Theorems · Theorem · convex and discrete geometry
openSegment_subset_Ioo
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : Semiring 𝕜] [inst_1 : PartialOrder 𝕜] [inst_2 : AddCommMonoid E]
[inst_3 : PartialOrder E] [IsOrderedCancelAddMonoid E] [inst_5 : Module 𝕜 E] [PosSMulStrictMono 𝕜 E] {x y : E},
x < y → openSegment 𝕜 x y ⊆ Set.Ioo x y- Defined in
- Mathlib.Analysis.Convex.Segment
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 17 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites13
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- PartialOrderstatement and proof · cited by 6,410
- Set.Ioostatement · cited by 1,214
- IsOrderedCancelAddMonoidstatement and proof · cited by 359
- PosSMulStrictMonostatement and proof · cited by 128
- openSegmentstatement and proof · cited by 102
- add_lt_add_rightproof · cited by 50
- add_lt_add_leftproof · cited by 47
- Convex.combo_selfproof · cited by 29
Cited by2
Results whose statement or proof uses this declaration.
- Set.OrdConnected.strictConvexproof · cited by 9
- openSegment_eq_Iooproof · cited by 4