Theorems · Definition · functional analysis

SeminormedGroup.ofMulDist

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

Construct a seminormed group from a multiplication-invariant distance.

Defined in: Mathlib.Analysis.Normed.Group.Defs
Cited by: 0 results in Mathlib
Foundations: Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: Norm Group PseudoMetricSpace

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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
PseudoMetricSpacestatement and proof · cited by 1,550
Dist.diststatement and proof · cited by 1,539
Normstatement and proof · cited by 512
SeminormedGroupstatement · cited by 250

Cited by2

Results whose statement or proof uses this declaration.

NormedGroup.ofMulDistproof · cited by 0
SeminormedCommGroup.ofMulDistproof · cited by 0