File: Mathlib/RingTheory/Ideal/Pointwise.lean

full_name: Ideal.pointwiseMulSemiringAction
code: /-- The action on an ideal corresponding to applying the action to every element.

This is available as an instance in the `Pointwise` locale. -/
protected def pointwiseMulSemiringAction : MulSemiringAction M (Ideal R) 
goal: MulSemiringAction M (Ideal R)

full_name: Ideal.pointwise_smul_def
code: theorem pointwise_smul_def {a : M} (S : Ideal R) :
    a • S = S.map (MulSemiringAction.toRingHom _ _ a) 
goal: a • S = Ideal.map (MulSemiringAction.toRingHom M R a) S

full_name: Ideal.smul_mem_pointwise_smul
code: theorem smul_mem_pointwise_smul (m : M) (r : R) (S : Ideal R) : r ∈ S → m • r ∈ m • S 
goal: r ∈ S → m • r ∈ m • S

full_name: Ideal.instCovariantClassHSMulLe
code: instance : CovariantClass M (Ideal R) HSMul.hSMul LE.le 
goal: CovariantClass M (Ideal R) HSMul.hSMul LE.le

full_name: Ideal.smul_bot
code: @[simp]
theorem smul_bot (a : M) : a • (⊥ : Ideal R) = ⊥ 
goal: a • ⊥ = ⊥

full_name: Ideal.smul_sup
code: theorem smul_sup (a : M) (S T : Ideal R) : a • (S ⊔ T) = a • S ⊔ a • T 
goal: a • (S ⊔ T) = a • S ⊔ a • T

full_name: Ideal.smul_closure
code: theorem smul_closure (a : M) (s : Set R) : a • span s = span (a • s) 
goal: a • Ideal.span s = Ideal.span (a • s)

full_name: Ideal.pointwise_central_scalar
code: instance pointwise_central_scalar [MulSemiringAction Mᵐᵒᵖ R] [IsCentralScalar M R] :
    IsCentralScalar M (Ideal R) 
goal: IsCentralScalar M (Ideal R)

full_name: Ideal.pointwise_smul_eq_comap
code: theorem pointwise_smul_eq_comap {a : M} (S : Ideal R) :
    a • S = S.comap (MulSemiringAction.toRingAut _ _ a).symm 
goal: a • S = Ideal.comap (RingEquiv.symm ((MulSemiringAction.toRingAut M R) a)) S

full_name: Ideal.smul_mem_pointwise_smul_iff
code: @[simp]
theorem smul_mem_pointwise_smul_iff {a : M} {S : Ideal R} {x : R} : a • x ∈ a • S ↔ x ∈ S 
goal: a • x ∈ a • S ↔ x ∈ S

full_name: Ideal.mem_pointwise_smul_iff_inv_smul_mem
code: theorem mem_pointwise_smul_iff_inv_smul_mem {a : M} {S : Ideal R} {x : R} :
    x ∈ a • S ↔ a⁻¹ • x ∈ S 
goal: x ∈ a • S ↔ a⁻¹ • x ∈ S

full_name: Ideal.pointwise_smul_toAddSubmonoid
code: @[simp]
theorem pointwise_smul_toAddSubmonoid (a : M) (S : Ideal R)
    (ha : Function.Surjective fun r : R => a • r) :
    (a • S).toAddSubmonoid = a • S.toAddSubmonoid 
goal: (a • S).toAddSubmonoid = a • S.toAddSubmonoid

full_name: Ideal.pointwise_smul_toAddSubGroup
code: @[simp]
theorem pointwise_smul_toAddSubGroup (a : M) (S : Ideal R)
    (ha : Function.Surjective fun r : R => a • r)  :
    (a • S).toAddSubgroup = a • S.toAddSubgroup 
goal: Submodule.toAddSubgroup (a • S) = a • Submodule.toAddSubgroup S

full_name: Ideal.mem_inv_pointwise_smul_iff
code: theorem mem_inv_pointwise_smul_iff {a : M} {S : Ideal R} {x : R} : x ∈ a⁻¹ • S ↔ a • x ∈ S 
goal: x ∈ a⁻¹ • S ↔ a • x ∈ S

full_name: Ideal.pointwise_smul_le_pointwise_smul_iff
code: @[simp]
theorem pointwise_smul_le_pointwise_smul_iff {a : M} {S T : Ideal R} : a • S ≤ a • T ↔ S ≤ T 
goal: a • S ≤ a • T ↔ S ≤ T

full_name: Ideal.pointwise_smul_subset_iff
code: theorem pointwise_smul_subset_iff {a : M} {S T : Ideal R} : a • S ≤ T ↔ S ≤ a⁻¹ • T 
goal: a • S ≤ T ↔ S ≤ a⁻¹ • T

full_name: Ideal.subset_pointwise_smul_iff
code: theorem subset_pointwise_smul_iff {a : M} {S T : Ideal R} : S ≤ a • T ↔ a⁻¹ • S ≤ T 
goal: S ≤ a • T ↔ a⁻¹ • S ≤ T