Theorems · Theorem · functional analysis
mem_sphere_iff_norm
∀ {E : Type u_5} [inst : SeminormedAddCommGroup E] {a b : E} {r : ℝ}, b ∈ Metric.sphere a r ↔ ‖b - a‖ = r- Defined in
- Mathlib.Analysis.Normed.Group.Basic
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext
- Assumes
- SeminormedAddCommGroup
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Norm.normstatement and proof · cited by 5,413
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Metric.spherestatement · cited by 371
- dist_eq_norm_subproof · cited by 29
- Metric.mem_sphereproof · cited by 9
Cited by9
Results whose statement or proof uses this declaration.
- circleAverage_re_herglotzRieszKernel_mul_log₀proof · cited by 1
- isPathConnected_sphereproof · cited by 1
- Complex.norm_max_aux₁proof · cited by 1
- Complex.cderiv_subproof · cited by 1
- circleIntegral.integral_sub_zpow_of_undefproof · cited by 1
- setOf_sub_mem_sphere_eq_sphereproof · cited by 0
- Complex.circleTransformDeriv_boundproof · cited by 0
- Complex.norm_iteratedDeriv_le_of_forall_mem_sphere_norm_leproof · cited by 0
- preimage_add_sphereproof · cited by 0