Theorems · Theorem · functional analysis
SeminormedAddCommGroup.dist_eq
∀ {E : Type u_8} [self : SeminormedAddCommGroup E] (x y : E), dist x y = ‖-x + y‖The distance function is induced by the norm.
- Defined in
- Mathlib.Analysis.Normed.Group.Defs
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses no axioms
- Assumes
- SeminormedAddCommGroup
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
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Dist.diststatement · cited by 1,539
Cited by7
Results whose statement or proof uses this declaration.
- NormedAddCommGroup.ofCoreproof · cited by 1
- NormedAddCommGroup.ofCoreReplaceAllproof · cited by 0
- NormedAddCommGroup.ofCoreReplaceTopologyproof · cited by 0
- NormedAddCommGroup.ofCoreReplaceUniformityproof · cited by 0
- NormedAddCommGroup.ofSeparationproof · cited by 0
- RingSeminorm.toSeminormedRingproof · cited by 0
- Equiv.seminormedAddCommGroupproof · cited by 0