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Theorem List for Metamath Proof Explorer - 24301-24400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmetucn 24301* Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24273. (Contributed by Thierry Arnoux, 26-Jan-2018.) (Revised by Thierry Arnoux, 11-Feb-2018.)
π‘ˆ = (metUnifβ€˜πΆ)    &   π‘‰ = (metUnifβ€˜π·)    &   (πœ‘ β†’ 𝑋 β‰  βˆ…)    &   (πœ‘ β†’ π‘Œ β‰  βˆ…)    &   (πœ‘ β†’ 𝐢 ∈ (PsMetβ€˜π‘‹))    &   (πœ‘ β†’ 𝐷 ∈ (PsMetβ€˜π‘Œ))    β‡’   (πœ‘ β†’ (𝐹 ∈ (π‘ˆ Cnu𝑉) ↔ (𝐹:π‘‹βŸΆπ‘Œ ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘ ∈ ℝ+ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 ((π‘₯𝐢𝑦) < 𝑐 β†’ ((πΉβ€˜π‘₯)𝐷(πΉβ€˜π‘¦)) < 𝑑))))
 
12.4.7  Examples of metric spaces
 
Theoremdscmet 24302* The discrete metric on any set 𝑋. Definition 1.1-8 of [Kreyszig] p. 8. (Contributed by FL, 12-Oct-2006.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(π‘₯ = 𝑦, 0, 1))    β‡’   (𝑋 ∈ 𝑉 β†’ 𝐷 ∈ (Metβ€˜π‘‹))
 
Theoremdscopn 24303* The discrete metric generates the discrete topology. In particular, the discrete topology is metrizable. (Contributed by Mario Carneiro, 29-Jan-2014.)
𝐷 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ if(π‘₯ = 𝑦, 0, 1))    β‡’   (𝑋 ∈ 𝑉 β†’ (MetOpenβ€˜π·) = 𝒫 𝑋)
 
Theoremnrmmetd 24304* 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 = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝐹:π‘‹βŸΆβ„)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((πΉβ€˜π‘₯) = 0 ↔ π‘₯ = 0 ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (πΉβ€˜(π‘₯ βˆ’ 𝑦)) ≀ ((πΉβ€˜π‘₯) + (πΉβ€˜π‘¦)))    β‡’   (πœ‘ β†’ (𝐹 ∘ βˆ’ ) ∈ (Metβ€˜π‘‹))
 
Theoremabvmet 24305 An absolute value 𝐹 generates a metric defined by 𝑑(π‘₯, 𝑦) = 𝐹(π‘₯ βˆ’ 𝑦), analogously to cnmet 24509. (In fact, the ring structure is not needed at all; the group properties abveq0 20578 and abvtri 20582, abvneg 20586 are sufficient.) (Contributed by Mario Carneiro, 9-Sep-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜π‘…)    &   π΄ = (AbsValβ€˜π‘…)    &    βˆ’ = (-gβ€˜π‘…)    β‡’   (𝐹 ∈ 𝐴 β†’ (𝐹 ∘ βˆ’ ) ∈ (Metβ€˜π‘‹))
 
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 24319: (π‘β€˜π΄) = (𝐴𝐷 0 )). By this definition, the norm function 𝑁 is actually a norm (satisfying the properties: being a function into the reals; subadditivity/triangle inequality (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)); 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 ∈ β„€, 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., (π‘β€˜( βˆ’ π‘₯) = π‘β€˜π‘₯). The definition df-ngp 24313 of a group norm follows the second aproach, see nminv 24351.] and positive definiteness/point-separation ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0)) if the structure is a metric space with a right-translation-invariant metric (see nmf 24345, nmtri 24356, nmvs 24414 and nmeq0 24348). An alternate definition of a normed group (i.e., a group equipped with a norm) not using the properties of a metric space is given by Theorem tngngp3 24394. The norm can be expressed as the distance to zero (nmfval 24318), so in a structure with a zero (a "pointed set", for instance a monoid or a group), the norm can be expressed as the distance restricted to the elements of the base set to zero (nmfval0 24320).

Usually, however, the norm of a normed structure is given, and the corresponding metric ("induced metric") is defined as the distance function based on the norm (the distance 𝐷 between two elements is the norm 𝑁 of their difference, see ngpds 24334: (𝐴𝐷𝐡) = (π‘β€˜(𝐴 βˆ’ 𝐡))). 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 for the difference of two elements to make sense) corresponding to a given norm as explained above: (distβ€˜π‘‡) = (𝑁 ∘ βˆ’ ), see also tngds 24385. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 24390) or a norm (see tngngpd 24391). If the norm is derived from a given metric, as done with df-nm 24312, the induced metric is the original metric restricted to the base set: (distβ€˜π‘‡) = ((distβ€˜πΊ) β†Ύ (𝑋 Γ— 𝑋)), see nrmtngdist 24395, and the norm remains the same: (normβ€˜π‘‡) = (normβ€˜πΊ), see nrmtngnrm 24396.

 
Syntaxcnm 24306 Norm of a normed ring.
class norm
 
Syntaxcngp 24307 The class of all normed groups.
class NrmGrp
 
Syntaxctng 24308 Make a normed group from a norm and a group.
class toNrmGrp
 
Syntaxcnrg 24309 Normed ring.
class NrmRing
 
Syntaxcnlm 24310 Normed module.
class NrmMod
 
Syntaxcnvc 24311 Normed vector space.
class NrmVec
 
Definitiondf-nm 24312* 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 = (𝑀 ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘€) ↦ (π‘₯(distβ€˜π‘€)(0gβ€˜π‘€))))
 
Definitiondf-ngp 24313 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) ∣ ((normβ€˜π‘”) ∘ (-gβ€˜π‘”)) βŠ† (distβ€˜π‘”)}
 
Definitiondf-tng 24314* 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 ⟨(distβ€˜ndx), (𝑓 ∘ (-gβ€˜π‘”))⟩) sSet ⟨(TopSetβ€˜ndx), (MetOpenβ€˜(𝑓 ∘ (-gβ€˜π‘”)))⟩))
 
Definitiondf-nrg 24315 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 = {𝑀 ∈ NrmGrp ∣ (normβ€˜π‘€) ∈ (AbsValβ€˜π‘€)}
 
Definitiondf-nlm 24316* 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 = {𝑀 ∈ (NrmGrp ∩ LMod) ∣ [(Scalarβ€˜π‘€) / 𝑓](𝑓 ∈ NrmRing ∧ βˆ€π‘₯ ∈ (Baseβ€˜π‘“)βˆ€π‘¦ ∈ (Baseβ€˜π‘€)((normβ€˜π‘€)β€˜(π‘₯( ·𝑠 β€˜π‘€)𝑦)) = (((normβ€˜π‘“)β€˜π‘₯) Β· ((normβ€˜π‘€)β€˜π‘¦)))}
 
Definitiondf-nvc 24317 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 24318* The value of the norm function as the distance to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    β‡’   π‘ = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐷 0 ))
 
Theoremnmval 24319 The value of the norm as the distance to zero. Problem 1 of [Kreyszig] p. 63. (Contributed by NM, 4-Dec-2006.) (Revised by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    β‡’   (𝐴 ∈ 𝑋 β†’ (π‘β€˜π΄) = (𝐴𝐷 0 ))
 
Theoremnmfval0 24320* The value of the norm function on a structure containing a zero as the distance restricted to the elements of the base set to zero. Examples of structures containing a "zero" are groups (see nmfval2 24321 proved from this theorem and grpidcl 18887) or more generally monoids (see mndidcl 18675), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24321. (Revised by BJ, 27-Aug-2024.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   πΈ = (𝐷 β†Ύ (𝑋 Γ— 𝑋))    β‡’   ( 0 ∈ 𝑋 β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
 
Theoremnmfval2 24321* The value of the norm function on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   πΈ = (𝐷 β†Ύ (𝑋 Γ— 𝑋))    β‡’   (π‘Š ∈ Grp β†’ 𝑁 = (π‘₯ ∈ 𝑋 ↦ (π‘₯𝐸 0 )))
 
Theoremnmval2 24322 The value of the norm on a group as the distance restricted to the elements of the base set to zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   πΈ = (𝐷 β†Ύ (𝑋 Γ— 𝑋))    β‡’   ((π‘Š ∈ Grp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) = (𝐴𝐸 0 ))
 
Theoremnmf2 24323 The norm on a metric group is a function from the base set into the reals. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜π‘Š)    &   π‘‹ = (Baseβ€˜π‘Š)    &   π· = (distβ€˜π‘Š)    &   πΈ = (𝐷 β†Ύ (𝑋 Γ— 𝑋))    β‡’   ((π‘Š ∈ Grp ∧ 𝐸 ∈ (Metβ€˜π‘‹)) β†’ 𝑁:π‘‹βŸΆβ„)
 
Theoremnmpropd 24324 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(πœ‘ β†’ (Baseβ€˜πΎ) = (Baseβ€˜πΏ))    &   (πœ‘ β†’ (+gβ€˜πΎ) = (+gβ€˜πΏ))    &   (πœ‘ β†’ (distβ€˜πΎ) = (distβ€˜πΏ))    β‡’   (πœ‘ β†’ (normβ€˜πΎ) = (normβ€˜πΏ))
 
Theoremnmpropd2 24325* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   (πœ‘ β†’ 𝐾 ∈ Grp)    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   (πœ‘ β†’ ((distβ€˜πΎ) β†Ύ (𝐡 Γ— 𝐡)) = ((distβ€˜πΏ) β†Ύ (𝐡 Γ— 𝐡)))    β‡’   (πœ‘ β†’ (normβ€˜πΎ) = (normβ€˜πΏ))
 
Theoremisngp 24326 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) βŠ† 𝐷))
 
Theoremisngp2 24327 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    &   πΈ = (𝐷 β†Ύ (𝑋 Γ— 𝑋))    β‡’   (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ∘ βˆ’ ) = 𝐸))
 
Theoremisngp3 24328* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 (π‘₯𝐷𝑦) = (π‘β€˜(π‘₯ βˆ’ 𝑦))))
 
Theoremngpgrp 24329 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp β†’ 𝐺 ∈ Grp)
 
Theoremngpms 24330 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp β†’ 𝐺 ∈ MetSp)
 
Theoremngpxms 24331 A normed group is an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp β†’ 𝐺 ∈ ∞MetSp)
 
Theoremngptps 24332 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝐺 ∈ NrmGrp β†’ 𝐺 ∈ TopSp)
 
Theoremngpmet 24333 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 = (Baseβ€˜πΊ)    &   π· = ((distβ€˜πΊ) β†Ύ (𝑋 Γ— 𝑋))    β‡’   (𝐺 ∈ NrmGrp β†’ 𝐷 ∈ (Metβ€˜π‘‹))
 
Theoremngpds 24334 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 = (normβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (π‘β€˜(𝐴 βˆ’ 𝐡)))
 
Theoremngpdsr 24335 Value of the distance function in terms of the norm of a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (normβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = (π‘β€˜(𝐡 βˆ’ 𝐴)))
 
Theoremngpds2 24336 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = ((𝐴 βˆ’ 𝐡)𝐷 0 ))
 
Theoremngpds2r 24337 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = ((𝐡 βˆ’ 𝐴)𝐷 0 ))
 
Theoremngpds3 24338 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = ( 0 𝐷(𝐴 βˆ’ 𝐡)))
 
Theoremngpds3r 24339 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝐷𝐡) = ( 0 𝐷(𝐡 βˆ’ 𝐴)))
 
Theoremngprcan 24340 Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴 + 𝐢)𝐷(𝐡 + 𝐢)) = (𝐴𝐷𝐡))
 
Theoremngplcan 24341 Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐢 + 𝐴)𝐷(𝐢 + 𝐡)) = (𝐴𝐷𝐡))
 
Theoremisngp4 24342* 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 = (Baseβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ βˆ€π‘₯ ∈ 𝑋 βˆ€π‘¦ ∈ 𝑋 βˆ€π‘§ ∈ 𝑋 ((π‘₯ + 𝑧)𝐷(𝑦 + 𝑧)) = (π‘₯𝐷𝑦)))
 
Theoremngpinvds 24343 Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ ((πΌβ€˜π΄)𝐷(πΌβ€˜π΅)) = (𝐴𝐷𝐡))
 
Theoremngpsubcan 24344 Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (distβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ ((𝐴 βˆ’ 𝐢)𝐷(𝐡 βˆ’ 𝐢)) = (𝐴𝐷𝐡))
 
Theoremnmf 24345 The norm on a normed group is a function into the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp β†’ 𝑁:π‘‹βŸΆβ„)
 
Theoremnmcl 24346 The norm of a normed group is closed in the reals. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜π΄) ∈ ℝ)
 
Theoremnmge0 24347 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 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) β†’ 0 ≀ (π‘β€˜π΄))
 
Theoremnmeq0 24348 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 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) β†’ ((π‘β€˜π΄) = 0 ↔ 𝐴 = 0 ))
 
Theoremnmne0 24349 The norm of a nonzero element is nonzero. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 β‰  0 ) β†’ (π‘β€˜π΄) β‰  0)
 
Theoremnmrpcl 24350 The norm of a nonzero element is a positive real. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐴 β‰  0 ) β†’ (π‘β€˜π΄) ∈ ℝ+)
 
Theoremnminv 24351 The norm of a negated element is the same as the norm of the original element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) β†’ (π‘β€˜(πΌβ€˜π΄)) = (π‘β€˜π΄))
 
Theoremnmmtri 24352 The triangle inequality for the norm of a subtraction. (Contributed by NM, 27-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 βˆ’ 𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅)))
 
Theoremnmsub 24353 The norm of the difference between two elements. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 βˆ’ 𝐡)) = (π‘β€˜(𝐡 βˆ’ 𝐴)))
 
Theoremnmrtri 24354 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 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (absβ€˜((π‘β€˜π΄) βˆ’ (π‘β€˜π΅))) ≀ (π‘β€˜(𝐴 βˆ’ 𝐡)))
 
Theoremnm2dif 24355 Inequality for the difference of norms. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ ((π‘β€˜π΄) βˆ’ (π‘β€˜π΅)) ≀ (π‘β€˜(𝐴 βˆ’ 𝐡)))
 
Theoremnmtri 24356 The triangle inequality for the norm of a sum. (Contributed by NM, 11-Nov-2006.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    + = (+gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (π‘β€˜(𝐴 + 𝐡)) ≀ ((π‘β€˜π΄) + (π‘β€˜π΅)))
 
Theoremnmtri2 24357 Triangle inequality for the norm of two subtractions. (Contributed by NM, 24-Feb-2008.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) β†’ (π‘β€˜(𝐴 βˆ’ 𝐢)) ≀ ((π‘β€˜(𝐴 βˆ’ 𝐡)) + (π‘β€˜(𝐡 βˆ’ 𝐢))))
 
Theoremngpi 24358* The properties of a normed group, which is a group accompanied by a norm. (Contributed by AV, 7-Oct-2021.)
𝑉 = (Baseβ€˜π‘Š)    &   π‘ = (normβ€˜π‘Š)    &    βˆ’ = (-gβ€˜π‘Š)    &    0 = (0gβ€˜π‘Š)    β‡’   (π‘Š ∈ NrmGrp β†’ (π‘Š ∈ Grp ∧ 𝑁:π‘‰βŸΆβ„ ∧ βˆ€π‘₯ ∈ 𝑉 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑉 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))))
 
Theoremnm0 24359 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp β†’ (π‘β€˜ 0 ) = 0)
 
Theoremnmgt0 24360 The norm of a nonzero element is a positive real. (Contributed by NM, 20-Nov-2007.) (Revised by AV, 8-Oct-2021.)
𝑋 = (Baseβ€˜πΊ)    &   π‘ = (normβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ 𝑋) β†’ (𝐴 β‰  0 ↔ 0 < (π‘β€˜π΄)))
 
Theoremsgrim 24361 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 β†Ύs π‘ˆ)    &   π· = (distβ€˜π‘‡)    &   πΈ = (distβ€˜π‘‹)    β‡’   (π‘ˆ ∈ 𝑆 β†’ 𝐸 = 𝐷)
 
Theoremsgrimval 24362 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 β†Ύs π‘ˆ)    &   π· = (distβ€˜π‘‡)    &   πΈ = (distβ€˜π‘‹)    &   π‘‡ = (𝐺 toNrmGrp 𝑁)    &   π‘ = (normβ€˜πΊ)    &   π‘† = (SubGrpβ€˜π‘‡)    β‡’   (((𝐺 ∈ NrmGrp ∧ π‘ˆ ∈ 𝑆) ∧ (𝐴 ∈ π‘ˆ ∧ 𝐡 ∈ π‘ˆ)) β†’ (𝐴𝐸𝐡) = (𝐴𝐷𝐡))
 
Theoremsubgnm 24363 The norm in a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺 β†Ύs 𝐴)    &   π‘ = (normβ€˜πΊ)    &   π‘€ = (normβ€˜π»)    β‡’   (𝐴 ∈ (SubGrpβ€˜πΊ) β†’ 𝑀 = (𝑁 β†Ύ 𝐴))
 
Theoremsubgnm2 24364 A substructure assigns the same values to the norms of elements of a subgroup. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺 β†Ύs 𝐴)    &   π‘ = (normβ€˜πΊ)    &   π‘€ = (normβ€˜π»)    β‡’   ((𝐴 ∈ (SubGrpβ€˜πΊ) ∧ 𝑋 ∈ 𝐴) β†’ (π‘€β€˜π‘‹) = (π‘β€˜π‘‹))
 
Theoremsubgngp 24365 A normed group restricted to a subgroup is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺 β†Ύs 𝐴)    β‡’   ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrpβ€˜πΊ)) β†’ 𝐻 ∈ NrmGrp)
 
Theoremngptgp 24366 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) β†’ 𝐺 ∈ TopGrp)
 
Theoremngppropd 24367* Property deduction for a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(πœ‘ β†’ 𝐡 = (Baseβ€˜πΎ))    &   (πœ‘ β†’ 𝐡 = (Baseβ€˜πΏ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝐡 ∧ 𝑦 ∈ 𝐡)) β†’ (π‘₯(+gβ€˜πΎ)𝑦) = (π‘₯(+gβ€˜πΏ)𝑦))    &   (πœ‘ β†’ ((distβ€˜πΎ) β†Ύ (𝐡 Γ— 𝐡)) = ((distβ€˜πΏ) β†Ύ (𝐡 Γ— 𝐡)))    &   (πœ‘ β†’ (TopOpenβ€˜πΎ) = (TopOpenβ€˜πΏ))    β‡’   (πœ‘ β†’ (𝐾 ∈ NrmGrp ↔ 𝐿 ∈ NrmGrp))
 
Theoremreldmtng 24368 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp
 
Theoremtngval 24369 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)    &    βˆ’ = (-gβ€˜πΊ)    &   π· = (𝑁 ∘ βˆ’ )    &   π½ = (MetOpenβ€˜π·)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ 𝑇 = ((𝐺 sSet ⟨(distβ€˜ndx), 𝐷⟩) sSet ⟨(TopSetβ€˜ndx), 𝐽⟩))
 
Theoremtnglem 24370 Lemma for tngbas 24372 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   πΈ = Slot (πΈβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (TopSetβ€˜ndx)    &   (πΈβ€˜ndx) β‰  (distβ€˜ndx)    β‡’   (𝑁 ∈ 𝑉 β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
 
TheoremtnglemOLD 24371 Obsolete version of tnglem 24370 as of 31-Oct-2024. Lemma for tngbas 24372 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   πΈ = Slot 𝐾    &   πΎ ∈ β„•    &   πΎ < 9    β‡’   (𝑁 ∈ 𝑉 β†’ (πΈβ€˜πΊ) = (πΈβ€˜π‘‡))
 
Theoremtngbas 24372 The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π‘‡))
 
TheoremtngbasOLD 24373 Obsolete proof of tngbas 24372 as of 31-Oct-2024. The base set of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π΅ = (Baseβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ 𝐡 = (Baseβ€˜π‘‡))
 
Theoremtngplusg 24374 The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    + = (+gβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ + = (+gβ€˜π‘‡))
 
TheoremtngplusgOLD 24375 Obsolete proof of tngplusg 24374 as of 31-Oct-2024. The group addition of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    + = (+gβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ + = (+gβ€˜π‘‡))
 
Theoremtng0 24376 The group identity of a structure augmented with a norm. (Contributed by Mario Carneiro, 4-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    0 = (0gβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ 0 = (0gβ€˜π‘‡))
 
Theoremtngmulr 24377 The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    Β· = (.rβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ Β· = (.rβ€˜π‘‡))
 
TheoremtngmulrOLD 24378 Obsolete proof of tngmulr 24377 as of 31-Oct-2024. The ring multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    Β· = (.rβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ Β· = (.rβ€˜π‘‡))
 
Theoremtngsca 24379 The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   πΉ = (Scalarβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ 𝐹 = (Scalarβ€˜π‘‡))
 
TheoremtngscaOLD 24380 Obsolete proof of tngsca 24379 as of 31-Oct-2024. The scalar ring of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   πΉ = (Scalarβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ 𝐹 = (Scalarβ€˜π‘‡))
 
Theoremtngvsca 24381 The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    Β· = ( ·𝑠 β€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ Β· = ( ·𝑠 β€˜π‘‡))
 
TheoremtngvscaOLD 24382 Obsolete proof of tngvsca 24381 as of 31-Oct-2024. The scalar multiplication of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    Β· = ( ·𝑠 β€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ Β· = ( ·𝑠 β€˜π‘‡))
 
Theoremtngip 24383 The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    , = (Β·π‘–β€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ , = (Β·π‘–β€˜π‘‡))
 
TheoremtngipOLD 24384 Obsolete proof of tngip 24383 as of 31-Oct-2024. The inner product operation of a structure augmented with a norm. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    , = (Β·π‘–β€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ , = (Β·π‘–β€˜π‘‡))
 
Theoremtngds 24385 The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof shortened by AV, 29-Oct-2024.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ (𝑁 ∘ βˆ’ ) = (distβ€˜π‘‡))
 
TheoremtngdsOLD 24386 Obsolete proof of tngds 24385 as of 29-Oct-2024. The metric function of a structure augmented with a norm. (Contributed by Mario Carneiro, 3-Oct-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &    βˆ’ = (-gβ€˜πΊ)    β‡’   (𝑁 ∈ 𝑉 β†’ (𝑁 ∘ βˆ’ ) = (distβ€˜π‘‡))
 
Theoremtngtset 24387 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π· = (distβ€˜π‘‡)    &   π½ = (MetOpenβ€˜π·)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ 𝐽 = (TopSetβ€˜π‘‡))
 
Theoremtngtopn 24388 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π· = (distβ€˜π‘‡)    &   π½ = (MetOpenβ€˜π·)    β‡’   ((𝐺 ∈ 𝑉 ∧ 𝑁 ∈ π‘Š) β†’ 𝐽 = (TopOpenβ€˜π‘‡))
 
Theoremtngnm 24389 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π‘‹ = (Baseβ€˜πΊ)    &   π΄ ∈ V    β‡’   ((𝐺 ∈ Grp ∧ 𝑁:π‘‹βŸΆπ΄) β†’ 𝑁 = (normβ€˜π‘‡))
 
Theoremtngngp2 24390 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)    &   π‘‹ = (Baseβ€˜πΊ)    &   π· = (distβ€˜π‘‡)    β‡’   (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Metβ€˜π‘‹))))
 
Theoremtngngpd 24391* 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)    &   π‘‹ = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &   (πœ‘ β†’ 𝐺 ∈ Grp)    &   (πœ‘ β†’ 𝑁:π‘‹βŸΆβ„)    &   ((πœ‘ ∧ π‘₯ ∈ 𝑋) β†’ ((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ))    &   ((πœ‘ ∧ (π‘₯ ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) β†’ (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦)))    β‡’   (πœ‘ β†’ 𝑇 ∈ NrmGrp)
 
Theoremtngngp 24392* 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)    &   π‘‹ = (Baseβ€˜πΊ)    &    βˆ’ = (-gβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    β‡’   (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ βˆ’ 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
 
Theoremtnggrpr 24393 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)    β‡’   ((𝑁 ∈ 𝑉 ∧ 𝑇 ∈ NrmGrp) β†’ 𝐺 ∈ Grp)
 
Theoremtngngp3 24394* 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)    &   π‘‹ = (Baseβ€˜πΊ)    &    0 = (0gβ€˜πΊ)    &    + = (+gβ€˜πΊ)    &   πΌ = (invgβ€˜πΊ)    β‡’   (𝑁:π‘‹βŸΆβ„ β†’ (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ βˆ€π‘₯ ∈ 𝑋 (((π‘β€˜π‘₯) = 0 ↔ π‘₯ = 0 ) ∧ (π‘β€˜(πΌβ€˜π‘₯)) = (π‘β€˜π‘₯) ∧ βˆ€π‘¦ ∈ 𝑋 (π‘β€˜(π‘₯ + 𝑦)) ≀ ((π‘β€˜π‘₯) + (π‘β€˜π‘¦))))))
 
Theoremnrmtngdist 24395 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 (normβ€˜πΊ))    &   π‘‹ = (Baseβ€˜πΊ)    β‡’   (𝐺 ∈ NrmGrp β†’ (distβ€˜π‘‡) = ((distβ€˜πΊ) β†Ύ (𝑋 Γ— 𝑋)))
 
Theoremnrmtngnrm 24396 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 (normβ€˜πΊ))    β‡’   (𝐺 ∈ NrmGrp β†’ (𝑇 ∈ NrmGrp ∧ (normβ€˜π‘‡) = (normβ€˜πΊ)))
 
Theoremtngngpim 24397 The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   π‘ = (normβ€˜πΊ)    &   π‘‹ = (Baseβ€˜πΊ)    &   π· = (distβ€˜π‘‡)    β‡’   (𝐺 ∈ NrmGrp β†’ 𝐷:(𝑋 Γ— 𝑋)βŸΆβ„)
 
Theoremisnrg 24398 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 = (normβ€˜π‘…)    &   π΄ = (AbsValβ€˜π‘…)    β‡’   (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁 ∈ 𝐴))
 
Theoremnrgabv 24399 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (normβ€˜π‘…)    &   π΄ = (AbsValβ€˜π‘…)    β‡’   (𝑅 ∈ NrmRing β†’ 𝑁 ∈ 𝐴)
 
Theoremnrgngp 24400 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing β†’ 𝑅 ∈ NrmGrp)
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