AffineSubspace.ext_of_direction_eq
∀ {k : Type u_1} {V : Type u_2} {P : Type u_3} [inst : Ring k] [inst_1 : AddCommGroup V] [inst_2 : Module k V]
[inst_3 : AddTorsor V P] {s₁ s₂ : AffineSubspace k P}, s₁.direction = s₂.direction → (↑s₁ ∩ ↑s₂).Nonempty → s₁ = s₂Two affine subspaces with the same direction and nonempty intersection are equal.
- Cited by
- 11 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites19
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
- AddCommGroupstatement and proof · cited by 12,871
- SetLike.coestatement and proof · cited by 8,199
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Set.Nonemptystatement and proof · cited by 2,627
- AddTorsorstatement and proof · cited by 1,657
- AffineSubspacestatement and proof · cited by 871
- VSub.vsubproof · cited by 817
- AffineSubspace.directionstatement and proof · cited by 339
- vsub_vaddproof · cited by 62
Cited by11
Results whose statement or proof uses this declaration.
- AffineSubspace.map_spanproof · cited by 16
- AffineSubspace.eq_iff_direction_eq_of_memproof · cited by 6
- AffineSubspace.eq_of_direction_eq_of_nonempty_of_leproof · cited by 3
- AffineSubspace.shift_zeroproof · cited by 2
- Affine.Triangle.altitude_replace_orthocenter_eq_affineSpanproof · cited by 1
- affineSpan_prod_eqproof · cited by 1
- affineSpan_singleton_union_vadd_eq_top_of_span_eq_topproof · cited by 1
- AffineSubspace.direction_eq_top_iff_of_nonemptyproof · cited by 1
- EuclideanGeometry.affineSpan_of_orthocentricSystemproof · cited by 1
- exists_affineIndependentproof · cited by 1
- AffineSubspace.mk'_eqproof · cited by 0