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Theorem List for Metamath Proof Explorer - 22401-22500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremnmcl 22401 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 22402 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 22403 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 22404 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 22405 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 22406 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 22407 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 22408 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 22409 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 22410 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 22411 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 22412 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 22413* 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 22414 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ NrmGrp → (𝑁0 ) = 0)

Theoremnmgt0 22415 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 22416 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 22417 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 22418 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 22419 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 22420 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 22421 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 22422* 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 22423 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp

Theoremtngval 22424 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 22425 Lemma for tngbas 22426 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 22426 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 22427 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 22428 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 22429 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 22430 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 22431 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 22432 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 22433 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 22434 The topology generated by a normed structure. (Contributed by Mario Carneiro, 3-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐷 = (dist‘𝑇)    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝐽 = (TopSet‘𝑇))

Theoremtngtopn 22435 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝐷 = (dist‘𝑇)    &   𝐽 = (MetOpen‘𝐷)       ((𝐺𝑉𝑁𝑊) → 𝐽 = (TopOpen‘𝑇))

Theoremtngnm 22436 The topology generated by a normed structure. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐴 ∈ V       ((𝐺 ∈ Grp ∧ 𝑁:𝑋𝐴) → 𝑁 = (norm‘𝑇))

Theoremtngngp2 22437 A norm turns a group into a normed group iff the generated metric is in fact a metric. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐷 ∈ (Met‘𝑋))))

Theoremtngngpd 22438* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ Grp)    &   (𝜑𝑁:𝑋⟶ℝ)    &   ((𝜑𝑥𝑋) → ((𝑁𝑥) = 0 ↔ 𝑥 = 0 ))    &   ((𝜑 ∧ (𝑥𝑋𝑦𝑋)) → (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦)))       (𝜑𝑇 ∈ NrmGrp)

Theoremtngngp 22439* Derive the axioms for a normed group from the axioms for a metric space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &    0 = (0g𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremtnggrpr 22440 If a structure equipped with a norm is a normed group, the structure itself must be a group. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)       ((𝑁𝑉𝑇 ∈ NrmGrp) → 𝐺 ∈ Grp)

Theoremtngngp3 22441* Alternate definition of a normed group (i.e. a group equipped with a norm) without using the properties of a metric space. This corresponds to the definition in N. H. Bingham, A. J. Ostaszewski: "Normed versus topological groups: dichotomy and duality", 2010, Dissertationes Mathematicae 472, pp. 1-138 and E. Deza, M.M. Deza: "Dictionary of Distances", Elsevier, 2006. (Contributed by AV, 16-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    + = (+g𝐺)    &   𝐼 = (invg𝐺)       (𝑁:𝑋⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ ∀𝑥𝑋 (((𝑁𝑥) = 0 ↔ 𝑥 = 0 ) ∧ (𝑁‘(𝐼𝑥)) = (𝑁𝑥) ∧ ∀𝑦𝑋 (𝑁‘(𝑥 + 𝑦)) ≤ ((𝑁𝑥) + (𝑁𝑦))))))

Theoremnrmtngdist 22442 The augmentation of a normed group by its own norm has the same distance function as the normed group (restricted to the base set). (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ NrmGrp → (dist‘𝑇) = ((dist‘𝐺) ↾ (𝑋 × 𝑋)))

Theoremnrmtngnrm 22443 The augmentation of a normed group by its own norm is a normed group with the same norm. (Contributed by AV, 15-Oct-2021.)
𝑇 = (𝐺 toNrmGrp (norm‘𝐺))       (𝐺 ∈ NrmGrp → (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) = (norm‘𝐺)))

Theoremtngngpim 22444 The induced metric of a normed group is a function. (Contributed by AV, 19-Oct-2021.)
𝑇 = (𝐺 toNrmGrp 𝑁)    &   𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &   𝐷 = (dist‘𝑇)       (𝐺 ∈ NrmGrp → 𝐷:(𝑋 × 𝑋)⟶ℝ)

Theoremisnrg 22445 A normed ring is a ring with a norm that makes it into a normed group, and such that the norm is an absolute value on the ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing ↔ (𝑅 ∈ NrmGrp ∧ 𝑁𝐴))

Theoremnrgabv 22446 The norm of a normed ring is an absolute value. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝐴 = (AbsVal‘𝑅)       (𝑅 ∈ NrmRing → 𝑁𝐴)

Theoremnrgngp 22447 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Theoremnrgring 22448 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ Ring)

Theoremnmmul 22449 The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝐴𝑋𝐵𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁𝐴) · (𝑁𝐵)))

Theoremnrgdsdi 22450 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝑁𝐴) · (𝐵𝐷𝐶)) = ((𝐴 · 𝐵)𝐷(𝐴 · 𝐶)))

Theoremnrgdsdir 22451 Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &    · = (.r𝑅)    &   𝐷 = (dist‘𝑅)       ((𝑅 ∈ NrmRing ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴𝐷𝐵) · (𝑁𝐶)) = ((𝐴 · 𝐶)𝐷(𝐵 · 𝐶)))

Theoremnm1 22452 The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &    1 = (1r𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁1 ) = 1)

Theoremunitnmn0 22453 The norm of a unit is nonzero in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴𝑈) → (𝑁𝐴) ≠ 0)

Theoremnminvr 22454 The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)       ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴𝑈) → (𝑁‘(𝐼𝐴)) = (1 / (𝑁𝐴)))

Theoremnmdvr 22455 The norm of a division in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑁 = (norm‘𝑅)    &   𝑈 = (Unit‘𝑅)    &    / = (/r𝑅)       (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴𝑋𝐵𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁𝐴) / (𝑁𝐵)))

Theoremnrgdomn 22456 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing))

Theoremnrgtgp 22457 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)

Theoremsubrgnrg 22458 A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐻 = (𝐺s 𝐴)       ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing)

Theoremtngnrg 22459 Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑇 = (𝑅 toNrmGrp 𝐹)    &   𝐴 = (AbsVal‘𝑅)       (𝐹𝐴𝑇 ∈ NrmRing)

Theoremisnlm 22460* 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.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥𝐾𝑦𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴𝑥) · (𝑁𝑦))))

Theoremnmvs 22461 Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &   𝑁 = (norm‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ 𝑋𝐾𝑌𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴𝑋) · (𝑁𝑌)))

Theoremnlmngp 22462 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Theoremnlmlmod 22463 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ LMod)

Theoremnlmnrg 22464 The scalar component of a left module is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ NrmMod → 𝐹 ∈ NrmRing)

Theoremnlmngp2 22465 The scalar component of a left module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ NrmMod → 𝐹 ∈ NrmGrp)

Theoremnlmdsdi 22466 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐴 = (norm‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝑉𝑍𝑉)) → ((𝐴𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)))

Theoremnlmdsdir 22467 Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝑁 = (norm‘𝑊)    &   𝐸 = (dist‘𝐹)       ((𝑊 ∈ NrmMod ∧ (𝑋𝐾𝑌𝐾𝑍𝑉)) → ((𝑋𝐸𝑌) · (𝑁𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)))

Theoremnlmmul0or 22468 If a scalar product is zero, one of its factors must be zero. (Contributed by NM, 6-Dec-2007.) (Revised by Mario Carneiro, 4-Oct-2015.)
𝑉 = (Base‘𝑊)    &    · = ( ·𝑠𝑊)    &    0 = (0g𝑊)    &   𝐹 = (Scalar‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝑂 = (0g𝐹)       ((𝑊 ∈ NrmMod ∧ 𝐴𝐾𝐵𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂𝐵 = 0 )))

Theoremsranlm 22469 The subring algebra over a normed ring is a normed left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐴 = ((subringAlg ‘𝑊)‘𝑆)       ((𝑊 ∈ NrmRing ∧ 𝑆 ∈ (SubRing‘𝑊)) → 𝐴 ∈ NrmMod)

Theoremnlmvscnlem2 22470 Lemma for nlmvscn 22472. Compare this proof with the similar elementary proof mulcn2 14307 for continuity of multiplication on . (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (dist‘𝐹)    &   𝑁 = (norm‘𝑊)    &   𝐴 = (norm‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)    &   (𝜑𝐶𝐾)    &   (𝜑𝑌𝑉)    &   (𝜑 → (𝐵𝐸𝐶) < 𝑈)    &   (𝜑 → (𝑋𝐷𝑌) < 𝑇)       (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)

Theoremnlmvscnlem1 22471* Lemma for nlmvscn 22472. (Contributed by Mario Carneiro, 5-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &   𝑉 = (Base‘𝑊)    &   𝐾 = (Base‘𝐹)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (dist‘𝐹)    &   𝑁 = (norm‘𝑊)    &   𝐴 = (norm‘𝐹)    &    · = ( ·𝑠𝑊)    &   𝑇 = ((𝑅 / 2) / ((𝐴𝐵) + 1))    &   𝑈 = ((𝑅 / 2) / ((𝑁𝑋) + 𝑇))    &   (𝜑𝑊 ∈ NrmMod)    &   (𝜑𝑅 ∈ ℝ+)    &   (𝜑𝐵𝐾)    &   (𝜑𝑋𝑉)       (𝜑 → ∃𝑟 ∈ ℝ+𝑥𝐾𝑦𝑉 (((𝐵𝐸𝑥) < 𝑟 ∧ (𝑋𝐷𝑦) < 𝑟) → ((𝐵 · 𝑋)𝐷(𝑥 · 𝑦)) < 𝑅))

Theoremnlmvscn 22472 The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 22475 and nlmtlm 22479. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)    &    · = ( ·sf𝑊)    &   𝐽 = (TopOpen‘𝑊)    &   𝐾 = (TopOpen‘𝐹)       (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))

Theoremrlmnlm 22473 The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)

Theoremrlmnm 22474 The norm function in the ring module. (Contributed by AV, 9-Oct-2021.)
(norm‘𝑅) = (norm‘(ringLMod‘𝑅))

Theoremnrgtrg 22475 A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)

Theoremnrginvrcnlem 22476* Lemma for nrginvrcn 22477. Compare this proof with reccn2 14308, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝑁 = (norm‘𝑅)    &   𝐷 = (dist‘𝑅)    &   (𝜑𝑅 ∈ NrmRing)    &   (𝜑𝑅 ∈ NzRing)    &   (𝜑𝐴𝑈)    &   (𝜑𝐵 ∈ ℝ+)    &   𝑇 = (if(1 ≤ ((𝑁𝐴) · 𝐵), 1, ((𝑁𝐴) · 𝐵)) · ((𝑁𝐴) / 2))       (𝜑 → ∃𝑥 ∈ ℝ+𝑦𝑈 ((𝐴𝐷𝑦) < 𝑥 → ((𝐼𝐴)𝐷(𝐼𝑦)) < 𝐵))

Theoremnrginvrcn 22477 The ring inverse function is continuous in a normed ring. (Note that this is true even in rings which are not division rings.) (Contributed by Mario Carneiro, 6-Oct-2015.)
𝑋 = (Base‘𝑅)    &   𝑈 = (Unit‘𝑅)    &   𝐼 = (invr𝑅)    &   𝐽 = (TopOpen‘𝑅)       (𝑅 ∈ NrmRing → 𝐼 ∈ ((𝐽t 𝑈) Cn (𝐽t 𝑈)))

Theoremnrgtdrg 22478 A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)

Theoremnlmtlm 22479 A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)

Theoremisnvc 22480 A normed vector space is just a normed module which is algebraically a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝑊 ∈ LVec))

Theoremnvcnlm 22481 A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)

Theoremnvclvec 22482 A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LVec)

Theoremnvclmod 22483 A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ LMod)

Theoremisnvc2 22484 A normed vector space is just a normed module whose scalar ring is a division ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ NrmVec ↔ (𝑊 ∈ NrmMod ∧ 𝐹 ∈ DivRing))

Theoremnvctvc 22485 A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)

Theoremlssnlm 22486 A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ NrmMod ∧ 𝑈𝑆) → 𝑋 ∈ NrmMod)

Theoremlssnvc 22487 A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (𝑊s 𝑈)    &   𝑆 = (LSubSp‘𝑊)       ((𝑊 ∈ NrmVec ∧ 𝑈𝑆) → 𝑋 ∈ NrmVec)

Theoremrlmnvc 22488 The ring module over a normed division ring is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → (ringLMod‘𝑅) ∈ NrmVec)

Theoremngpocelbl 22489 Membership of an off-center vector in a ball in a normed module. (Contributed by NM, 27-Dec-2007.) (Revised by AV, 14-Oct-2021.)
𝑁 = (norm‘𝐺)    &   𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))       ((𝐺 ∈ NrmMod ∧ 𝑅 ∈ ℝ* ∧ (𝑃𝑋𝐴𝑋)) → ((𝑃 + 𝐴) ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑁𝐴) < 𝑅))

12.4.9  Normed space homomorphisms (bounded linear operators)

Syntaxcnmo 22490 The operator norm function.
class normOp

Syntaxcnghm 22491 The class of normed group homomorphisms.
class NGHom

Syntaxcnmhm 22492 The class of normed module homomorphisms.
class NMHom

Definitiondf-nmo 22493* Define the norm of an operator between two normed groups (usually vector spaces). This definition produces an operator norm function for each pair of groups 𝑠, 𝑡. Equivalent to the definition of linear operator norm in [AkhiezerGlazman] p. 39. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 25-Sep-2020.)
normOp = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ (𝑓 ∈ (𝑠 GrpHom 𝑡) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ (Base‘𝑠)((norm‘𝑡)‘(𝑓𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))

Definitiondf-nghm 22494* Define the set of normed group homomorphisms between two normed groups. A normed group homomorphism is a group homomorphism which additionally bounds the increase of norm by a fixed real operator. In vector spaces these are also known as bounded linear operators. (Contributed by Mario Carneiro, 18-Oct-2015.)
NGHom = (𝑠 ∈ NrmGrp, 𝑡 ∈ NrmGrp ↦ ((𝑠 normOp 𝑡) “ ℝ))

Definitiondf-nmhm 22495* Define a normed module homomorphism, also known as a bounded linear operator. This is a module homomorphism (a linear function) such that the operator norm is finite, or equivalently there is a constant 𝑐 such that... (Contributed by Mario Carneiro, 18-Oct-2015.)
NMHom = (𝑠 ∈ NrmMod, 𝑡 ∈ NrmMod ↦ ((𝑠 LMHom 𝑡) ∩ (𝑠 NGHom 𝑡)))

Theoremnmoffn 22496 The function producing operator norm functions is a function on normed groups. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.)
normOp Fn (NrmGrp × NrmGrp)

Theoremreldmnghm 22497 Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NGHom

Theoremreldmnmhm 22498 Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
Rel dom NMHom

Theoremnmofval 22499* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 26-Sep-2020.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁 = (𝑓 ∈ (𝑆 GrpHom 𝑇) ↦ inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝑓𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < )))

Theoremnmoval 22500* Value of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Revised by AV, 26-Sep-2020.)
𝑁 = (𝑆 normOp 𝑇)    &   𝑉 = (Base‘𝑆)    &   𝐿 = (norm‘𝑆)    &   𝑀 = (norm‘𝑇)       ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁𝐹) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥𝑉 (𝑀‘(𝐹𝑥)) ≤ (𝑟 · (𝐿𝑥))}, ℝ*, < ))

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330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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