Theorems · Theorem · commutative algebra

Submodule.neg_restrictScalars

∀ {R : Type u_2} {M : Type u_3} [inst : Semiring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {S : Type u_4}
  [inst_3 : Semiring S] [inst_4 : SMul S R] [inst_5 : Module S M] [inst_6 : IsScalarTower S R M] (p : Submodule R M),
  -Submodule.restrictScalars S p = Submodule.restrictScalars S (-p)

Defined in: Mathlib.Algebra.Module.Submodule.Pointwise
Cited by: 3 results in Mathlib
Foundations: Depth 19 from the axioms · uses propext, Quot.sound
Assumes: Semiring AddCommGroup Module Semiring SMul Module IsScalarTower

Mathlib docs Source Loogle LeanSearch

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
Semiringstatement and proof · cited by 13,802
AddCommGroupstatement and proof · cited by 12,871
Submodulestatement and proof · cited by 7,192
IsScalarTowerstatement and proof · cited by 3,896
Submodule.extproof · cited by 204
Submodule.restrictScalarsstatement · cited by 180
Submodule.pointwiseNegstatement · cited by 20

Cited by3

Results whose statement or proof uses this declaration.

PointedCone.neg_ofSubmoduleproof · cited by 0
PointedCone.sup_inf_assoc_of_le_submoduleproof · cited by 0
PointedCone.inf_sup_assoc_of_le_of_submodule_leproof · cited by 0