File: Mathlib/Algebra/Ring/Subring/Pointwise.lean

full_name: Subring.pointwiseMulAction
code: /-- The action on a subring corresponding to applying the action to every element.

This is available as an instance in the `Pointwise` locale. -/
protected def pointwiseMulAction : MulAction M (Subring R) 
goal: MulAction M (Subring R)

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

full_name: Subring.coe_pointwise_smul
code: @[simp]
theorem coe_pointwise_smul (m : M) (S : Subring R) : ↑(m • S) = m • (S : Set R) 
goal: ↑(m • S) = m • ↑S

full_name: Subring.pointwise_smul_toAddSubgroup
code: @[simp]
theorem pointwise_smul_toAddSubgroup (m : M) (S : Subring R) :
    (m • S).toAddSubgroup = m • S.toAddSubgroup 
goal: (m • S).toAddSubgroup = m • S.toAddSubgroup

full_name: Subring.pointwise_smul_toSubsemiring
code: @[simp]
theorem pointwise_smul_toSubsemiring (m : M) (S : Subring R) :
    (m • S).toSubsemiring = m • S.toSubsemiring 
goal: (m • S).toSubsemiring = m • S.toSubsemiring

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

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

full_name: Subring.mem_smul_pointwise_iff_exists
code: theorem mem_smul_pointwise_iff_exists (m : M) (r : R) (S : Subring R) :
    r ∈ m • S ↔ ∃ s : R, s ∈ S ∧ m • s = r 
goal: r ∈ m • S ↔ ∃ s ∈ S, m • s = r

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

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

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

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

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

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

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

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

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

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

full_name: Subring.smul_mem_pointwise_smul_iff₀
code: @[simp]
theorem smul_mem_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    a • x ∈ a • S ↔ x ∈ S 
goal: a • x ∈ a • S ↔ x ∈ S

full_name: Subring.mem_pointwise_smul_iff_inv_smul_mem₀
code: theorem mem_pointwise_smul_iff_inv_smul_mem₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    x ∈ a • S ↔ a⁻¹ • x ∈ S 
goal: x ∈ a • S ↔ a⁻¹ • x ∈ S

full_name: Subring.mem_inv_pointwise_smul_iff₀
code: theorem mem_inv_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) (S : Subring R) (x : R) :
    x ∈ a⁻¹ • S ↔ a • x ∈ S 
goal: x ∈ a⁻¹ • S ↔ a • x ∈ S

full_name: Subring.pointwise_smul_le_pointwise_smul_iff₀
code: @[simp]
theorem pointwise_smul_le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} :
    a • S ≤ a • T ↔ S ≤ T 
goal: a • S ≤ a • T ↔ S ≤ T

full_name: Subring.pointwise_smul_le_iff₀
code: theorem pointwise_smul_le_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} : a • S ≤ T ↔ S ≤ a⁻¹ • T 
goal: a • S ≤ T ↔ S ≤ a⁻¹ • T

full_name: Subring.le_pointwise_smul_iff₀
code: theorem le_pointwise_smul_iff₀ {a : M} (ha : a ≠ 0) {S T : Subring R} : S ≤ a • T ↔ a⁻¹ • S ≤ T 
goal: S ≤ a • T ↔ a⁻¹ • S ≤ T