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	rlmnlm 24601	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 24602	The norm function in the ring module. (Contributed by AV, 9-Oct-2021.)
⊢ (norm‘𝑅) = (norm‘(ringLMod‘𝑅))
Theorem	nrgtrg 24603	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 24604*	Lemma for nrginvrcn 24605. Compare this proof with reccn2 15501, 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 24605	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 24606	A normed division ring is a topological division ring. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ ((𝑅 ∈ NrmRing ∧ 𝑅 ∈ DivRing) → 𝑅 ∈ TopDRing)
Theorem	nlmtlm 24607	A normed module is a topological module. (Contributed by Mario Carneiro, 6-Oct-2015.)
⊢ (𝑊 ∈ NrmMod → 𝑊 ∈ TopMod)
Theorem	isnvc 24608	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 24609	A normed vector space is a normed module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ NrmMod)
Theorem	nvclvec 24610	A normed vector space is a left vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LVec)
Theorem	nvclmod 24611	A normed vector space is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ LMod)
Theorem	isnvc2 24612	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 24613	A normed vector space is a topological vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ (𝑊 ∈ NrmVec → 𝑊 ∈ TopVec)
Theorem	lssnlm 24614	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 24615	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 24616	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 24617	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 24618	The operator norm function.
class normOp
Syntax	cnghm 24619	The class of normed group homomorphisms.
class NGHom
Syntax	cnmhm 24620	The class of normed module homomorphisms.
class NMHom
Definition	df-nmo 24621*	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 24622*	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 24623*	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 24624	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 24625	Lemma for normed group homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NGHom
Theorem	reldmnmhm 24626	Lemma for module homomorphisms. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ Rel dom NMHom
Theorem	nmofval 24627*	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 24628*	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 24629*	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 24630*	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 24631*	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 24632	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*)
Theorem	nmocl 24633	The operator norm of an operator is an extended real. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝑁‘𝐹) ∈ ℝ*)
Theorem	nmoge0 24634	The operator norm of an operator is nonnegative. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → 0 ≤ (𝑁‘𝐹))
Theorem	nghmfval 24635	A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (𝑆 NGHom 𝑇) = (◡𝑁 “ ℝ)
Theorem	isnghm 24636	A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) ∧ (𝐹 ∈ (𝑆 GrpHom 𝑇) ∧ (𝑁‘𝐹) ∈ ℝ)))
Theorem	isnghm2 24637	A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ))
Theorem	isnghm3 24638	A normed group homomorphism is a group homomorphism with bounded norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → (𝐹 ∈ (𝑆 NGHom 𝑇) ↔ (𝑁‘𝐹) < +∞))
Theorem	bddnghm 24639	A bounded group homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝐴 ∈ ℝ ∧ (𝑁‘𝐹) ≤ 𝐴)) → 𝐹 ∈ (𝑆 NGHom 𝑇))
Theorem	nghmcl 24640	A normed group homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
Theorem	nmoi 24641	The operator norm achieves the minimum of the set of upper bounds, if the operator is bounded. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) · (𝐿‘𝑋)))
Theorem	nmoix 24642	The operator norm is a bound on the size of an operator, even when it is infinite (using extended real multiplication). (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    ⇒   ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ 𝑋 ∈ 𝑉) → (𝑀‘(𝐹‘𝑋)) ≤ ((𝑁‘𝐹) ·e (𝐿‘𝑋)))
Theorem	nmoi2 24643	The operator norm is a bound on the growth of a vector under the action of the operator. (Contributed by Mario Carneiro, 19-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐿 = (norm‘𝑆)    &   ⊢ 𝑀 = (norm‘𝑇)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ (((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) ∧ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ 0 )) → ((𝑀‘(𝐹‘𝑋)) / (𝐿‘𝑋)) ≤ (𝑁‘𝐹))
Theorem	nmoleub 24644*	The operator norm, defined as an infimum of upper bounds, can also be defined as a supremum of norms of 𝐹(𝑥) away from zero. (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 → ((𝑁‘𝐹) ≤ 𝐴 ↔ ∀𝑥 ∈ 𝑉 (𝑥 ≠ 0 → ((𝑀‘(𝐹‘𝑥)) / (𝐿‘𝑥)) ≤ 𝐴)))
Theorem	nghmrcl1 24645	Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑆 ∈ NrmGrp)
Theorem	nghmrcl2 24646	Reverse closure for a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝑇 ∈ NrmGrp)
Theorem	nghmghm 24647	A normed group homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	nmo0 24648	The operator norm of the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑁‘(𝑉 × { 0 })) = 0)
Theorem	nmoeq0 24649	The operator norm is zero only for the zero operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp ∧ 𝐹 ∈ (𝑆 GrpHom 𝑇)) → ((𝑁‘𝐹) = 0 ↔ 𝐹 = (𝑉 × { 0 })))
Theorem	nmoco 24650	An upper bound on the operator norm of a composition. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑈)    &   ⊢ 𝐿 = (𝑇 normOp 𝑈)    &   ⊢ 𝑀 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘ 𝐺)) ≤ ((𝐿‘𝐹) · (𝑀‘𝐺)))
Theorem	nghmco 24651	The composition of normed group homomorphisms is a normed group homomorphism. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ ((𝐹 ∈ (𝑇 NGHom 𝑈) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NGHom 𝑈))
Theorem	nmotri 24652	Triangle inequality for the operator norm. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝑁‘(𝐹 ∘f + 𝐺)) ≤ ((𝑁‘𝐹) + (𝑁‘𝐺)))
Theorem	nghmplusg 24653	The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝑇 ∈ Abel ∧ 𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐺 ∈ (𝑆 NGHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NGHom 𝑇))
Theorem	0nghm 24654	The zero operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ 𝑇 ∈ NrmGrp) → (𝑉 × { 0 }) ∈ (𝑆 NGHom 𝑇))
Theorem	nmoid 24655	The operator norm of the identity function on a nontrivial group. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑆)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑆)    ⇒   ⊢ ((𝑆 ∈ NrmGrp ∧ { 0 } ⊊ 𝑉) → (𝑁‘( I ↾ 𝑉)) = 1)
Theorem	idnghm 24656	The identity operator is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    ⇒   ⊢ (𝑆 ∈ NrmGrp → ( I ↾ 𝑉) ∈ (𝑆 NGHom 𝑆))
Theorem	nmods 24657	Upper bound for the distance between the values of a bounded linear operator. (Contributed by Mario Carneiro, 22-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    &   ⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 𝐶 = (dist‘𝑆)    &   ⊢ 𝐷 = (dist‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 NGHom 𝑇) ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ((𝐹‘𝐴)𝐷(𝐹‘𝐵)) ≤ ((𝑁‘𝐹) · (𝐴𝐶𝐵)))
Theorem	nghmcn 24658	A normed group homomorphism is a continuous function. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝐽 = (TopOpen‘𝑆)    &   ⊢ 𝐾 = (TopOpen‘𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NGHom 𝑇) → 𝐹 ∈ (𝐽 Cn 𝐾))
Theorem	isnmhm 24659	A normed module homomorphism is a left module homomorphism which is also a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod) ∧ (𝐹 ∈ (𝑆 LMHom 𝑇) ∧ 𝐹 ∈ (𝑆 NGHom 𝑇))))
Theorem	nmhmrcl1 24660	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑆 ∈ NrmMod)
Theorem	nmhmrcl2 24661	Reverse closure for a normed module homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝑇 ∈ NrmMod)
Theorem	nmhmlmhm 24662	A normed module homomorphism is a left module homomorphism (a linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 LMHom 𝑇))
Theorem	nmhmnghm 24663	A normed module homomorphism is a normed group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 NGHom 𝑇))
Theorem	nmhmghm 24664	A normed module homomorphism is a group homomorphism. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → 𝐹 ∈ (𝑆 GrpHom 𝑇))
Theorem	isnmhm2 24665	A normed module homomorphism is a left module homomorphism with bounded norm (a bounded linear operator). (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 ∈ (𝑆 LMHom 𝑇)) → (𝐹 ∈ (𝑆 NMHom 𝑇) ↔ (𝑁‘𝐹) ∈ ℝ))
Theorem	nmhmcl 24666	A normed module homomorphism has a real operator norm. (Contributed by Mario Carneiro, 18-Oct-2015.)
⊢ 𝑁 = (𝑆 normOp 𝑇)    ⇒   ⊢ (𝐹 ∈ (𝑆 NMHom 𝑇) → (𝑁‘𝐹) ∈ ℝ)
Theorem	idnmhm 24667	The identity operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    ⇒   ⊢ (𝑆 ∈ NrmMod → ( I ↾ 𝑉) ∈ (𝑆 NMHom 𝑆))
Theorem	0nmhm 24668	The zero operator is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ 𝑉 = (Base‘𝑆)    &   ⊢ 0 = (0g‘𝑇)    &   ⊢ 𝐹 = (Scalar‘𝑆)    &   ⊢ 𝐺 = (Scalar‘𝑇)    ⇒   ⊢ ((𝑆 ∈ NrmMod ∧ 𝑇 ∈ NrmMod ∧ 𝐹 = 𝐺) → (𝑉 × { 0 }) ∈ (𝑆 NMHom 𝑇))
Theorem	nmhmco 24669	The composition of bounded linear operators is a bounded linear operator. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ ((𝐹 ∈ (𝑇 NMHom 𝑈) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘ 𝐺) ∈ (𝑆 NMHom 𝑈))
Theorem	nmhmplusg 24670	The sum of two bounded linear operators is bounded linear. (Contributed by Mario Carneiro, 20-Oct-2015.)
⊢ + = (+g‘𝑇)    ⇒   ⊢ ((𝐹 ∈ (𝑆 NMHom 𝑇) ∧ 𝐺 ∈ (𝑆 NMHom 𝑇)) → (𝐹 ∘f + 𝐺) ∈ (𝑆 NMHom 𝑇))
12.4.10  Topology on the reals
Theorem	qtopbaslem 24671	The set of open intervals with endpoints in a subset forms a basis for a topology. (Contributed by Mario Carneiro, 17-Jun-2014.)
⊢ 𝑆 ⊆ ℝ*    ⇒   ⊢ ((,) “ (𝑆 × 𝑆)) ∈ TopBases
Theorem	qtopbas 24672	The set of open intervals with rational endpoints forms a basis for a topology. (Contributed by NM, 8-Mar-2007.)
⊢ ((,) “ (ℚ × ℚ)) ∈ TopBases
Theorem	retopbas 24673	A basis for the standard topology on the reals. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 17-Jun-2014.)
⊢ ran (,) ∈ TopBases
Theorem	retop 24674	The standard topology on the reals. (Contributed by FL, 4-Jun-2007.)
⊢ (topGen‘ran (,)) ∈ Top
Theorem	uniretop 24675	The underlying set of the standard topology on the reals is the reals. (Contributed by FL, 4-Jun-2007.)
⊢ ℝ = ∪ (topGen‘ran (,))
Theorem	retopon 24676	The standard topology on the reals is a topology on the reals. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (topGen‘ran (,)) ∈ (TopOn‘ℝ)
Theorem	retps 24677	The standard topological space on the reals. (Contributed by NM, 19-Oct-2012.)
⊢ 𝐾 = {⟨(Base‘ndx), ℝ⟩, ⟨(TopSet‘ndx), (topGen‘ran (,))⟩}    ⇒   ⊢ 𝐾 ∈ TopSp
Theorem	iooretop 24678	Open intervals are open sets of the standard topology on the reals . (Contributed by FL, 18-Jun-2007.)
⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
Theorem	icccld 24679	Closed intervals are closed sets of the standard topology on ℝ. (Contributed by FL, 14-Sep-2007.)
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	icopnfcld 24680	Right-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ (𝐴 ∈ ℝ → (𝐴[,)+∞) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	iocmnfcld 24681	Left-unbounded closed intervals are closed sets of the standard topology on ℝ. (Contributed by Mario Carneiro, 17-Feb-2015.)
⊢ (𝐴 ∈ ℝ → (-∞(,]𝐴) ∈ (Clsd‘(topGen‘ran (,))))
Theorem	qdensere 24682	ℚ is dense in the standard topology on ℝ. (Contributed by NM, 1-Mar-2007.)
⊢ ((cls‘(topGen‘ran (,)))‘ℚ) = ℝ
Theorem	cnmetdval 24683	Value of the distance function of the metric space of complex numbers. (Contributed by NM, 9-Dec-2006.) (Revised by Mario Carneiro, 27-Dec-2014.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴𝐷𝐵) = (abs‘(𝐴 − 𝐵)))
Theorem	cnmet 24684	The absolute value metric determines a metric space on the complex numbers. This theorem provides a link between complex numbers and metrics spaces, making metric space theorems available for use with complex numbers. (Contributed by FL, 9-Oct-2006.)
⊢ (abs ∘ − ) ∈ (Met‘ℂ)
Theorem	cnxmet 24685	The absolute value metric is an extended metric. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ (abs ∘ − ) ∈ (∞Met‘ℂ)
Theorem	cnbl0 24686	Two ways to write the open ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,)𝑅)) = (0(ball‘𝐷)𝑅))
Theorem	cnblcld 24687*	Two ways to write the closed ball centered at zero. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐷 = (abs ∘ − )    ⇒   ⊢ (𝑅 ∈ ℝ* → (◡abs “ (0[,]𝑅)) = {𝑥 ∈ ℂ ∣ (0𝐷𝑥) ≤ 𝑅})
Theorem	cnfldms 24688	The complex number field is a metric space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ MetSp
Theorem	cnfldxms 24689	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ ∞MetSp
Theorem	cnfldtps 24690	The complex number field is a topological space. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ ℂfld ∈ TopSp
Theorem	cnfldnm 24691	The norm of the field of complex numbers. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ abs = (norm‘ℂfld)
Theorem	cnngp 24692	The complex numbers form a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmGrp
Theorem	cnnrg 24693	The complex numbers form a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
⊢ ℂfld ∈ NrmRing
Theorem	cnfldtopn 24694	The topology of the complex numbers. (Contributed by Mario Carneiro, 28-Aug-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 = (MetOpen‘(abs ∘ − ))
Theorem	cnfldtopon 24695	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ (TopOn‘ℂ)
Theorem	cnfldtop 24696	The topology of the complex numbers is a topology. (Contributed by Mario Carneiro, 2-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Top
Theorem	cnfldhaus 24697	The topology of the complex numbers is Hausdorff. (Contributed by Mario Carneiro, 8-Sep-2015.)
⊢ 𝐽 = (TopOpen‘ℂfld)    ⇒   ⊢ 𝐽 ∈ Haus
Theorem	unicntop 24698	The underlying set of the standard topology on the complex numbers is the set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ = ∪ (TopOpen‘ℂfld)
Theorem	cnopn 24699	The set of complex numbers is open with respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
⊢ ℂ ∈ (TopOpen‘ℂfld)
Theorem	cnn0opn 24700	The set of nonzero complex numbers is open with respect to the standard topology on complex numbers. (Contributed by SN, 7-Oct-2025.)
⊢ (ℂ ∖ {0}) ∈ (TopOpen‘ℂfld)