Documentation

Mathlib.RingTheory.Subsemiring.Pointwise

Pointwise instances on Subsemirings #

This file provides the action Subsemiring.PointwiseMulAction which matches the action of MulActionSet.

This actions is available in the Pointwise locale.

Implementation notes #

This file is almost identical to GroupTheory/Submonoid/Pointwise.lean. Where possible, try to keep them in sync.

source
def Subsemiring.pointwiseMulAction {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] :
MulAction M (Subsemiring R)

The action on a subsemiring corresponding to applying the action to every element.

This is available as an instance in the Pointwise locale.

Equations
Instances For
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    theorem Subsemiring.pointwise_smul_def {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] {a : M} (S : Subsemiring R) :
    a • S = Subsemiring.map (MulSemiringAction.toRingHom M R a) S
    source
    @[simp]
    theorem Subsemiring.coe_pointwise_smul {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (m : M) (S : Subsemiring R) :
    ↑(m • S) = m • ↑S
    source
    @[simp]
    theorem Subsemiring.pointwise_smul_toAddSubmonoid {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (m : M) (S : Subsemiring R) :
    Subsemiring.toAddSubmonoid (m • S) = m • Subsemiring.toAddSubmonoid S
    source
    theorem Subsemiring.smul_mem_pointwise_smul {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (m : M) (r : R) (S : Subsemiring R) :
    r ∈ S → m • r ∈ m • S
    source
    instance Subsemiring.instCovariantClassSubsemiringToNonAssocSemiringHSMulLeToLEToPreorderToPartialOrderToCompleteSemilatticeInfInstCompleteLatticeSubsemiring {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] :
    CovariantClass M (Subsemiring R) HSMul.hSMul LE.le
    Equations
    • ⋯ = ⋯
    source
    theorem Subsemiring.mem_smul_pointwise_iff_exists {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (m : M) (r : R) (S : Subsemiring R) :
    r ∈ m • S ↔ ∃ s ∈ S, m • s = r
    source
    @[simp]
    theorem Subsemiring.smul_bot {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) :
    a • ⊥ = ⊥
    source
    theorem Subsemiring.smul_sup {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (S : Subsemiring R) (T : Subsemiring R) :
    a • (S ⊔ T) = a • S ⊔ a • T
    source
    theorem Subsemiring.smul_closure {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] (a : M) (s : Set R) :
    a • Subsemiring.closure s = Subsemiring.closure (a • s)
    source
    instance Subsemiring.pointwise_central_scalar {M : Type u_1} {R : Type u_2} [Monoid M] [Semiring R] [MulSemiringAction M R] [MulSemiringAction Mᵐᵒᵖ R] [IsCentralScalar M R] :
    IsCentralScalar M (Subsemiring R)
    Equations
    • ⋯ = ⋯
    source
    @[simp]
    theorem Subsemiring.smul_mem_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {x : R} :
    a • x ∈ a • S ↔ x ∈ S
    source
    theorem Subsemiring.mem_pointwise_smul_iff_inv_smul_mem {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {x : R} :
    x ∈ a • S ↔ a⁻¹ • x ∈ S
    source
    theorem Subsemiring.mem_inv_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {x : R} :
    x ∈ a⁻¹ • S ↔ a • x ∈ S
    source
    @[simp]
    theorem Subsemiring.pointwise_smul_le_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {T : Subsemiring R} :
    a • S ≤ a • T ↔ S ≤ T
    source
    theorem Subsemiring.pointwise_smul_subset_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {T : Subsemiring R} :
    a • S ≤ T ↔ S ≤ a⁻¹ • T
    source
    theorem Subsemiring.subset_pointwise_smul_iff {M : Type u_1} {R : Type u_2} [Group M] [Semiring R] [MulSemiringAction M R] {a : M} {S : Subsemiring R} {T : Subsemiring R} :
    S ≤ a • T ↔ a⁻¹ • S ≤ T

    TODO: add equiv_smul like we have for subgroup.

    source
    @[simp]
    theorem Subsemiring.smul_mem_pointwise_smul_iff₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
    a • x ∈ a • S ↔ x ∈ S
    source
    theorem Subsemiring.mem_pointwise_smul_iff_inv_smul_mem₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
    x ∈ a • S ↔ a⁻¹ • x ∈ S
    source
    theorem Subsemiring.mem_inv_pointwise_smul_iff₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) (S : Subsemiring R) (x : R) :
    x ∈ a⁻¹ • S ↔ a • x ∈ S
    source
    @[simp]
    theorem Subsemiring.pointwise_smul_le_pointwise_smul_iff₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S : Subsemiring R} {T : Subsemiring R} :
    a • S ≤ a • T ↔ S ≤ T
    source
    theorem Subsemiring.pointwise_smul_le_iff₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S : Subsemiring R} {T : Subsemiring R} :
    a • S ≤ T ↔ S ≤ a⁻¹ • T
    source
    theorem Subsemiring.le_pointwise_smul_iff₀ {M : Type u_1} {R : Type u_2} [GroupWithZero M] [Semiring R] [MulSemiringAction M R] {a : M} (ha : a ≠ 0) {S : Subsemiring R} {T : Subsemiring R} :
    S ≤ a • T ↔ a⁻¹ • S ≤ T