Theorems · Theorem · commutative algebra

IsLocalizedModule.map_LocalizedModules

∀ {R : Type u_1} [inst : CommSemiring R] (S : Submonoid R) {M₀ : Type u_6} [inst_1 : AddCommMonoid M₀]
  [inst_2 : Module R M₀] {M₁ : Type u_7} [inst_3 : AddCommMonoid M₁] [inst_4 : Module R M₁] (g : M₀ →ₗ[R] M₁) (m : M₀)
  (s : ↥S),
  ((IsLocalizedModule.map S (LocalizedModule.mkLinearMap S M₀) (LocalizedModule.mkLinearMap S M₁)) g)
      (LocalizedModule.mk m s) =
    LocalizedModule.mk (g m) s

Formula for IsLocalizedModule.map when each localized module is a LocalizedModule.

Defined in: Mathlib.Algebra.Module.LocalizedModule.Basic
Cited by: 1 results in Mathlib
Foundations: Depth 56 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommSemiring AddCommMonoid Module AddCommMonoid Module

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Cites29

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
Modulestatement and proof · cited by 20,661
RingHom.idstatement and proof · cited by 18,349
AddCommMonoidstatement and proof · cited by 12,281
CommSemiringstatement and proof · cited by 10,911
LinearMapstatement and proof · cited by 10,215
LinearEquivproof · cited by 3,317
Submonoidstatement and proof · cited by 3,086
Units.valproof · cited by 1,966
LinearMap.compproof · cited by 1,642
LinearEquiv.symmproof · cited by 1,461
LinearEquiv.toLinearMapproof · cited by 1,171

Cited by1

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LocalizedModule.map_exactproof · cited by 1