P. 250 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 24901-25000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	nmrpcl 24901	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 24902	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 24903	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 24904	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 24905	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 24906	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 24907	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 24908	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 24909*	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 24910	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 24911	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 24912	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 24913	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 24914	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 24915	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 24916	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 24917	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 24918*	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 24919	The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
⊢ Rel dom toNrmGrp
Theorem	tngval 24920	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 24921	Lemma for tngbas 24922 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	tngbas 24922	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	tngplusg 24923	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	tng0 24924	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 24925	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	tngsca 24926	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	tngvsca 24927	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	tngip 24928	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	tngds 24929	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	tngtset 24930	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 24931	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 24932	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 24933	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 24934*	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 24935*	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 24936	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 24937*	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 24938	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 24939	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 24940	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 24941	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 24942	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 24943	A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)
Theorem	nrgring 24944	A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ Ring)
Theorem	nmmul 24945	The norm of a product in a normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ · = (.r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NrmRing ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝑁‘(𝐴 · 𝐵)) = ((𝑁‘𝐴) · (𝑁‘𝐵)))
Theorem	nrgdsdi 24946	Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐷 = (dist‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NrmRing ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝑁‘𝐴) · (𝐵𝐷𝐶)) = ((𝐴 · 𝐵)𝐷(𝐴 · 𝐶)))
Theorem	nrgdsdir 24947	Distribute a distance calculation. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ · = (.r‘𝑅)    &   ⊢ 𝐷 = (dist‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NrmRing ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋 ∧ 𝐶 ∈ 𝑋)) → ((𝐴𝐷𝐵) · (𝑁‘𝐶)) = ((𝐴 · 𝐶)𝐷(𝐵 · 𝐶)))
Theorem	nm1 24948	The norm of one in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 1 = (1r‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) → (𝑁‘ 1 ) = 1)
Theorem	unitnmn0 24949	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 ∧ 𝐴 ∈ 𝑈) → (𝑁‘𝐴) ≠ 0)
Theorem	nminvr 24950	The norm of an inverse in a nonzero normed ring. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    ⇒   ⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing ∧ 𝐴 ∈ 𝑈) → (𝑁‘(𝐼‘𝐴)) = (1 / (𝑁‘𝐴)))
Theorem	nmdvr 24951	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)    &   ⊢ / = (/r‘𝑅)    ⇒   ⊢ (((𝑅 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑈)) → (𝑁‘(𝐴 / 𝐵)) = ((𝑁‘𝐴) / (𝑁‘𝐵)))
Theorem	nrgdomn 24952	A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing))
Theorem	nrgtgp 24953	A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)
Theorem	subrgnrg 24954	A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝐻 = (𝐺 ↾s 𝐴)    ⇒   ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing)
Theorem	tngnrg 24955	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)    ⇒   ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing)
Theorem	isnlm 24956*	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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐴 = (norm‘𝐹)    ⇒   ⊢ (𝑊 ∈ NrmMod ↔ ((𝑊 ∈ NrmGrp ∧ 𝑊 ∈ LMod ∧ 𝐹 ∈ NrmRing) ∧ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (𝑁‘(𝑥 · 𝑦)) = ((𝐴‘𝑥) · (𝑁‘𝑦))))
Theorem	nmvs 24957	Defining property of a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐴 = (norm‘𝐹)    ⇒   ⊢ ((𝑊 ∈ NrmMod ∧ 𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉) → (𝑁‘(𝑋 · 𝑌)) = ((𝐴‘𝑋) · (𝑁‘𝑌)))
Theorem	nlmngp 24958	A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)
Theorem	nlmlmod 24959	A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ LMod)
Theorem	nlmnrg 24960	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)
Theorem	nlmngp2 24961	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)
Theorem	nlmdsdi 24962	Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝐴 = (norm‘𝐹)    ⇒   ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝑉 ∧ 𝑍 ∈ 𝑉)) → ((𝐴‘𝑋) · (𝑌𝐷𝑍)) = ((𝑋 · 𝑌)𝐷(𝑋 · 𝑍)))
Theorem	nlmdsdir 24963	Distribute a distance calculation. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑉 = (Base‘𝑊)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝐷 = (dist‘𝑊)    &   ⊢ 𝑁 = (norm‘𝑊)    &   ⊢ 𝐸 = (dist‘𝐹)    ⇒   ⊢ ((𝑊 ∈ NrmMod ∧ (𝑋 ∈ 𝐾 ∧ 𝑌 ∈ 𝐾 ∧ 𝑍 ∈ 𝑉)) → ((𝑋𝐸𝑌) · (𝑁‘𝑍)) = ((𝑋 · 𝑍)𝐷(𝑌 · 𝑍)))
Theorem	nlmmul0or 24964	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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 0 = (0g‘𝑊)    &   ⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ 𝐾 = (Base‘𝐹)    &   ⊢ 𝑂 = (0g‘𝐹)    ⇒   ⊢ ((𝑊 ∈ NrmMod ∧ 𝐴 ∈ 𝐾 ∧ 𝐵 ∈ 𝑉) → ((𝐴 · 𝐵) = 0 ↔ (𝐴 = 𝑂 ∨ 𝐵 = 0 )))
Theorem	sranlm 24965	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)
Theorem	nlmvscnlem2 24966	Lemma for nlmvscn 24968. Compare this proof with the similar elementary proof mulcn2 15731 for continuity of multiplication on ℂ. (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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1))    &   ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))    &   ⊢ (𝜑 → 𝑊 ∈ NrmMod)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    &   ⊢ (𝜑 → 𝐶 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑌 ∈ 𝑉)    &   ⊢ (𝜑 → (𝐵𝐸𝐶) < 𝑈)    &   ⊢ (𝜑 → (𝑋𝐷𝑌) < 𝑇)    ⇒   ⊢ (𝜑 → ((𝐵 · 𝑋)𝐷(𝐶 · 𝑌)) < 𝑅)
Theorem	nlmvscnlem1 24967*	Lemma for nlmvscn 24968. (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)    &   ⊢ · = ( ·𝑠 ‘𝑊)    &   ⊢ 𝑇 = ((𝑅 / 2) / ((𝐴‘𝐵) + 1))    &   ⊢ 𝑈 = ((𝑅 / 2) / ((𝑁‘𝑋) + 𝑇))    &   ⊢ (𝜑 → 𝑊 ∈ NrmMod)    &   ⊢ (𝜑 → 𝑅 ∈ ℝ+)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐾)    &   ⊢ (𝜑 → 𝑋 ∈ 𝑉)    ⇒   ⊢ (𝜑 → ∃𝑟 ∈ ℝ+ ∀𝑥 ∈ 𝐾 ∀𝑦 ∈ 𝑉 (((𝐵𝐸𝑥) < 𝑟 ∧ (𝑋𝐷𝑦) < 𝑟) → ((𝐵 · 𝑋)𝐷(𝑥 · 𝑦)) < 𝑅))
Theorem	nlmvscn 24968	The scalar multiplication of a normed module is continuous. Lemma for nrgtrg 24971 and nlmtlm 24975. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝐹 = (Scalar‘𝑊)    &   ⊢ · = ( ·sf ‘𝑊)    &   ⊢ 𝐽 = (TopOpen‘𝑊)    &   ⊢ 𝐾 = (TopOpen‘𝐹)    ⇒   ⊢ (𝑊 ∈ NrmMod → · ∈ ((𝐾 ×t 𝐽) Cn 𝐽))
Theorem	rlmnlm 24969	The ring module over a normed ring is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑅 ∈ NrmRing → (ringLMod‘𝑅) ∈ NrmMod)
Theorem	rlmnm 24970	The norm function in the ring module. (Contributed by AV, 9-Oct-2021.)
⊢ (norm‘𝑅) = (norm‘(ringLMod‘𝑅))
Theorem	nrgtrg 24971	A normed ring is a topological ring. (Contributed by Mario Carneiro, 4-Oct-2015.) (Proof shortened by AV, 31-Oct-2024.)
⊢ (𝑅 ∈ NrmRing → 𝑅 ∈ TopRing)
Theorem	nrginvrcnlem 24972*	Lemma for nrginvrcn 24973. Compare this proof with reccn2 15732, the elementary proof of continuity of division. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ 𝑋 = (Base‘𝑅)    &   ⊢ 𝑈 = (Unit‘𝑅)    &   ⊢ 𝐼 = (invr‘𝑅)    &   ⊢ 𝑁 = (norm‘𝑅)    &   ⊢ 𝐷 = (dist‘𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ NrmRing)    &   ⊢ (𝜑 → 𝑅 ∈ NzRing)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ+)    &   ⊢ 𝑇 = (if(1 ≤ ((𝑁‘𝐴) · 𝐵), 1, ((𝑁‘𝐴) · 𝐵)) · ((𝑁‘𝐴) / 2))    ⇒   ⊢ (𝜑 → ∃𝑥 ∈ ℝ+ ∀𝑦 ∈ 𝑈 ((𝐴𝐷𝑦) < 𝑥 → ((𝐼‘𝐴)𝐷(𝐼‘𝑦)) < 𝐵))
Theorem	nrginvrcn 24973	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 = (invr‘𝑅)    &   ⊢ 𝐽 = (TopOpen‘𝑅)    ⇒   ⊢ (𝑅 ∈ NrmRing → 𝐼 ∈ ((𝐽 ↾t 𝑈) Cn (𝐽 ↾t 𝑈)))
Theorem	nrgtdrg 24974	A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)
Theorem	nlmtlm 24975	A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)
Theorem	isnvc 24976	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))
Theorem	nvcnlm 24977	A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
Theorem	nvclvec 24978	A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)
Theorem	nvclmod 24979	A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
Theorem	isnvc2 24980	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))
Theorem	nvctvc 24981	A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)
Theorem	lssnlm 24982	A subspace of a normed module is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ NrmMod ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmMod)
Theorem	lssnvc 24983	A subspace of a normed vector space is a normed vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ 𝑋 = (𝑊 ↾s 𝑈)    &   ⊢ 𝑆 = (LSubSp‘𝑊)    ⇒   ⊢ ((𝑊 ∈ NrmVec ∧ 𝑈 ∈ 𝑆) → 𝑋 ∈ NrmVec)
Theorem	rlmnvc 24984	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)
Theorem	ngpocelbl 24985	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)    &   ⊢ + = (+g‘𝐺)    &   ⊢ 𝐷 = ((dist‘𝐺) ↾ (𝑋 × 𝑋))    ⇒   ⊢ ((𝐺 ∈ NrmMod ∧ 𝑅 ∈ ℝ* ∧ (𝑃 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋)) → ((𝑃 + 𝐴) ∈ (𝑃(ball‘𝐷)𝑅) ↔ (𝑁‘𝐴) < 𝑅))
12.4.9  Normed space homomorphisms (bounded linear operators)
Syntax	cnmo 24986	The operator norm function.
class normOp
Syntax	cnghm 24987	The class of normed group homomorphisms.
class NGHom
Syntax	cnmhm 24988	The class of normed module homomorphisms.
class NMHom
Definition	df-nmo 24989*	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)‘(𝑓‘𝑥)) ≤ (𝑟 · ((norm‘𝑠)‘𝑥))}, ℝ*, < )))
Definition	df-nghm 24990*	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))
Definition	df-nmhm 24991*	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)))
Theorem	nmoffn 24992	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)
Theorem	reldmnghm 24993	Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NGHom
Theorem	reldmnmhm 24994	Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NMHom
Theorem	nmofval 24995*	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[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝑓‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < )))
Theorem	nmoval 24996*	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)) → (𝑁‘𝐹) = inf({𝑟 ∈ (0[,)+∞) ∣ ∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥))}, ℝ*, < ))
Theorem	nmogelb 24997*	Property of the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened 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)) ∧ 𝐴 ∈ ℝ*) → (𝐴 ≤ (𝑁‘𝐹) ↔ ∀𝑟 ∈ (0[,)+∞)(∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝑟 · (𝐿‘𝑥)) → 𝐴 ≤ 𝑟)))
Theorem	nmolb 24998*	Any upper bound on the values of a linear operator translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened 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)) ∧ 𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (∀𝑥 ∈ 𝑉 (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)) → (𝑁‘𝐹) ≤ 𝐴))
Theorem	nmolb2d 24999*	Any upper bound on the values of a linear operator at nonzero vectors translates to an upper bound on the operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    &   ⊢ (𝜑 → 𝑆 ∈ NrmGrp)    &   ⊢ (𝜑 → 𝑇 ∈ NrmGrp)    &   ⊢ (𝜑 → 𝐹 ∈ (𝑆 GrpHom 𝑇))    &   ⊢ (𝜑 → 𝐴 ∈ ℝ)    &   ⊢ (𝜑 → 0 ≤ 𝐴)    &   ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑉 ∧ 𝑥 ≠ 0 )) → (𝑀‘(𝐹‘𝑥)) ≤ (𝐴 · (𝐿‘𝑥)))    ⇒   ⊢ (𝜑 → (𝑁‘𝐹) ≤ 𝐴)
Theorem	nmof 25000	The operator norm is a function into the extended reals. (Contributed by Mario Carneiro, 18-Oct-2015.) (Proof shortened by AV, 26-Sep-2020.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → 𝑁:(𝑆 GrpHom 𝑇)⟶ℝ*)