P. 247 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24601-24700   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	psmetutop 24601	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 24602	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 24603	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 24604	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 24605*	Uniform continuity in metric spaces. Compare the order of the quantifiers with metcn 24577. (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))))
12.4.7  Examples of metric spaces
Theorem	dscmet 24606*	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 24607*	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 24608*	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 24609	An absolute value 𝐹 generates a metric defined by 𝑑(𝑥, 𝑦) = 𝐹(𝑥 − 𝑦), analogously to cnmet 24813. (In fact, the ring structure is not needed at all; the group properties abveq0 20841 and abvtri 20845, abvneg 20849 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))
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 24623: (𝑁‘𝐴) = (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 (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 ∈ ℤ, 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 24617 of a group norm follows the second approach, see nminv 24655.] and positive definiteness/point-separation ((𝑁‘𝑥) = 0 ↔ 𝑥 = 0)) if the structure is a metric space with a right-translation-invariant metric (see nmf 24649, nmtri 24660, nmvs 24718 and nmeq0 24652). 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 24698. The norm can be expressed as the distance to zero (nmfval 24622), 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 24624). 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 24638: (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 for the difference of two elements to make sense) corresponding to a given norm as explained above: (dist‘𝑇) = (𝑁 ∘ − ), see also tngds 24689. By this, the enhanced structure becomes a normed structure if the induced metric is in fact a metric (see tngngp2 24694) or a norm (see tngngpd 24695). If the norm is derived from a given metric, as done with df-nm 24616, the induced metric is the original metric restricted to the base set: (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)), see nrmtngdist 24699, and the norm remains the same: (norm‘𝑇) = (norm‘𝐺), see nrmtngnrm 24700.
Syntax	cnm 24610	Norm of a normed ring.
class norm
Syntax	cngp 24611	The class of all normed groups.
class NrmGrp
Syntax	ctng 24612	Make a normed group from a norm and a group.
class toNrmGrp
Syntax	cnrg 24613	Normed ring.
class NrmRing
Syntax	cnlm 24614	Normed module.
class NrmMod
Syntax	cnvc 24615	Normed vector space.
class NrmVec
Definition	df-nm 24616*	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 24617	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 24618*	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 24619	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 24620*	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 24621	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 24622*	The value of the norm function as the distance to zero. (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 24623	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 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    ⇒   ⊢ (𝐴 ∈ 𝑋 → (𝑁‘𝐴) = (𝐴𝐷 0 ))
Theorem	nmfval0 24624*	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 24625 proved from this theorem and grpidcl 19005) or more generally monoids (see mndidcl 18787), or pointed sets). (Contributed by Mario Carneiro, 2-Oct-2015.) Extract this result from the proof of nmfval2 24625. (Revised by BJ, 27-Aug-2024.)
⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ ( 0 ∈ 𝑋 → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
Theorem	nmfval2 24625*	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 = (norm‘𝑊)    &   ⊢ 𝑋 = (Base‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐸 = (𝐷 ↾ (𝑋 × 𝑋))    ⇒   ⊢ (𝑊 ∈ Grp → 𝑁 = (𝑥 ∈ 𝑋 ↦ (𝑥𝐸 0 )))
Theorem	nmval2 24626	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 = (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 24627	The norm on a metric group 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 24628	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 24629*	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 24630	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 24631	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 24632*	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 24633	A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)
Theorem	ngpms 24634	A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)
Theorem	ngpxms 24635	A normed group is an extended metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)
Theorem	ngptps 24636	A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)
Theorem	ngpmet 24637	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 24638	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 24639	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 24640	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 24641	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 24642	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 24643	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 24644	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 24645	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 24646*	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 24647	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 24648	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 24649	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 24650	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 24651	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 24652	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 24653	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 24654	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 24655	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 24656	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 24657	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 24658	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 24659	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 24660	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 24661	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 24662*	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 24663	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 24664	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 24665	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 24666	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 24667	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 24668	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 24669	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 24670	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 24671*	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 24672	The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ Rel dom toNrmGrp
Theorem	tngval 24673	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 24674	Lemma for tngbas 24676 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (Revised by AV, 31-Oct-2024.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    &   ⊢ 𝐸 = Slot (𝐸‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (TopSet‘ndx)    &   ⊢ (𝐸‘ndx) ≠ (dist‘ndx)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇))
Theorem	tnglemOLD 24675	Obsolete version of tnglem 24674 as of 31-Oct-2024. Lemma for tngbas 24676 and similar theorems. (Contributed by Mario Carneiro, 2-Oct-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝑇 = (𝐺 toNrmGrp 𝑁)    &   ⊢ 𝐸 = Slot 𝐾    &   ⊢ 𝐾 ∈ ℕ    &   ⊢ 𝐾 < 9    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝐸‘𝐺) = (𝐸‘𝑇))
Theorem	tngbas 24676	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)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐵 = (Base‘𝑇))
Theorem	tngbasOLD 24677	Obsolete proof of tngbas 24676 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)    &   ⊢ 𝐵 = (Base‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐵 = (Base‘𝑇))
Theorem	tngplusg 24678	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)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → + = (+g‘𝑇))
Theorem	tngplusgOLD 24679	Obsolete proof of tngplusg 24678 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)    &   ⊢ + = (+g‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → + = (+g‘𝑇))
Theorem	tng0 24680	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 = (0g‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 0 = (0g‘𝑇))
Theorem	tngmulr 24681	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)    &   ⊢ · = (.r‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → · = (.r‘𝑇))
Theorem	tngmulrOLD 24682	Obsolete proof of tngmulr 24681 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)    &   ⊢ · = (.r‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → · = (.r‘𝑇))
Theorem	tngsca 24683	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)    &   ⊢ 𝐹 = (Scalar‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐹 = (Scalar‘𝑇))
Theorem	tngscaOLD 24684	Obsolete proof of tngsca 24683 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)    &   ⊢ 𝐹 = (Scalar‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → 𝐹 = (Scalar‘𝑇))
Theorem	tngvsca 24685	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)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘𝑇))
Theorem	tngvscaOLD 24686	Obsolete proof of tngvsca 24685 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)    &   ⊢ · = ( ·𝑠 ‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → · = ( ·𝑠 ‘𝑇))
Theorem	tngip 24687	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)    &   ⊢ , = (·𝑖‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇))
Theorem	tngipOLD 24688	Obsolete proof of tngip 24687 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)    &   ⊢ , = (·𝑖‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → , = (·𝑖‘𝑇))
Theorem	tngds 24689	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)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇))
Theorem	tngdsOLD 24690	Obsolete proof of tngds 24689 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)    &   ⊢ − = (-g‘𝐺)    ⇒   ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∘ − ) = (dist‘𝑇))
Theorem	tngtset 24691	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 24692	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 24693	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 24694	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 24695*	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 24696*	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 24697	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 24698*	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 24699	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 24700	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)))