Theorems · Theorem · functional analysis
frontier_closedBall
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] [NormedSpace ℝ E] (x : E) {r : ℝ},
r ≠ 0 → frontier (Metric.closedBall x r) = Metric.sphere x r- Cited by
- 4 results in Mathlib
- Foundations
- Depth 164 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- NormedSpacestatement and proof · cited by 12,499
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- interiorproof · cited by 714
- Metric.closedBallstatement and proof · cited by 704
- Metric.spherestatement and proof · cited by 371
- frontierstatement · cited by 214
- Metric.closedBall_sdiff_ballproof · cited by 10
- interior_closedBallproof · cited by 8
- Metric.closure_closedBallproof · cited by 3
Cited by4
Results whose statement or proof uses this declaration.
- interior_sphereproof · cited by 1
- StrictConvexSpace.of_norm_combo_ne_oneproof · cited by 1
- threeAPFree_sphereproof · cited by 1
- StrictConvexSpace.sphere_subset_extremePoints_closedBallproof · cited by 0