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Theorem List for Metamath Proof Explorer - 22201-22300   *Has distinct variable group(s)
TypeLabelDescription
Statement

Definitiondf-nlm 22201* 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( ·𝑠𝑤)𝑦)) = (((norm‘𝑓)‘𝑥) · ((norm‘𝑤)‘𝑦)))}

Definitiondf-nvc 22202 A normed vector space is a normed module which is also a vector space. (Contributed by Mario Carneiro, 4-Oct-2015.)
NrmVec = (NrmMod ∩ LVec)

Theoremnmfval 22203* The value of the norm function. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       𝑁 = (𝑥𝑋 ↦ (𝑥𝐷 0 ))

Theoremnmval 22204 The value of the norm function. 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 = (0g𝑊)    &   𝐷 = (dist‘𝑊)       (𝐴𝑋 → (𝑁𝐴) = (𝐴𝐷 0 ))

Theoremnmfval2 22205* The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       (𝑊 ∈ Grp → 𝑁 = (𝑥𝑋 ↦ (𝑥𝐸 0 )))

Theoremnmval2 22206 The value of the norm function using a restricted metric. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝑊)    &   𝑋 = (Base‘𝑊)    &    0 = (0g𝑊)    &   𝐷 = (dist‘𝑊)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       ((𝑊 ∈ Grp ∧ 𝐴𝑋) → (𝑁𝐴) = (𝐴𝐸 0 ))

Theoremnmf2 22207 The norm 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)

Theoremnmpropd 22208 Weak property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑 → (Base‘𝐾) = (Base‘𝐿))    &   (𝜑 → (+g𝐾) = (+g𝐿))    &   (𝜑 → (dist‘𝐾) = (dist‘𝐿))       (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Theoremnmpropd2 22209* Strong property deduction for a norm. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝜑𝐵 = (Base‘𝐾))    &   (𝜑𝐵 = (Base‘𝐿))    &   (𝜑𝐾 ∈ Grp)    &   ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))    &   (𝜑 → ((dist‘𝐾) ↾ (𝐵 × 𝐵)) = ((dist‘𝐿) ↾ (𝐵 × 𝐵)))       (𝜑 → (norm‘𝐾) = (norm‘𝐿))

Theoremisngp 22210 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) ⊆ 𝐷))

Theoremisngp2 22211 The property of being a normed group. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)    &   𝑋 = (Base‘𝐺)    &   𝐸 = (𝐷 ↾ (𝑋 × 𝑋))       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ (𝑁 ) = 𝐸))

Theoremisngp3 22212* The property of being a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)    &   𝑋 = (Base‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥𝑋𝑦𝑋 (𝑥𝐷𝑦) = (𝑁‘(𝑥 𝑦))))

Theoremngpgrp 22213 A normed group is a group. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ Grp)

Theoremngpms 22214 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ MetSp)

Theoremngpxms 22215 A normed group is a metric space. (Contributed by Mario Carneiro, 2-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ ∞MetSp)

Theoremngptps 22216 A normed group is a topological space. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝐺 ∈ NrmGrp → 𝐺 ∈ TopSp)

Theoremngpmet 22217 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))

Theoremngpds 22218 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)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐴 𝐵)))

Theoremngpdsr 22219 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)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = (𝑁‘(𝐵 𝐴)))

Theoremngpds2 22220 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ((𝐴 𝐵)𝐷 0 ))

Theoremngpds2r 22221 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ((𝐵 𝐴)𝐷 0 ))

Theoremngpds3 22222 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐴 𝐵)))

Theoremngpds3r 22223 Write the distance between two points in terms of distance from zero. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    0 = (0g𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ 𝐴𝑋𝐵𝑋) → (𝐴𝐷𝐵) = ( 0 𝐷(𝐵 𝐴)))

Theoremngprcan 22224 Cancel right addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴 + 𝐶)𝐷(𝐵 + 𝐶)) = (𝐴𝐷𝐵))

Theoremngplcan 22225 Cancel left addition inside a distance calculation. (Contributed by Mario Carneiro, 2-Oct-2015.)
𝑋 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐶 + 𝐴)𝐷(𝐶 + 𝐵)) = (𝐴𝐷𝐵))

Theoremisngp4 22226* 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)    &    + = (+g𝐺)    &   𝐷 = (dist‘𝐺)       (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ∀𝑥𝑋𝑦𝑋𝑧𝑋 ((𝑥 + 𝑧)𝐷(𝑦 + 𝑧)) = (𝑥𝐷𝑦)))

Theoremngpinvds 22227 Two elements are the same distance apart as their inverses. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &   𝐼 = (invg𝐺)    &   𝐷 = (dist‘𝐺)       (((𝐺 ∈ NrmGrp ∧ 𝐺 ∈ Abel) ∧ (𝐴𝑋𝐵𝑋)) → ((𝐼𝐴)𝐷(𝐼𝐵)) = (𝐴𝐷𝐵))

Theoremngpsubcan 22228 Cancel right subtraction inside a distance calculation. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑋 = (Base‘𝐺)    &    = (-g𝐺)    &   𝐷 = (dist‘𝐺)       ((𝐺 ∈ NrmGrp ∧ (𝐴𝑋𝐵𝑋𝐶𝑋)) → ((𝐴 𝐶)𝐷(𝐵 𝐶)) = (𝐴𝐷𝐵))

Theoremnmf 22229 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)

Theoremnmcl 22230 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 22231 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 22232 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 22233 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 22234 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 22235 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 22236 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 22237 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 22238 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 22239 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 22240 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 22241 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 22242* 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 22243 Norm of the identity element. (Contributed by Mario Carneiro, 4-Oct-2015.)
𝑁 = (norm‘𝐺)    &    0 = (0g𝐺)       (𝐺 ∈ NrmGrp → (𝑁0 ) = 0)

Theoremnmgt0 22244 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 22245 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 22246 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 22247 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 22248 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 22249 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 22250 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 22251* 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 22252 The function toNrmGrp is a two-argument function. (Contributed by Mario Carneiro, 8-Oct-2015.)
Rel dom toNrmGrp

Theoremtngval 22253 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 22254 Lemma for tngbas 22255 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 22255 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 22256 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 22257 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 22258 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 22259 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 22260 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 22261 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 22262 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 22263 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 22264 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 22265 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 22266 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 22267* 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 22268* 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 22269 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 22270* 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 22271 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 22272 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 22273 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 22274 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 22275 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 22276 A normed ring is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ NrmGrp)

Theoremnrgring 22277 A normed ring is a ring. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ Ring)

Theoremnmmul 22278 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 22279 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 22280 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 22281 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 22282 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 22283 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 22284 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 22285 A nonzero normed ring is a domain. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑅 ∈ NrmRing → (𝑅 ∈ Domn ↔ 𝑅 ∈ NzRing))

Theoremnrgtgp 22286 A normed ring is a topological group. (Contributed by Mario Carneiro, 5-Oct-2015.)
(𝑅 ∈ NrmRing → 𝑅 ∈ TopGrp)

Theoremsubrgnrg 22287 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 22288 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 22289* 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 22290 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 22291 A normed module is a normed group. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ NrmGrp)

Theoremnlmlmod 22292 A normed module is a left module. (Contributed by Mario Carneiro, 4-Oct-2015.)
(𝑊 ∈ NrmMod → 𝑊 ∈ LMod)

Theoremnlmnrg 22293 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 22294 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 22295 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 22296 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 22297 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 22298 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 22299 Lemma for nlmvscn 22301. Compare this proof with the similar elementary proof mulcn2 14174 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 22300* Lemma for nlmvscn 22301. (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))

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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 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