Pointwise instances on `Subgroup` and `AddSubgroup`s #

This file provides the actions

which matches the action of Set.mulActionSet.

These actions are available in the Pointwise locale.

Implementation notes #

The pointwise section of this file is almost identical to the file Mathlib.Algebra.Group.Submonoid.Pointwise. Where possible, try to keep them in sync.

source

@[simp]

theorem inv_coe_set {G : Type u_2} {S : Type u_4} [InvolutiveInv G] [SetLike S G] [InvMemClass S G] {H : S} :

(↑H)⁻¹ = ↑H

source

@[simp]

theorem neg_coe_set {G : Type u_2} {S : Type u_4} [InvolutiveNeg G] [SetLike S G] [NegMemClass S G] {H : S} :

-↑H = ↑H

source

@[simp]

theorem smul_coe_set {G : Type u_2} {S : Type u_4} [Group G] [SetLike S G] [SubgroupClass S G] {s : S} {a : G} (ha : a ∈ s) :

a • ↑s = ↑s

source

@[simp]

theorem vadd_coe_set {G : Type u_2} {S : Type u_4} [AddGroup G] [SetLike S G] [AddSubgroupClass S G] {s : S} {a : G} (ha : a ∈ s) :

a +ᵥ ↑s = ↑s

source

theorem coe_set_eq_one {G : Type u_2} [Group G] {s : Subgroup G} :

↑s = 1 ↔ s = ⊥

source

theorem coe_set_eq_zero {G : Type u_2} [AddGroup G] {s : AddSubgroup G} :

↑s = 0 ↔ s = ⊥

source

@[simp]

theorem op_smul_coe_set {G : Type u_2} {S : Type u_4} [Group G] [SetLike S G] [SubgroupClass S G] {s : S} {a : G} (ha : a ∈ s) :

MulOpposite.op a • ↑s = ↑s

source

@[simp]

theorem op_vadd_coe_set {G : Type u_2} {S : Type u_4} [AddGroup G] [SetLike S G] [AddSubgroupClass S G] {s : S} {a : G} (ha : a ∈ s) :

AddOpposite.op a +ᵥ ↑s = ↑s

source

@[simp]

theorem coe_div_coe {G : Type u_2} {S : Type u_4} [SetLike S G] [DivisionMonoid G] [SubgroupClass S G] (H : S) :

↑H / ↑H = ↑H

source

@[simp]

theorem coe_sub_coe {G : Type u_2} {S : Type u_4} [SetLike S G] [SubtractionMonoid G] [AddSubgroupClass S G] (H : S) :

↑H - ↑H = ↑H

source

@[simp]

theorem Set.mul_subgroupClosure {G : Type u_2} [Group G] {s : Set G} (hs : s.Nonempty) :

s * ↑(Subgroup.closure s) = ↑(Subgroup.closure s)

source

@[simp]

theorem Set.add_addSubgroupClosure {G : Type u_2} [AddGroup G] {s : Set G} (hs : s.Nonempty) :

s + ↑(AddSubgroup.closure s) = ↑(AddSubgroup.closure s)

source

@[simp]

theorem Set.subgroupClosure_mul {G : Type u_2} [Group G] {s : Set G} (hs : s.Nonempty) :

↑(Subgroup.closure s) * s = ↑(Subgroup.closure s)

source

@[simp]

theorem Set.addSubgroupClosure_add {G : Type u_2} [AddGroup G] {s : Set G} (hs : s.Nonempty) :

↑(AddSubgroup.closure s) + s = ↑(AddSubgroup.closure s)

source

@[simp]

theorem Set.pow_mul_subgroupClosure {G : Type u_2} [Group G] {s : Set G} (hs : s.Nonempty) (n : ℕ) :

s ^ n * ↑(Subgroup.closure s) = ↑(Subgroup.closure s)

source

@[simp]

theorem Set.nsmul_add_addSubgroupClosure {G : Type u_2} [AddGroup G] {s : Set G} (hs : s.Nonempty) (n : ℕ) :

n • s + ↑(AddSubgroup.closure s) = ↑(AddSubgroup.closure s)

source

@[simp]

theorem Set.subgroupClosure_mul_pow {G : Type u_2} [Group G] {s : Set G} (hs : s.Nonempty) (n : ℕ) :

↑(Subgroup.closure s) * s ^ n = ↑(Subgroup.closure s)

source

@[simp]

theorem Set.addSubgroupClosure_add_nsmul {G : Type u_2} [AddGroup G] {s : Set G} (hs : s.Nonempty) (n : ℕ) :

↑(AddSubgroup.closure s) + n • s = ↑(AddSubgroup.closure s)

source

@[simp]

theorem Subgroup.inv_subset_closure {G : Type u_2} [Group G] (S : Set G) :

S⁻¹ ⊆ ↑(closure S)

source

@[simp]

theorem AddSubgroup.neg_subset_closure {G : Type u_2} [AddGroup G] (S : Set G) :

-S ⊆ ↑(closure S)

source

theorem Subgroup.closure_toSubmonoid {G : Type u_2} [Group G] (S : Set G) :

(closure S).toSubmonoid = Submonoid.closure (S ∪ S⁻¹)

source

theorem AddSubgroup.closure_toAddSubmonoid {G : Type u_2} [AddGroup G] (S : Set G) :

(closure S).toAddSubmonoid = AddSubmonoid.closure (S ∪ -S)

source

theorem Subgroup.closure_induction_left {G : Type u_2} [Group G] {s : Set G} {p : (x : G) → x ∈ closure s → Prop} (one : p 1 ⋯) (mul_left : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ closure s), p y hy → p (x * y) ⋯) (inv_mul_cancel : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ closure s), p y hy → p (x⁻¹ * y) ⋯) {x : G} (h : x ∈ closure s) :

p x h

For subgroups generated by a single element, see the simpler zpow_induction_left.

source

theorem AddSubgroup.closure_induction_left {G : Type u_2} [AddGroup G] {s : Set G} {p : (x : G) → x ∈ closure s → Prop} (one : p 0 ⋯) (mul_left : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ closure s), p y hy → p (x + y) ⋯) (inv_mul_cancel : ∀ (x : G) (hx : x ∈ s) (y : G) (hy : y ∈ closure s), p y hy → p (-x + y) ⋯) {x : G} (h : x ∈ closure s) :

p x h

For additive subgroups generated by a single element, see the simpler zsmul_induction_left.

source

theorem Subgroup.closure_induction_right {G : Type u_2} [Group G] {s : Set G} {p : (x : G) → x ∈ closure s → Prop} (one : p 1 ⋯) (mul_right : ∀ (x : G) (hx : x ∈ closure s) (y : G) (hy : y ∈ s), p x hx → p (x * y) ⋯) (mul_inv_cancel : ∀ (x : G) (hx : x ∈ closure s) (y : G) (hy : y ∈ s), p x hx → p (x * y⁻¹) ⋯) {x : G} (h : x ∈ closure s) :

p x h

For subgroups generated by a single element, see the simpler zpow_induction_right.

source

theorem AddSubgroup.closure_induction_right {G : Type u_2} [AddGroup G] {s : Set G} {p : (x : G) → x ∈ closure s → Prop} (one : p 0 ⋯) (mul_right : ∀ (x : G) (hx : x ∈ closure s) (y : G) (hy : y ∈ s), p x hx → p (x + y) ⋯) (mul_inv_cancel : ∀ (x : G) (hx : x ∈ closure s) (y : G) (hy : y ∈ s), p x hx → p (x + -y) ⋯) {x : G} (h : x ∈ closure s) :

p x h

For additive subgroups generated by a single element, see the simpler zsmul_induction_right.

source

@[simp]

theorem Subgroup.closure_inv {G : Type u_2} [Group G] (s : Set G) :

closure s⁻¹ = closure s

source

@[simp]

theorem AddSubgroup.closure_neg {G : Type u_2} [AddGroup G] (s : Set G) :

closure (-s) = closure s

source

@[simp]

theorem Subgroup.closure_singleton_inv {G : Type u_2} [Group G] (x : G) :

closure {x⁻¹ } = closure {x}

source

@[simp]

theorem AddSubgroup.closure_singleton_neg {G : Type u_2} [AddGroup G] (x : G) :

closure {-x} = closure {x}

source

theorem Subgroup.closure_induction'' {G : Type u_2} [Group G] {s : Set G} {p : (g : G) → g ∈ closure s → Prop} (mem : ∀ (x : G) (hx : x ∈ s), p x ⋯) (inv_mem : ∀ (x : G) (hx : x ∈ s), p x⁻¹ ⋯) (one : p 1 ⋯) (mul : ∀ (x y : G) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x * y) ⋯) {x : G} (h : x ∈ closure s) :

p x h

An induction principle for closure membership. If p holds for 1 and all elements of k and their inverse, and is preserved under multiplication, then p holds for all elements of the closure of k.

source

theorem AddSubgroup.closure_induction'' {G : Type u_2} [AddGroup G] {s : Set G} {p : (g : G) → g ∈ closure s → Prop} (mem : ∀ (x : G) (hx : x ∈ s), p x ⋯) (inv_mem : ∀ (x : G) (hx : x ∈ s), p (-x) ⋯) (one : p 0 ⋯) (mul : ∀ (x y : G) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯) {x : G} (h : x ∈ closure s) :

p x h

An induction principle for additive closure membership. If p holds for 0 and all elements of k and their negation, and is preserved under addition, then p holds for all elements of the additive closure of k.

source

theorem Subgroup.iSup_induction {G : Type u_2} [Group G] {ι : Sort u_5} (S : ι → Subgroup G) {C : G → Prop} {x : G} (hx : x ∈ ⨆ (i : ι), S i) (mem : ∀ (i : ι), ∀ x ∈ S i, C x) (one : C 1) (mul : ∀ (x y : G), C x → C y → C (x * y)) :

C x

An induction principle for elements of ⨆ i, S i. If C holds for 1 and all elements of S i for all i, and is preserved under multiplication, then it holds for all elements of the supremum of S.

source

theorem AddSubgroup.iSup_induction {G : Type u_2} [AddGroup G] {ι : Sort u_5} (S : ι → AddSubgroup G) {C : G → Prop} {x : G} (hx : x ∈ ⨆ (i : ι), S i) (mem : ∀ (i : ι), ∀ x ∈ S i, C x) (one : C 0) (mul : ∀ (x y : G), C x → C y → C (x + y)) :

C x

An induction principle for elements of ⨆ i, S i. If C holds for 0 and all elements of S i for all i, and is preserved under addition, then it holds for all elements of the supremum of S.

source

theorem Subgroup.iSup_induction' {G : Type u_2} [Group G] {ι : Sort u_5} (S : ι → Subgroup G) {C : (x : G) → x ∈ ⨆ (i : ι), S i → Prop} (hp : ∀ (i : ι) (x : G) (hx : x ∈ S i), C x ⋯) (h1 : C 1 ⋯) (hmul : ∀ (x y : G) (hx : x ∈ ⨆ (i : ι), S i) (hy : y ∈ ⨆ (i : ι), S i), C x hx → C y hy → C (x * y) ⋯) {x : G} (hx : x ∈ ⨆ (i : ι), S i) :

C x hx

A dependent version of Subgroup.iSup_induction.

source

theorem AddSubgroup.iSup_induction' {G : Type u_2} [AddGroup G] {ι : Sort u_5} (S : ι → AddSubgroup G) {C : (x : G) → x ∈ ⨆ (i : ι), S i → Prop} (hp : ∀ (i : ι) (x : G) (hx : x ∈ S i), C x ⋯) (h1 : C 0 ⋯) (hmul : ∀ (x y : G) (hx : x ∈ ⨆ (i : ι), S i) (hy : y ∈ ⨆ (i : ι), S i), C x hx → C y hy → C (x + y) ⋯) {x : G} (hx : x ∈ ⨆ (i : ι), S i) :

C x hx

A dependent version of AddSubgroup.iSup_induction.

source

theorem Subgroup.closure_mul_le {G : Type u_2} [Group G] (S T : Set G) :

closure (S * T) ≤ closure S ⊔ closure T

source

theorem AddSubgroup.closure_add_le {G : Type u_2} [AddGroup G] (S T : Set G) :

closure (S + T) ≤ closure S ⊔ closure T

source

theorem Subgroup.closure_pow_le {G : Type u_2} [Group G] {s : Set G} {n : ℕ} :

n ≠ 0 → closure (s ^ n) ≤ closure s

source

theorem AddSubgroup.closure_nsmul_le {G : Type u_2} [AddGroup G] {s : Set G} {n : ℕ} :

n ≠ 0 → closure (n • s) ≤ closure s

source

theorem Subgroup.closure_pow {G : Type u_2} [Group G] {s : Set G} {n : ℕ} (hs : 1 ∈ s) (hn : n ≠ 0) :

closure (s ^ n) = closure s

source

theorem AddSubgroup.closure_nsmul {G : Type u_2} [AddGroup G] {s : Set G} {n : ℕ} (hs : 0 ∈ s) (hn : n ≠ 0) :

closure (n • s) = closure s

source

theorem Subgroup.sup_eq_closure_mul {G : Type u_2} [Group G] (H K : Subgroup G) :

H ⊔ K = closure (↑H * ↑K)

source

theorem AddSubgroup.sup_eq_closure_add {G : Type u_2} [AddGroup G] (H K : AddSubgroup G) :

H ⊔ K = closure (↑H + ↑K)

source

theorem Subgroup.set_mul_normal_comm {G : Type u_2} [Group G] (s : Set G) (N : Subgroup G) [hN : N.Normal] :

s * ↑N = ↑N * s

source

theorem AddSubgroup.set_add_normal_comm {G : Type u_2} [AddGroup G] (s : Set G) (N : AddSubgroup G) [hN : N.Normal] :

s + ↑N = ↑N + s

source

theorem Subgroup.mul_normal {G : Type u_2} [Group G] (H N : Subgroup G) [hN : N.Normal] :

↑(H ⊔ N) = ↑H * ↑N

The carrier of H ⊔ N is just ↑H * ↑N (pointwise set product) when N is normal.

source

theorem AddSubgroup.add_normal {G : Type u_2} [AddGroup G] (H N : AddSubgroup G) [hN : N.Normal] :

↑(H ⊔ N) = ↑H + ↑N

The carrier of H ⊔ N is just ↑H + ↑N (pointwise set addition) when N is normal.

source

theorem Subgroup.normal_mul {G : Type u_2} [Group G] (N H : Subgroup G) [N.Normal] :

↑(N ⊔ H) = ↑N * ↑H

The carrier of N ⊔ H is just ↑N * ↑H (pointwise set product) when N is normal.

source

theorem AddSubgroup.normal_add {G : Type u_2} [AddGroup G] (N H : AddSubgroup G) [N.Normal] :

↑(N ⊔ H) = ↑N + ↑H

The carrier of N ⊔ H is just ↑N + ↑H (pointwise set addition) when N is normal.

source

theorem Subgroup.mul_inf_assoc {G : Type u_2} [Group G] (A B C : Subgroup G) (h : A ≤ C) :

↑A * ↑(B ⊓ C) = ↑A * ↑B ∩ ↑C

source

theorem AddSubgroup.add_inf_assoc {G : Type u_2} [AddGroup G] (A B C : AddSubgroup G) (h : A ≤ C) :

↑A + ↑(B ⊓ C) = (↑A + ↑B) ∩ ↑C

source

theorem Subgroup.inf_mul_assoc {G : Type u_2} [Group G] (A B C : Subgroup G) (h : C ≤ A) :

↑(A ⊓ B) * ↑C = ↑A ∩ (↑B * ↑C)

source

theorem AddSubgroup.inf_add_assoc {G : Type u_2} [AddGroup G] (A B C : AddSubgroup G) (h : C ≤ A) :

↑(A ⊓ B) + ↑C = ↑A ∩ (↑B + ↑C)

source

instance Subgroup.sup_normal {G : Type u_2} [Group G] (H K : Subgroup G) [hH : H.Normal] [hK : K.Normal] :

(H ⊔ K).Normal

source

instance AddSubgroup.sup_normal {G : Type u_2} [AddGroup G] (H K : AddSubgroup G) [hH : H.Normal] [hK : K.Normal] :

(H ⊔ K).Normal

source

theorem Subgroup.smul_mem_of_mem_closure_of_mem {G : Type u_2} [Group G] {X : Type u_5} [MulAction G X] {s : Set G} {t : Set X} (hs : ∀ g ∈ s, g⁻¹ ∈ s) (hst : ∀ g ∈ s, ∀ x ∈ t, g • x ∈ t) {g : G} (hg : g ∈ closure s) {x : X} (hx : x ∈ t) :

g • x ∈ t

source

theorem AddSubgroup.vadd_mem_of_mem_closure_of_mem {G : Type u_2} [AddGroup G] {X : Type u_5} [AddAction G X] {s : Set G} {t : Set X} (hs : ∀ g ∈ s, -g ∈ s) (hst : ∀ g ∈ s, ∀ x ∈ t, g +ᵥ x ∈ t) {g : G} (hg : g ∈ closure s) {x : X} (hx : x ∈ t) :

g +ᵥ x ∈ t

source

theorem Subgroup.smul_opposite_image_mul_preimage' {G : Type u_2} [Group G] (g : G) (h : Gᵐᵒᵖ) (s : Set G) :

(fun (y : G) => h • y) '' ((fun (x : G) => g * x) ⁻¹' s) = (fun (x : G) => g * x) ⁻¹' ((fun (y : G) => h • y) '' s)

source

theorem AddSubgroup.vadd_opposite_image_add_preimage' {G : Type u_2} [AddGroup G] (g : G) (h : Gᵃᵒᵖ) (s : Set G) :

(fun (y : G) => h +ᵥ y) '' ((fun (x : G) => g + x) ⁻¹' s) = (fun (x : G) => g + x) ⁻¹' ((fun (y : G) => h +ᵥ y) '' s)

source

theorem Subgroup.smul_opposite_image_mul_preimage {G : Type u_2} [Group G] {H : Subgroup G} (g : G) (h : ↥H.op) (s : Set G) :

(fun (y : G) => h • y) '' ((fun (x : G) => g * x) ⁻¹' s) = (fun (x : G) => g * x) ⁻¹' ((fun (y : G) => h • y) '' s)

source

theorem AddSubgroup.vadd_opposite_image_add_preimage {G : Type u_2} [AddGroup G] {H : AddSubgroup G} (g : G) (h : ↥H.op) (s : Set G) :

(fun (y : G) => h +ᵥ y) '' ((fun (x : G) => g + x) ⁻¹' s) = (fun (x : G) => g + x) ⁻¹' ((fun (y : G) => h +ᵥ y) '' s)

Pointwise action #

source

def Subgroup.pointwiseMulAction {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] :

MulAction α (Subgroup G)

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

This is available as an instance in the Pointwise locale.

Equations

Subgroup.pointwiseMulAction = MulAction.mk ⋯ ⋯

Instances For

source

theorem Subgroup.pointwise_smul_def {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] {a : α} (S : Subgroup G) :

a • S = map ((MulDistribMulAction.toMonoidEnd α G) a) S

source

@[simp]

theorem Subgroup.coe_pointwise_smul {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (S : Subgroup G) :

↑(a • S) = a • ↑S

source

@[simp]

theorem Subgroup.pointwise_smul_toSubmonoid {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (S : Subgroup G) :

(a • S).toSubmonoid = a • S.toSubmonoid

source

theorem Subgroup.smul_mem_pointwise_smul {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (m : G) (a : α) (S : Subgroup G) :

m ∈ S → a • m ∈ a • S

source

instance Subgroup.instCovariantClassHSMulLe {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] :

CovariantClass α (Subgroup G) HSMul.hSMul LE.le

source

theorem Subgroup.mem_smul_pointwise_iff_exists {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (m : G) (a : α) (S : Subgroup G) :

m ∈ a • S ↔ ∃ s ∈ S, a • s = m

source

@[simp]

theorem Subgroup.smul_bot {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) :

a • ⊥ = ⊥

source

theorem Subgroup.smul_sup {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (S T : Subgroup G) :

a • (S ⊔ T) = a • S ⊔ a • T

source

theorem Subgroup.smul_closure {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (s : Set G) :

a • closure s = closure (a • s)

source

instance Subgroup.pointwise_isCentralScalar {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] [MulDistribMulAction αᵐᵒᵖ G] [IsCentralScalar α G] :

IsCentralScalar α (Subgroup G)

source

theorem Subgroup.conj_smul_le_of_le {G : Type u_2} [Group G] {P H : Subgroup G} (hP : P ≤ H) (h : ↥H) :

MulAut.conj ↑h • P ≤ H

source

theorem Subgroup.conj_smul_subgroupOf {G : Type u_2} [Group G] {P H : Subgroup G} (hP : P ≤ H) (h : ↥H) :

MulAut.conj h • P.subgroupOf H = (MulAut.conj ↑h • P).subgroupOf H

source

@[simp]

theorem Subgroup.smul_mem_pointwise_smul_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S : Subgroup G} {x : G} :

a • x ∈ a • S ↔ x ∈ S

source

theorem Subgroup.mem_pointwise_smul_iff_inv_smul_mem {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S : Subgroup G} {x : G} :

x ∈ a • S ↔ a⁻¹ • x ∈ S

source

theorem Subgroup.mem_inv_pointwise_smul_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S : Subgroup G} {x : G} :

x ∈ a⁻¹ • S ↔ a • x ∈ S

source

@[simp]

theorem Subgroup.pointwise_smul_le_pointwise_smul_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S T : Subgroup G} :

a • S ≤ a • T ↔ S ≤ T

source

theorem Subgroup.pointwise_smul_subset_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S T : Subgroup G} :

a • S ≤ T ↔ S ≤ a⁻¹ • T

source

theorem Subgroup.subset_pointwise_smul_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S T : Subgroup G} :

S ≤ a • T ↔ a⁻¹ • S ≤ T

source

@[simp]

theorem Subgroup.smul_inf {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (S T : Subgroup G) :

a • (S ⊓ T) = a • S ⊓ a • T

source

def Subgroup.equivSMul {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (H : Subgroup G) :

↥H ≃* ↥(a • H)

Applying a MulDistribMulAction results in an isomorphic subgroup

Equations

Subgroup.equivSMul a H = (MulDistribMulAction.toMulEquiv G a).subgroupMap H

Instances For

source

@[simp]

theorem Subgroup.equivSMul_symm_apply_coe {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (H : Subgroup G) (y : ↑(⇑↑(MulDistribMulAction.toMulEquiv G a) '' ↑H.toSubmonoid)) :

↑((equivSMul a H).symm y) = a⁻¹ • ↑y

source

@[simp]

theorem Subgroup.equivSMul_apply_coe {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (H : Subgroup G) (x : ↑↑H.toSubmonoid) :

↑((equivSMul a H) x) = a • ↑x

source

theorem Subgroup.subgroup_mul_singleton {G : Type u_2} [Group G] {H : Subgroup G} {h : G} (hh : h ∈ H) :

↑H * {h} = ↑H

source

theorem Subgroup.singleton_mul_subgroup {G : Type u_2} [Group G] {H : Subgroup G} {h : G} (hh : h ∈ H) :

{h} * ↑H = ↑H

source

theorem Subgroup.Normal.conjAct {G : Type u_2} [Group G] {H : Subgroup G} (hH : H.Normal) (g : ConjAct G) :

g • H = H

source

@[simp]

theorem Subgroup.Normal.conj_smul_eq_self {G : Type u_2} [Group G] (g : G) (H : Subgroup G) [h : H.Normal] :

MulAut.conj g • H = H

source

@[deprecated Subgroup.Normal.conj_smul_eq_self (since := "2025-03-01")]

theorem Subgroup.smul_normal {G : Type u_2} [Group G] (g : G) (H : Subgroup G) [h : H.Normal] :

MulAut.conj g • H = H

Alias of Subgroup.Normal.conj_smul_eq_self.

source

theorem Subgroup.Normal.of_conjugate_fixed {G : Type u_2} [Group G] {H : Subgroup G} (h : ∀ (g : G), MulAut.conj g • H = H) :

H.Normal

source

theorem Subgroup.normalCore_eq_iInf_conjAct {G : Type u_2} [Group G] (H : Subgroup G) :

H.normalCore = ⨅ (g : ConjAct G), g • H

source

@[simp]

theorem Subgroup.smul_mem_pointwise_smul_iff₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) (S : Subgroup G) (x : G) :

a • x ∈ a • S ↔ x ∈ S

source

theorem Subgroup.mem_pointwise_smul_iff_inv_smul_mem₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) (S : Subgroup G) (x : G) :

x ∈ a • S ↔ a⁻¹ • x ∈ S

source

theorem Subgroup.mem_inv_pointwise_smul_iff₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) (S : Subgroup G) (x : G) :

x ∈ a⁻¹ • S ↔ a • x ∈ S

source

@[simp]

theorem Subgroup.pointwise_smul_le_pointwise_smul_iff₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) {S T : Subgroup G} :

a • S ≤ a • T ↔ S ≤ T

source

theorem Subgroup.pointwise_smul_le_iff₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) {S T : Subgroup G} :

a • S ≤ T ↔ S ≤ a⁻¹ • T

source

theorem Subgroup.le_pointwise_smul_iff₀ {α : Type u_1} {G : Type u_2} [Group G] [GroupWithZero α] [MulDistribMulAction α G] {a : α} (ha : a ≠ 0) {S T : Subgroup G} :

S ≤ a • T ↔ a⁻¹ • S ≤ T

source

def AddSubgroup.pointwiseMulAction {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] :

MulAction α (AddSubgroup A)

The action on an additive subgroup corresponding to applying the action to every element.

This is available as an instance in the Pointwise locale.

Equations

AddSubgroup.pointwiseMulAction = MulAction.mk ⋯ ⋯

Instances For

source

theorem AddSubgroup.pointwise_smul_def {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] {a : α} (S : AddSubgroup A) :

a • S = map ((DistribMulAction.toAddMonoidEnd α A) a) S

source

@[simp]

theorem AddSubgroup.coe_pointwise_smul {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] (a : α) (S : AddSubgroup A) :

↑(a • S) = a • ↑S

source

@[simp]

theorem AddSubgroup.pointwise_smul_toAddSubmonoid {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] (a : α) (S : AddSubgroup A) :

(a • S).toAddSubmonoid = a • S.toAddSubmonoid

source

theorem AddSubgroup.smul_mem_pointwise_smul {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] (m : A) (a : α) (S : AddSubgroup A) :

m ∈ S → a • m ∈ a • S

source

theorem AddSubgroup.mem_smul_pointwise_iff_exists {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] (m : A) (a : α) (S : AddSubgroup A) :

m ∈ a • S ↔ ∃ s ∈ S, a • s = m

source

instance AddSubgroup.pointwise_isCentralScalar {α : Type u_1} {A : Type u_3} [AddGroup A] [Monoid α] [DistribMulAction α A] [DistribMulAction αᵐᵒᵖ A] [IsCentralScalar α A] :

IsCentralScalar α (AddSubgroup A)

source

@[simp]

theorem AddSubgroup.smul_mem_pointwise_smul_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S : AddSubgroup A} {x : A} :

a • x ∈ a • S ↔ x ∈ S

source

theorem AddSubgroup.mem_pointwise_smul_iff_inv_smul_mem {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S : AddSubgroup A} {x : A} :

x ∈ a • S ↔ a⁻¹ • x ∈ S

source

theorem AddSubgroup.mem_inv_pointwise_smul_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S : AddSubgroup A} {x : A} :

x ∈ a⁻¹ • S ↔ a • x ∈ S

source

@[simp]

theorem AddSubgroup.pointwise_smul_le_pointwise_smul_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S T : AddSubgroup A} :

a • S ≤ a • T ↔ S ≤ T

source

theorem AddSubgroup.pointwise_smul_le_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S T : AddSubgroup A} :

a • S ≤ T ↔ S ≤ a⁻¹ • T

source

theorem AddSubgroup.le_pointwise_smul_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S T : AddSubgroup A} :

S ≤ a • T ↔ a⁻¹ • S ≤ T

source

@[simp]

theorem AddSubgroup.smul_mem_pointwise_smul_iff₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) (S : AddSubgroup A) (x : A) :

a • x ∈ a • S ↔ x ∈ S

source

theorem AddSubgroup.mem_pointwise_smul_iff_inv_smul_mem₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) (S : AddSubgroup A) (x : A) :

x ∈ a • S ↔ a⁻¹ • x ∈ S

source

theorem AddSubgroup.mem_inv_pointwise_smul_iff₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) (S : AddSubgroup A) (x : A) :

x ∈ a⁻¹ • S ↔ a • x ∈ S

source

@[simp]

theorem AddSubgroup.pointwise_smul_le_pointwise_smul_iff₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) {S T : AddSubgroup A} :

a • S ≤ a • T ↔ S ≤ T

source

theorem AddSubgroup.pointwise_smul_le_iff₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) {S T : AddSubgroup A} :

a • S ≤ T ↔ S ≤ a⁻¹ • T

source

theorem AddSubgroup.le_pointwise_smul_iff₀ {α : Type u_1} {A : Type u_3} [AddGroup A] [GroupWithZero α] [DistribMulAction α A] {a : α} (ha : a ≠ 0) {S T : AddSubgroup A} :

S ≤ a • T ↔ a⁻¹ • S ≤ T

source

@[simp]

theorem AddSubgroup.zero_smul {R : Type u_5} {M : Type u_6} [Semiring R] [AddCommGroup M] [Module R M] (s : AddSubgroup M) :

0 • s = ⊥

source

def AddSubgroup.mul {R : Type u_5} [NonUnitalNonAssocRing R] :

Mul (AddSubgroup R)

For additive subgroups S and T of a ring, the product of S and T as submonoids is automatically a subgroup, which we define as the product of S and T as subgroups.

Equations

AddSubgroup.mul = { mul := fun (M N : AddSubgroup R) => let __spread.0 := M.toAddSubmonoid * N.toAddSubmonoid; { toAddSubmonoid := __spread.0, neg_mem' := ⋯ } }

Instances For

source

theorem AddSubgroup.mul_toAddSubmonoid {R : Type u_5} [NonUnitalNonAssocRing R] (M N : AddSubgroup R) :

(M * N).toAddSubmonoid = M.toAddSubmonoid * N.toAddSubmonoid

Documentation

Mathlib.Algebra.Group.Subgroup.Pointwise

Pointwise instances on `Subgroup` and `AddSubgroup`s #

Implementation notes #

Pointwise action #

Pointwise instances on Subgroup and AddSubgroups #

Implementation notes #

Pointwise action #

Pointwise instances on `Subgroup` and `AddSubgroup`s #