Documentation

Mathlib.GroupTheory.Subgroup.Pointwise

Pointwise instances on Subgroup and AddSubgroups #

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 GroupTheory/Submonoid/Pointwise.lean. Where possible, try to keep them in sync.

@[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
@[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
@[simp]
@[simp]
theorem Subgroup.inv_subset_closure {G : Type u_2} [Group G] (S : Set G) :
theorem Subgroup.closure_toSubmonoid {G : Type u_2} [Group G] (S : Set G) :
theorem AddSubgroup.closure_induction_left {G : Type u_2} [AddGroup G] {s : Set G} {p : GProp} {x : G} (h : x AddSubgroup.closure s) (H1 : p 0) (Hmul : (x : G) → x s(y : G) → p yp (x + y)) (Hinv : (x : G) → x s(y : G) → p yp (-x + y)) :
p x

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

theorem Subgroup.closure_induction_left {G : Type u_2} [Group G] {s : Set G} {p : GProp} {x : G} (h : x Subgroup.closure s) (H1 : p 1) (Hmul : (x : G) → x s(y : G) → p yp (x * y)) (Hinv : (x : G) → x s(y : G) → p yp (x⁻¹ * y)) :
p x

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

theorem AddSubgroup.closure_induction_right {G : Type u_2} [AddGroup G] {s : Set G} {p : GProp} {x : G} (h : x AddSubgroup.closure s) (H1 : p 0) (Hmul : (x y : G) → y sp xp (x + y)) (Hinv : (x y : G) → y sp xp (x + -y)) :
p x

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

theorem Subgroup.closure_induction_right {G : Type u_2} [Group G] {s : Set G} {p : GProp} {x : G} (h : x Subgroup.closure s) (H1 : p 1) (Hmul : (x y : G) → y sp xp (x * y)) (Hinv : (x y : G) → y sp xp (x * y⁻¹)) :
p x

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

theorem AddSubgroup.closure_induction'' {G : Type u_2} [AddGroup G] {s : Set G} {p : GProp} {x : G} (h : x AddSubgroup.closure s) (Hk : (x : G) → x sp x) (Hk_inv : (x : G) → x sp (-x)) (H1 : p 0) (Hmul : (x y : G) → p xp yp (x + y)) :
p x

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.

theorem Subgroup.closure_induction'' {G : Type u_2} [Group G] {s : Set G} {p : GProp} {x : G} (h : x Subgroup.closure s) (Hk : (x : G) → x sp x) (Hk_inv : (x : G) → x sp x⁻¹) (H1 : p 1) (Hmul : (x y : G) → p xp yp (x * y)) :
p x

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.

theorem AddSubgroup.iSup_induction {G : Type u_2} [AddGroup G] {ι : Sort u_5} (S : ιAddSubgroup G) {C : GProp} {x : G} (hx : x ⨆ (i : ι), S i) (hp : (i : ι) → (x : G) → x S iC x) (h1 : C 0) (hmul : (x y : G) → C xC yC (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.

theorem Subgroup.iSup_induction {G : Type u_2} [Group G] {ι : Sort u_5} (S : ιSubgroup G) {C : GProp} {x : G} (hx : x ⨆ (i : ι), S i) (hp : (i : ι) → (x : G) → x S iC x) (h1 : C 1) (hmul : (x y : G) → C xC yC (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.

theorem AddSubgroup.iSup_induction' {G : Type u_2} [AddGroup G] {ι : Sort u_5} (S : ιAddSubgroup G) {C : (x : G) → x ⨆ (i : ι), S iProp} (hp : (i : ι) → (x : G) → (hx : x S i) → C x (_ : x ⨆ (i : ι), S i)) (h1 : C 0 (_ : 0 ⨆ (i : ι), S i)) (hmul : (x y : G) → (hx : x ⨆ (i : ι), S i) → (hy : y ⨆ (i : ι), S i) → C x hxC y hyC (x + y) (_ : x + y ⨆ (i : ι), S i)) {x : G} (hx : x ⨆ (i : ι), S i) :
C x hx

A dependent version of AddSubgroup.iSup_induction.

theorem Subgroup.iSup_induction' {G : Type u_2} [Group G] {ι : Sort u_5} (S : ιSubgroup G) {C : (x : G) → x ⨆ (i : ι), S iProp} (hp : (i : ι) → (x : G) → (hx : x S i) → C x (_ : x ⨆ (i : ι), S i)) (h1 : C 1 (_ : 1 ⨆ (i : ι), S i)) (hmul : (x y : G) → (hx : x ⨆ (i : ι), S i) → (hy : y ⨆ (i : ι), S i) → C x hxC y hyC (x * y) (_ : x * y ⨆ (i : ι), S i)) {x : G} (hx : x ⨆ (i : ι), S i) :
C x hx

A dependent version of Subgroup.iSup_induction.

abbrev AddSubgroup.closure_add_le.match_1 {G : Type u_1} [AddGroup G] (S : Set G) (T : Set G) (_x : G) (motive : _x S + TProp) :
(x : _x S + T) → ((_s _t : G) → (hs : _s S) → (ht : _t T) → (hx : (fun x x_1 => x + x_1) _s _t = _x) → motive (_ : a b, a S b T (fun x x_1 => x + x_1) a b = _x)) → motive x
Equations
  • One or more equations did not get rendered due to their size.
Instances For
    theorem AddSubgroup.sup_eq_closure_add {G : Type u_2} [AddGroup G] (H : AddSubgroup G) (K : AddSubgroup G) :
    H K = AddSubgroup.closure (H + K)
    theorem Subgroup.sup_eq_closure_mul {G : Type u_2} [Group G] (H : Subgroup G) (K : Subgroup G) :
    H K = Subgroup.closure (H * K)
    theorem AddSubgroup.set_add_normal_comm {G : Type u_2} [AddGroup G] (s : Set G) (N : AddSubgroup G) [hN : AddSubgroup.Normal N] :
    s + N = N + s
    theorem Subgroup.set_mul_normal_comm {G : Type u_2} [Group G] (s : Set G) (N : Subgroup G) [hN : Subgroup.Normal N] :
    s * N = N * s
    theorem AddSubgroup.add_normal {G : Type u_2} [AddGroup G] (H : AddSubgroup G) (N : AddSubgroup G) [hN : AddSubgroup.Normal N] :
    ↑(H N) = H + N

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

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

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

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

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

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

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

    theorem AddSubgroup.add_inf_assoc {G : Type u_2} [AddGroup G] (A : AddSubgroup G) (B : AddSubgroup G) (C : AddSubgroup G) (h : A C) :
    A + ↑(B C) = (A + B) C
    theorem Subgroup.mul_inf_assoc {G : Type u_2} [Group G] (A : Subgroup G) (B : Subgroup G) (C : Subgroup G) (h : A C) :
    A * ↑(B C) = A * B C
    theorem AddSubgroup.inf_add_assoc {G : Type u_2} [AddGroup G] (A : AddSubgroup G) (B : AddSubgroup G) (C : AddSubgroup G) (h : C A) :
    ↑(A B) + C = A (B + C)
    theorem Subgroup.inf_mul_assoc {G : Type u_2} [Group G] (A : Subgroup G) (B : Subgroup G) (C : Subgroup G) (h : C A) :
    ↑(A B) * C = A B * C
    theorem AddSubgroup.vadd_opposite_image_add_preimage' {G : Type u_2} [AddGroup G] (g : G) (h : Gᵃᵒᵖ) (s : Set G) :
    (fun y => h +ᵥ y) '' ((fun x => g + x) ⁻¹' s) = (fun x => g + x) ⁻¹' ((fun y => h +ᵥ y) '' s)
    theorem Subgroup.smul_opposite_image_mul_preimage' {G : Type u_2} [Group G] (g : G) (h : Gᵐᵒᵖ) (s : Set G) :
    (fun y => h y) '' ((fun x => g * x) ⁻¹' s) = (fun x => g * x) ⁻¹' ((fun y => h y) '' s)
    theorem AddSubgroup.vadd_opposite_image_add_preimage {G : Type u_2} [AddGroup G] {H : AddSubgroup G} (g : G) (h : { x // x AddSubgroup.op H }) (s : Set G) :
    (fun y => h +ᵥ y) '' ((fun x => g + x) ⁻¹' s) = (fun x => g + x) ⁻¹' ((fun y => h +ᵥ y) '' s)
    theorem Subgroup.smul_opposite_image_mul_preimage {G : Type u_2} [Group G] {H : Subgroup G} (g : G) (h : { x // x Subgroup.op H }) (s : Set G) :
    (fun y => h y) '' ((fun x => g * x) ⁻¹' s) = (fun x => g * x) ⁻¹' ((fun y => h y) '' s)

    Pointwise action #

    def Subgroup.pointwiseMulAction {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α 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
    • One or more equations did not get rendered due to their size.
    Instances For
      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) :
      @[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
      @[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
      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 Sa m a S
      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 S a s = m
      @[simp]
      theorem Subgroup.smul_bot {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) :
      theorem Subgroup.smul_sup {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (S : Subgroup G) (T : Subgroup G) :
      a (S T) = a S a T
      theorem Subgroup.smul_closure {α : Type u_1} {G : Type u_2} [Group G] [Monoid α] [MulDistribMulAction α G] (a : α) (s : Set G) :
      theorem Subgroup.conj_smul_le_of_le {G : Type u_2} [Group G] {P : Subgroup G} {H : Subgroup G} (hP : P H) (h : { x // x H }) :
      MulAut.conj h P H
      theorem Subgroup.conj_smul_subgroupOf {G : Type u_2} [Group G] {P : Subgroup G} {H : Subgroup G} (hP : P H) (h : { x // x H }) :
      MulAut.conj h Subgroup.subgroupOf P H = Subgroup.subgroupOf (MulAut.conj h P) H
      @[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
      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
      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
      @[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 : Subgroup G} {T : Subgroup G} :
      a S a T S T
      theorem Subgroup.pointwise_smul_subset_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S : Subgroup G} {T : Subgroup G} :
      a S T S a⁻¹ T
      theorem Subgroup.subset_pointwise_smul_iff {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] {a : α} {S : Subgroup G} {T : Subgroup G} :
      S a T a⁻¹ S T
      @[simp]
      theorem Subgroup.smul_inf {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (S : Subgroup G) (T : Subgroup G) :
      a (S T) = a S a T
      @[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) :
      ∀ (a : H.toSubmonoid), ↑(↑(Subgroup.equivSMul a H) a) = a a
      @[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) :
      ∀ (a : ↑(↑(MulDistribMulAction.toMulEquiv G a) '' H.toSubmonoid)), ↑(↑(MulEquiv.symm (Subgroup.equivSMul a H)) a) = a⁻¹ a
      def Subgroup.equivSMul {α : Type u_1} {G : Type u_2} [Group G] [Group α] [MulDistribMulAction α G] (a : α) (H : Subgroup G) :
      { x // x H } ≃* { x // x a H }

      Applying a MulDistribMulAction results in an isomorphic subgroup

      Equations
      Instances For
        theorem Subgroup.subgroup_mul_singleton {G : Type u_2} [Group G] {H : Subgroup G} {h : G} (hh : h H) :
        H * {h} = H
        theorem Subgroup.singleton_mul_subgroup {G : Type u_2} [Group G] {H : Subgroup G} {h : G} (hh : h H) :
        {h} * H = H
        theorem Subgroup.Normal.conjAct {G : Type u_5} [Group G] {H : Subgroup G} (hH : Subgroup.Normal H) (g : ConjAct G) :
        g H = H
        @[simp]
        theorem Subgroup.smul_normal {G : Type u_2} [Group G] (g : G) (H : Subgroup G) [h : Subgroup.Normal H] :
        MulAut.conj g H = H
        @[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
        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
        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
        @[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 : Subgroup G} {T : Subgroup G} :
        a S a T S T
        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 : Subgroup G} {T : Subgroup G} :
        a S T S a⁻¹ T
        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 : Subgroup G} {T : Subgroup G} :
        S a T a⁻¹ S T

        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
        • One or more equations did not get rendered due to their size.
        Instances For
          @[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
          @[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
          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 Sa m a S
          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 S a s = m
          @[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
          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
          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
          @[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 : AddSubgroup A} {T : AddSubgroup A} :
          a S a T S T
          theorem AddSubgroup.pointwise_smul_le_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S : AddSubgroup A} {T : AddSubgroup A} :
          a S T S a⁻¹ T
          theorem AddSubgroup.le_pointwise_smul_iff {α : Type u_1} {A : Type u_3} [AddGroup A] [Group α] [DistribMulAction α A] {a : α} {S : AddSubgroup A} {T : AddSubgroup A} :
          S a T a⁻¹ S T
          @[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
          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
          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
          @[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 : AddSubgroup A} {T : AddSubgroup A} :
          a S a T S T
          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 : AddSubgroup A} {T : AddSubgroup A} :
          a S T S a⁻¹ T
          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 : AddSubgroup A} {T : AddSubgroup A} :
          S a T a⁻¹ S T