Theorems · Theorem · global analysis
sub_mem_posTangentConeAt_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 NNReal s xIf a subset of a real vector space contains an open segment, then the direction of this segment belongs to the positive tangent cone at its endpoints.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 121 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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
- NNRealstatement and proof · cited by 4,310
- Nat.cast_oneproof · cited by 2,501
- Nat.cast_zeroproof · cited by 1,870
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- add_commproof · cited by 1,535
- NNReal.toRealproof · cited by 1,260
Cited by2
Results whose statement or proof uses this declaration.
- mem_tangentConeAt_of_openSegment_subsetproof · cited by 2
- sub_mem_posTangentConeAt_of_segment_subsetproof · cited by 2