Theorems · Theorem · commutative algebra
neg_smul_neg
∀ {R : Type u_1} {M : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (r : R) (x : M),
-r • -x = r • x- Defined in
- Mathlib.Algebra.Module.Defs
- Cited by
- 9 results in Mathlib
- Foundations
- Depth 15 from the axioms · uses propext
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
Cited by9
Results whose statement or proof uses this declaration.
- Affine.Simplex.centroid_vsub_point_eq_smul_vsubproof · cited by 5
- SymplecticGroup.inv_left_mul_auxproof · cited by 2
- ConvexOn.smul''proof · cited by 2
- ConcaveOn.smul''proof · cited by 2
- isSelfAdjoint_smul_of_mem_skewAdjointproof · cited by 1
- wbtw_smul_vadd_smul_vadd_of_nonpos_of_leproof · cited by 1
- Affine.Simplex.point_vsub_faceOppositeCentroid_eq_smul_vsubproof · cited by 1
- smul_nonneg_iff_neg_imp_nonposproof · cited by 0
- Affine.Simplex.collinear_point_centroid_faceOppositeCentroidproof · cited by 0