Theorems · Theorem · convex and discrete geometry
Convex.openSegment_subset
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : SMul 𝕜 E] {s : Set E}, Convex 𝕜 s → ∀ {x y : E}, x ∈ s → y ∈ s → openSegment 𝕜 x y ⊆ s- Defined in
- Mathlib.Analysis.Convex.Basic
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 60 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites9
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
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- PartialOrderstatement and proof · cited by 6,410
- LE.le.transproof · cited by 3,151
- Convexstatement and proof · cited by 551
- openSegmentstatement · cited by 102
- Convex.segment_subsetproof · cited by 22
- openSegment_subset_segmentproof · cited by 8
Cited by3
Results whose statement or proof uses this declaration.
- setOfPred_gauge_lt_one_subset_selfproof · cited by 5
- Convex.Ioo_subset_of_mem_closureproof · cited by 3
- IsExtreme.convex_sdiffproof · cited by 2