474 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 22475 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 22476 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 22477 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 22478 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 22479* 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 22480 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ NrmGrp → (𝑁0 ) = 0)
 
Theoremnmgt0 22481 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 22482 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 22483 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 22484 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 22485 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 22486 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 22487 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 22488* 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 22489 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp
 
Theoremtngval 22490 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 22491 Lemma for tngbas 22492 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 22492 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 22493 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 22494 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 22495 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 22496 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 22497 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 22498 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 22499 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 22500 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))
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