Theorems · Theorem · functional analysis
Ioo_smul_sphere_zero
∀ {E : Type u_2} [inst : SeminormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {a b r : ℝ},
0 ≤ a → 0 < r → Set.Ioo a b • Metric.sphere 0 r = Metric.ball 0 (b * r) \ Metric.closedBall 0 (a * r)- Cited by
- 1 results in Mathlib
- Foundations
- Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites23
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
- Set.imageproof · cited by 5,609
- Norm.normproof · cited by 5,413
- Set.preimageproof · cited by 4,946
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- Set.extproof · cited by 2,266
- Set.Ioostatement and proof · cited by 1,214
- LE.le.trans_ltproof · cited by 795
- Metric.ballstatement and proof · cited by 735
- Metric.closedBallstatement and proof · cited by 704
Cited by1
Results whose statement or proof uses this declaration.
- MeasureTheory.Measure.toSphere_apply_univ'proof · cited by 1