Theorems · Theorem · commutative algebra
Units.neg_smul
∀ {R : Type u_2} {M : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (u : Rˣ) (x : M),
-u • x = -(u • x)- Defined in
- Mathlib.Algebra.Module.Basic
- Cited by
- 27 results in Mathlib
- Foundations
- Depth 16 from the axioms · uses propext
- Assumes
- RingAddCommGroupModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
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
- AddCommGroupstatement and proof · cited by 12,871
- Ringstatement and proof · cited by 7,463
- Unitsstatement and proof · cited by 2,804
- Units.valproof · cited by 1,966
- neg_smulproof · cited by 306
- Units.smul_defproof · cited by 24
- Units.val_negproof · cited by 6
Cited by27
Results whose statement or proof uses this declaration.
- AlternatingMap.map_permproof · cited by 6
- CochainComplex.HomComplex.δ_zero_cochain_vproof · cited by 5
- CochainComplex.HomComplex.δ_compproof · cited by 5
- CliffordAlgebra.map_mul_map_of_isOrtho_of_mem_evenOddproof · cited by 3
- CochainComplex.mappingCocone.liftCochain_v_fst_fproof · cited by 3
- Set.Countable.isPathConnected_compl_of_one_lt_rankproof · cited by 2
- Module.Basis.adjustToOrientation_apply_eq_or_eq_negproof · cited by 2
- HomologicalComplex₂.D₂_D₁proof · cited by 2
- Polynomial.ascPochhammer_smeval_neg_eq_descPochhammerproof · cited by 1
- HomologicalComplex.mapBifunctorMapHomotopy.comm₁_auxproof · cited by 1
- CochainComplex.mappingCocone.inl_v_fst_fproof · cited by 1
- Matrix.submatrix_succAbove_det_eq_negOnePow_submatrix_succAbove_detproof · cited by 1