Theorems · Theorem · group theory
vsub_eq_zero_iff_eq
∀ {G : Type u_1} {P : Type u_2} [inst : AddGroup G] [T : AddTorsor G P] {p₁ p₂ : P}, p₁ -ᵥ p₂ = 0 ↔ p₁ = p₂Subtracting two points produces 0 if and only if they are equal.
- Defined in
- Mathlib.Algebra.Torsor.Defs
- Cited by
- 45 results in Mathlib
- Foundations
- Depth 9 from the axioms · uses no axioms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- AddGroupstatement and proof · cited by 4,410
- AddTorsorstatement and proof · cited by 1,657
- VSub.vsubstatement · cited by 817
- vsub_selfproof · cited by 74
- eq_of_vsub_eq_zeroproof · cited by 1
Cited by45
Results whose statement or proof uses this declaration.
- vsub_ne_zeroproof · cited by 39
- AffineIndependent.injectiveproof · cited by 21
- EuclideanGeometry.angle_add_angle_add_angle_eq_piproof · cited by 10
- vsub_left_cancelproof · cited by 6
- AffineSubspace.wSameSide_iff_exists_leftproof · cited by 5
- AffineSubspace.wOppSide_iff_exists_leftproof · cited by 5
- affineIndependent_iff_indicator_eq_of_affineCombination_eqproof · cited by 4
- AffineSubspace.WSameSide.trans_wOppSideproof · cited by 4
- AffineMap.lineMap_eq_lineMap_iffproof · cited by 3
- AffineSubspace.mem_affineSpan_insert_iffproof · cited by 3
- Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_oneproof · cited by 2
- vsub_vadd_commproof · cited by 2