Theorems · Theorem · convex and discrete geometry
PointedCone.IsFaceOf.mem_of_sum_mem
∀ {R : Type u_1} {M : Type u_2} [inst : Semiring R] [inst_1 : PartialOrder R] [inst_2 : IsOrderedRing R]
[inst_3 : AddCommGroup M] [inst_4 : Module R M] {C F : PointedCone R M} {ι : Type u_4} [inst_5 : Fintype ι]
{f : ι → M}, F.IsFaceOf C → (∀ (i : ι), f i ∈ C) → ∑ i, f i ∈ F → ∀ (i : ι), f i ∈ FIf the sum of points of a cone is in a face, then all the points are in the face.
- Defined in
- Mathlib.Geometry.Convex.Cone.Face.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites15
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommGroupstatement and proof · cited by 12,871
- Fintypestatement and proof · cited by 7,736
- PartialOrderstatement and proof · cited by 6,410
- Finset.sumstatement and proof · cited by 5,195
- Finset.univstatement and proof · cited by 3,473
- IsOrderedRingstatement and proof · cited by 777
- Finset.eraseproof · cited by 455
- add_sub_cancelproof · cited by 195
- PointedConestatement and proof · cited by 151
- sum_memproof · cited by 64
Cited by1
Results whose statement or proof uses this declaration.
- PointedCone.IsFaceOf.sum_mem_iff_memproof · cited by 0