Theorems · Theorem · functional analysis
SeminormedGroup.dist_eq
∀ {E : Type u_8} [self : SeminormedGroup E] (x y : E), dist x y = ‖x⁻¹ * y‖The distance function is induced by the norm.
- Defined in
- Mathlib.Analysis.Normed.Group.Defs
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses no axioms
- Assumes
- SeminormedGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement · cited by 25,697
- Norm.normstatement · cited by 5,413
- Dist.diststatement · cited by 1,539
- SeminormedGroupstatement and proof · cited by 250
Cited by11
Results whose statement or proof uses this declaration.
- dist_eq_norm_inv_mulproof · cited by 41
- GroupSeminormClass.toSeminormedCommGroupproof · cited by 1
- GroupNormClass.toNormedGroupproof · cited by 1
- NormedGroup.ofMulDistproof · cited by 0
- NormedGroup.ofMulDist'proof · cited by 0
- GroupNorm.toNormedGroupproof · cited by 0
- NormedGroup.ofSeparationproof · cited by 0
- SeminormedCommGroup.ofMulDistproof · cited by 0
- SeminormedCommGroup.ofMulDist'proof · cited by 0
- CauchySeq.mul_norm_bddAboveproof · cited by 0
- GroupSeminorm.toSeminormedCommGroupproof · cited by 0