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Type | Label | Description |
---|---|---|
Statement | ||
Theorem | blval2 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) ∧ 𝑃 ∈ 𝑋 ∧ 𝑅 ∈ ℝ+) → (𝑃(ball‘𝐷)𝑅) = ((◡𝐷 “ (0[,)𝑅)) “ {𝑃})) | ||
Theorem | elbl4 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+) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐵 ∈ (𝐴(ball‘𝐷)𝑅) ↔ 𝐵(◡𝐷 “ (0[,)𝑅))𝐴)) | ||
Theorem | metuel 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)))𝑤 ⊆ 𝑉))) | ||
Theorem | metuel2 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)) → (𝑉 ∈ 𝑈 ↔ (𝑉 ⊆ (𝑋 × 𝑋) ∧ ∃𝑑 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐷𝑦) < 𝑑 → 𝑥𝑉𝑦)))) | ||
Theorem | metustbl 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) ∧ 𝑃 ∈ 𝑋) → ∃𝑎 ∈ ran (ball‘𝐷)(𝑃 ∈ 𝑎 ∧ 𝑎 ⊆ (𝑉 “ {𝑃}))) | ||
Theorem | psmetutop 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))) | ||
Theorem | xmetutop 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)) | ||
Theorem | xmsusp 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) | ||
Theorem | restmetu 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) ∧ 𝐴 ⊆ 𝑋) → ((metUnif‘𝐷) ↾t (𝐴 × 𝐴)) = (metUnif‘(𝐷 ↾ (𝐴 × 𝐴)))) | ||
Theorem | metucn 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) & ⊢ (𝜑 → 𝑋 ≠ ∅) & ⊢ (𝜑 → 𝑌 ≠ ∅) & ⊢ (𝜑 → 𝐶 ∈ (PsMet‘𝑋)) & ⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑌)) ⇒ ⊢ (𝜑 → (𝐹 ∈ (𝑈 Cnu𝑉) ↔ (𝐹:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ+ ∃𝑐 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ((𝑥𝐶𝑦) < 𝑐 → ((𝐹‘𝑥)𝐷(𝐹‘𝑦)) < 𝑑)))) | ||
Theorem | dscmet 23111* | The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.) |
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1)) ⇒ ⊢ (𝑋 ∈ 𝑉 → 𝐷 ∈ (Met‘𝑋)) | ||
Theorem | dscopn 23112* | The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.) |
⊢ 𝐷 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(𝑥 = 𝑦, 0, 1)) ⇒ ⊢ (𝑋 ∈ 𝑉 → (MetOpen‘𝐷) = 𝒫 𝑋) | ||
Theorem | nrmmetd 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) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝐹:𝑋⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝐹‘𝑥) = 0 ↔ 𝑥 = 0 )) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝐹‘(𝑥 − 𝑦)) ≤ ((𝐹‘𝑥) + (𝐹‘𝑦))) ⇒ ⊢ (𝜑 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) | ||
Theorem | abvmet 23114 | An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥 − 𝑦), 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) & ⊢ − = (-g‘𝑅) ⇒ ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ − ) ∈ (Met‘𝑋)) | ||
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: (𝑁‘𝐴) = (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 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. | ||
Syntax | cnm 23115 | Norm of a normed ring. |
class norm | ||
Syntax | cngp 23116 | The class of all normed groups. |
class NrmGrp | ||
Syntax | ctng 23117 | Make a normed group from a norm and a group. |
class toNrmGrp | ||
Syntax | cnrg 23118 | Normed ring. |
class NrmRing | ||
Syntax | cnlm 23119 | Normed module. |
class NrmMod | ||
Syntax | cnvc 23120 | Normed vector space. |
class NrmVec | ||
Definition | df-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)(0g‘𝑤)))) | ||
Definition | df-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) ∘ (-g‘𝑔)) ⊆ (dist‘𝑔)} | ||
Definition | df-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 ∘ (-g‘𝑔))〉) sSet 〈(TopSet‘ndx), (MetOpen‘(𝑓 ∘ (-g‘𝑔)))〉)) | ||
Definition | df-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)} | ||
Definition | df-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( ·𝑠 ‘𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))} | ||
Definition | df-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) | ||
Theorem | nmfval 23127* | The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑁 = (norm‘𝑊) & ⊢ 𝑋 = (Base‘𝑊) & ⊢ 0 = (0g‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) ⇒ ⊢ 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐷 0 )) | ||
Theorem | nmval 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 = (0g‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) ⇒ ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 )) | ||
Theorem | nmfval2 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 = (0g‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 ))) | ||
Theorem | nmval2 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 = (0g‘𝑊) & ⊢ 𝐷 = (dist‘𝑊) & ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) ⇒ ⊢ ((𝑊 ∈ Grp ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) = (𝐴𝐸 0 )) | ||
Theorem | nmf2 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) | ||
Theorem | nmpropd 23132 | Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → (Base‘𝐾) = (Base‘𝐿)) & ⊢ (𝜑 → (+g‘𝐾) = (+g‘𝐿)) & ⊢ (𝜑 → (dist‘𝐾) = (dist‘𝐿)) ⇒ ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) | ||
Theorem | nmpropd2 23133* | Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ (𝜑 → 𝐾 ∈ Grp) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) ⇒ ⊢ (𝜑 → (norm‘𝐾) = (norm‘𝐿)) | ||
Theorem | isngp 23134 | The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) ⊆ 𝐷)) | ||
Theorem | isngp2 23135 | The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋)) ⇒ ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ − ) = 𝐸)) | ||
Theorem | isngp3 23136* | The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) & ⊢ 𝑋 = (Base‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 (𝑥𝐷𝑦) = (𝑁‘(𝑥 − 𝑦)))) | ||
Theorem | ngpgrp 23137 | A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp) | ||
Theorem | ngpms 23138 | A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp) | ||
Theorem | ngpxms 23139 | A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp) | ||
Theorem | ngptps 23140 | A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.) |
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp) | ||
Theorem | ngpmet 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)) | ||
Theorem | ngpds 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) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 − 𝐵))) | ||
Theorem | ngpdsr 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) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐵 − 𝐴))) | ||
Theorem | ngpds2 23144 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐴 − 𝐵)𝐷 0 )) | ||
Theorem | ngpds2r 23145 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ((𝐵 − 𝐴)𝐷 0 )) | ||
Theorem | ngpds3 23146 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐴 − 𝐵))) | ||
Theorem | ngpds3r 23147 | Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐵 − 𝐴))) | ||
Theorem | ngprcan 23148 | Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 + 𝐶)𝐷(𝐵 + 𝐶)) = (𝐴𝐷𝐵)) | ||
Theorem | ngplcan 23149 | Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐶 + 𝐴)𝐷(𝐶 + 𝐵)) = (𝐴𝐷𝐵)) | ||
Theorem | isngp4 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) & ⊢ + = (+g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ 𝑋 ∀𝑧 ∈ 𝑋 ((𝑥 + 𝑧)𝐷(𝑦 + 𝑧)) = (𝑥𝐷𝑦))) | ||
Theorem | ngpinvds 23151 | Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → ((𝐼‘𝐴)𝐷(𝐼‘𝐵)) = (𝐴𝐷𝐵)) | ||
Theorem | ngpsubcan 23152 | Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (dist‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴 − 𝐶)𝐷(𝐵 − 𝐶)) = (𝐴𝐷𝐵)) | ||
Theorem | nmf 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) | ||
Theorem | nmcl 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 ∧ 𝐴 ∈ 𝑋) → (𝑁‘𝐴) ∈ ℝ) | ||
Theorem | nmge0 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 ∧ 𝐴 ∈ 𝑋) → 0 ≤ (𝑁‘𝐴)) | ||
Theorem | nmeq0 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 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → ((𝑁‘𝐴) = 0 ↔ 𝐴 = 0 )) | ||
Theorem | nmne0 23157 | The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → (𝑁‘𝐴) ≠ 0) | ||
Theorem | nmrpcl 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 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 ≠ 0 ) → (𝑁‘𝐴) ∈ ℝ+) | ||
Theorem | nminv 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 = (invg‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝑁‘(𝐼‘𝐴)) = (𝑁‘𝐴)) | ||
Theorem | nmmtri 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) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
Theorem | nmsub 23161 | The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 − 𝐵)) = (𝑁‘(𝐵 − 𝐴))) | ||
Theorem | nmrtri 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) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (abs‘((𝑁‘𝐴) − (𝑁‘𝐵))) ≤ (𝑁‘(𝐴 − 𝐵))) | ||
Theorem | nm2dif 23163 | Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.) |
⊢ 𝑋 = (Base‘𝐺) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → ((𝑁‘𝐴) − (𝑁‘𝐵)) ≤ (𝑁‘(𝐴 − 𝐵))) | ||
Theorem | nmtri 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) & ⊢ + = (+g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 + 𝐵)) ≤ ((𝑁‘𝐴) + (𝑁‘𝐵))) | ||
Theorem | nmtri2 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) & ⊢ − = (-g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → (𝑁‘(𝐴 − 𝐶)) ≤ ((𝑁‘(𝐴 − 𝐵)) + (𝑁‘(𝐵 − 𝐶)))) | ||
Theorem | ngpi 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) & ⊢ − = (-g‘𝑊) & ⊢ 0 = (0g‘𝑊) ⇒ ⊢ (𝑊 ∈ NrmGrp → (𝑊 ∈ Grp ∧ 𝑁:𝑉⟶ℝ ∧ ∀𝑥 ∈ 𝑉 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))))) | ||
Theorem | nm0 23167 | Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑁 = (norm‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝐺 ∈ NrmGrp → (𝑁‘ 0 ) = 0) | ||
Theorem | nmgt0 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 = (0g‘𝐺) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) → (𝐴 ≠ 0 ↔ 0 < (𝑁‘𝐴))) | ||
Theorem | sgrim 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 = (𝑇 ↾s 𝑈) & ⊢ 𝐷 = (dist‘𝑇) & ⊢ 𝐸 = (dist‘𝑋) ⇒ ⊢ (𝑈 ∈ 𝑆 → 𝐸 = 𝐷) | ||
Theorem | sgrimval 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 = (𝑇 ↾s 𝑈) & ⊢ 𝐷 = (dist‘𝑇) & ⊢ 𝐸 = (dist‘𝑋) & ⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑆 = (SubGrp‘𝑇) ⇒ ⊢ (((𝐺 ∈ NrmGrp ∧ 𝑈 ∈ 𝑆) ∧ (𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈)) → (𝐴𝐸𝐵) = (𝐴𝐷𝐵)) | ||
Theorem | subgnm 23171 | The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑀 = (norm‘𝐻) ⇒ ⊢ (𝐴 ∈ (SubGrp‘𝐺) → 𝑀 = (𝑁 ↾ 𝐴)) | ||
Theorem | subgnm2 23172 | A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) & ⊢ 𝑁 = (norm‘𝐺) & ⊢ 𝑀 = (norm‘𝐻) ⇒ ⊢ ((𝐴 ∈ (SubGrp‘𝐺) ∧ 𝑋 ∈ 𝐴) → (𝑀‘𝑋) = (𝑁‘𝑋)) | ||
Theorem | subgngp 23173 | A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝐻 = (𝐺 ↾s 𝐴) ⇒ ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ NrmGrp) | ||
Theorem | ngptgp 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) | ||
Theorem | ngppropd 23175* | Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝜑 → 𝐵 = (Base‘𝐾)) & ⊢ (𝜑 → 𝐵 = (Base‘𝐿)) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)) & ⊢ (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵))) & ⊢ (𝜑 → (TopOpen‘𝐾) = (TopOpen‘𝐿)) ⇒ ⊢ (𝜑 → (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp)) | ||
Theorem | reldmtng 23176 | The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.) |
⊢ Rel dom toNrmGrp | ||
Theorem | tngval 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) & ⊢ − = (-g‘𝐺) & ⊢ 𝐷 = (𝑁 ∘ − ) & ⊢ 𝐽 = (MetOpen‘𝐷) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝑇 = ((𝐺 sSet 〈(dist‘ndx), 𝐷〉) sSet 〈(TopSet‘ndx), 𝐽〉)) | ||
Theorem | tnglem 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 ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇)) | ||
Theorem | tngbas 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) ⇒ ⊢ (𝑁 ∈ 𝑉 → 𝐵 = (Base‘𝑇)) | ||
Theorem | tngplusg 23180 | The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ + = (+g‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → + = (+g‘𝑇)) | ||
Theorem | tng0 23181 | The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇)) | ||
Theorem | tngmulr 23182 | The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ · = (.r‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → · = (.r‘𝑇)) | ||
Theorem | tngsca 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) ⇒ ⊢ (𝑁 ∈ 𝑉 → 𝐹 = (Scalar‘𝑇)) | ||
Theorem | tngvsca 23184 | The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ · = ( ·𝑠 ‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘𝑇)) | ||
Theorem | tngip 23185 | The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ , = (·𝑖‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇)) | ||
Theorem | tngds 23186 | The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) |
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁) & ⊢ − = (-g‘𝐺) ⇒ ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇)) | ||
Theorem | tngtset 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) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝐽 = (TopSet‘𝑇)) | ||
Theorem | tngtopn 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) ⇒ ⊢ ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ 𝑊) → 𝐽 = (TopOpen‘𝑇)) | ||
Theorem | tngnm 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:𝑋⟶𝐴) → 𝑁 = (norm‘𝑇)) | ||
Theorem | tngngp2 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)))) | ||
Theorem | tngngpd 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) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) & ⊢ (𝜑 → 𝐺 ∈ Grp) & ⊢ (𝜑 → 𝑁:𝑋⟶ℝ) & ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 )) & ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦))) ⇒ ⊢ (𝜑 → 𝑇 ∈ NrmGrp) | ||
Theorem | tngngp 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) & ⊢ − = (-g‘𝐺) & ⊢ 0 = (0g‘𝐺) ⇒ ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 − 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))) | ||
Theorem | tnggrpr 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) ⇒ ⊢ ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp) | ||
Theorem | tngngp3 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 = (0g‘𝐺) & ⊢ + = (+g‘𝐺) & ⊢ 𝐼 = (invg‘𝐺) ⇒ ⊢ (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥 ∈ 𝑋 (((𝑁‘𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼‘𝑥)) = (𝑁‘𝑥) ∧ ∀𝑦 ∈ 𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁‘𝑥) + (𝑁‘𝑦)))))) | ||
Theorem | nrmtngdist 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))) | ||
Theorem | nrmtngnrm 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))) | ||
Theorem | tngngpim 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) | ||
Theorem | isnrg 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 ∧ 𝑁 ∈ 𝐴)) | ||
Theorem | nrgabv 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 → 𝑁 ∈ 𝐴) | ||
Theorem | nrgngp 23200 | A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.) |
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp) |
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