Theorems · Theorem · commutative algebra
neg_one_smul
∀ (R : Type u_1) {M : Type u_3} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] (x : M), -1 • x = -x- Defined in
- Mathlib.Algebra.Module.Defs
- Cited by
- 23 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.
Cites5
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
- one_smulproof · cited by 1,374
- neg_smulproof · cited by 306
Cited by23
Results whose statement or proof uses this declaration.
- inner_neg_leftproof · cited by 33
- intervalIntegral.intervalIntegral_eq_integral_uIocproof · cited by 17
- Matrix.det_negproof · cited by 14
- MeasureTheory.condExp_negproof · cited by 10
- iteratedFDerivWithin_comp_negproof · cited by 3
- Sbtw.sOppSide_of_notMem_of_memproof · cited by 3
- sbtw_iff_mem_image_Ioo_and_neproof · cited by 2
- eq_zero_of_sameRay_self_negproof · cited by 2
- EuclideanGeometry.oangle_swap₁₂_signproof · cited by 2
- AlternatingMap.alternatizeUncurryFin_alternatizeUncurryFinLM_comp_applyproof · cited by 2
- IntermediateField.adjoin_inductionproof · cited by 2
- InnerProductSpace.Core.inner_neg_leftproof · cited by 2