collinear_insert_iff_of_mem_affineSpan
∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : DivisionRing k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
[inst_3 : AddTorsor V P] {s : Set P} {p : P}, p ∈ affineSpan k s → (Collinear k (insert p s) ↔ Collinear k s)Adding a point in the affine span of a set does not change whether that set is collinear.
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- Submoduleproof · cited by 7,192
- AddTorsorstatement and proof · cited by 1,657
- DivisionRingstatement and proof · cited by 1,062
- AffineSubspacestatement · cited by 871
- Module.rankproof · cited by 496
- affineSpanstatement and proof · cited by 417
- vectorSpanproof · cited by 123
- Collinearstatement and proof · cited by 79
- vectorSpan_insert_eq_vectorSpanproof · cited by 2
Cited by5
Results whose statement or proof uses this declaration.
- collinear_insert_of_mem_affineSpan_pairproof · cited by 5
- Collinear.oangle_sign_of_sameRay_vsubproof · cited by 2
- collinear_insert_insert_of_mem_affineSpan_pairproof · cited by 1
- EuclideanGeometry.Sphere.secondInter_collinearproof · cited by 1
- collinear_insert_insert_insert_of_mem_affineSpan_pairproof · cited by 1