Theorems · Definition · functional analysis

NormedGroup.ofMulDist

{E : Type u_5} →
  [inst : Norm E] →
    [inst_1 : Group E] →
      [inst_2 : MetricSpace E] →
        (∀ (x : E), ‖x‖ = dist 1 x) → (∀ (x y z : E), dist x y ≤ dist (z * x) (z * y)) → NormedGroup E

Construct a normed group from a multiplication-invariant distance.

Defined in: Mathlib.Analysis.Normed.Group.Defs
Cited by: 0 results in Mathlib
Foundations: Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Norm Group MetricSpace

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites10

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement · cited by 25,697
Groupstatement and proof · cited by 6,238
Norm.normstatement and proof · cited by 5,413
MetricSpacestatement and proof · cited by 1,684
Dist.diststatement and proof · cited by 1,539
Normstatement and proof · cited by 512
SeminormedGroupproof · cited by 250
NormedGroupstatement · cited by 18
SeminormedGroup.dist_eqproof · cited by 2
SeminormedGroup.ofMulDistproof · cited by 0

Cited by1

Results whose statement or proof uses this declaration.

NormedCommGroup.ofMulDistproof · cited by 0