Theorems · Theorem · functional analysis
segment_subset_closedBall_left
∀ {E : Type u_1} [inst : SeminormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] (x y : E),
segment ℝ x y ⊆ Metric.closedBall x (dist x y)The segment from x to y is contained in the closed ball centered at x with radius
dist x y.
- Defined in
- Mathlib.Analysis.Normed.Module.Convex
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- NormedSpacestatement and proof · cited by 12,499
- SeminormedAddCommGroupstatement and proof · cited by 2,671
- le_reflproof · cited by 2,061
- Dist.diststatement and proof · cited by 1,539
- Metric.closedBallstatement · cited by 704
- dist_commproof · cited by 188
- dist_nonnegproof · cited by 126
- segmentstatement · cited by 120
- Metric.mem_closedBallproof · cited by 42
- Convex.segment_subsetproof · cited by 22
Cited by1
Results whose statement or proof uses this declaration.
- segment_subset_closedBall_rightproof · cited by 0