Theorems · Theorem · global analysis
mem_tangentConeAt_of_openSegment_subset
∀ {E : Type u_1} [inst : AddCommGroup E] [inst_1 : Module ℝ E] [inst_2 : TopologicalSpace E] [ContinuousSMul ℝ E]
{s : Set E} {x y : E}, openSegment ℝ x y ⊆ s → y - x ∈ tangentConeAt ℝ s xIf a subset of a real vector space contains an open segment, then the direction of this segment belongs to the tangent cone at its endpoints.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 122 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- ContinuousSMulstatement and proof · cited by 1,016
- openSegmentstatement and proof · cited by 102
- tangentConeAtstatement · cited by 54
- tangentConeAt_mono_fieldproof · cited by 3
- sub_mem_posTangentConeAt_of_openSegment_subsetproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Convex.span_tangentConeAtproof · cited by 1
- mem_tangentConeAt_of_segment_subsetproof · cited by 0