File: Mathlib/Algebra/Group/Subgroup/Actions.lean

full_name: Subgroup.instMulAction
code: /-- The action by a subgroup is the action by the underlying group. -/
@[to_additive "The additive action by an add_subgroup is the action by the underlying `AddGroup`. "]
instance instMulAction : MulAction S α 
goal: MulAction (↥S) α

full_name: AddSubgroup.instAddAction
code: /-- The action by a subgroup is the action by the underlying group. -/
@[to_additive "The additive action by an add_subgroup is the action by the underlying `AddGroup`. "]
instance instMulAction : MulAction S α 
goal: AddAction (↥S) α

full_name: Subgroup.smul_def
code: @[to_additive] lemma smul_def (g : S) (m : α) : g • m = (g : G) • m 
goal: g • m = ↑g • m

full_name: AddSubgroup.vadd_def
code: @[to_additive] lemma smul_def (g : S) (m : α) : g • m = (g : G) • m 
goal: g +ᵥ m = ↑g +ᵥ m

full_name: AddSubgroup.mk_vadd
code: @[to_additive (attr := simp)]
lemma mk_smul (g : G) (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a 
goal: ⟨g, hg⟩ +ᵥ a = g +ᵥ a

full_name: Subgroup.mk_smul
code: @[to_additive (attr := simp)]
lemma mk_smul (g : G) (hg : g ∈ S) (a : α) : (⟨g, hg⟩ : S) • a = g • a 
goal: ⟨g, hg⟩ • a = g • a

full_name: AddSubgroup.vaddCommClass_left
code: @[to_additive]
instance smulCommClass_left [MulAction G β] [SMul α β] [SMulCommClass G α β] (S : Subgroup G) :
    SMulCommClass S α β 
goal: VAddCommClass (↥S) α β

full_name: Subgroup.smulCommClass_left
code: @[to_additive]
instance smulCommClass_left [MulAction G β] [SMul α β] [SMulCommClass G α β] (S : Subgroup G) :
    SMulCommClass S α β 
goal: SMulCommClass (↥S) α β

full_name: AddSubgroup.vaddCommClass_right
code: @[to_additive]
instance smulCommClass_right [SMul α β] [MulAction G β] [SMulCommClass α G β] (S : Subgroup G) :
    SMulCommClass α S β 
goal: VAddCommClass α (↥S) β

full_name: Subgroup.smulCommClass_right
code: @[to_additive]
instance smulCommClass_right [SMul α β] [MulAction G β] [SMulCommClass α G β] (S : Subgroup G) :
    SMulCommClass α S β 
goal: SMulCommClass α (↥S) β

full_name: Subgroup.instIsScalarTowerSubtypeMem
code: /-- Note that this provides `IsScalarTower S G G` which is needed by `smul_mul_assoc`. -/
instance [SMul α β] [MulAction G α] [MulAction G β] [IsScalarTower G α β] (S : Subgroup G) :
    IsScalarTower S α β 
goal: IsScalarTower (↥S) α β

full_name: Subgroup.instFaithfulSMulSubtypeMem
code: instance [MulAction G α] [FaithfulSMul G α] (S : Subgroup G) : FaithfulSMul S α 
goal: FaithfulSMul (↥S) α

full_name: Subgroup.instDistribMulActionSubtypeMem
code: /-- The action by a subgroup is the action by the underlying group. -/
instance [AddMonoid α] [DistribMulAction G α] (S : Subgroup G) : DistribMulAction S α 
goal: DistribMulAction (↥S) α

full_name: Subgroup.instMulDistribMulActionSubtypeMem
code: /-- The action by a subgroup is the action by the underlying group. -/
instance [Monoid α] [MulDistribMulAction G α] (S : Subgroup G) : MulDistribMulAction S α 
goal: MulDistribMulAction (↥S) α

full_name: Subgroup.center.smulCommClass_left
code: /-- The center of a group acts commutatively on that group. -/
instance center.smulCommClass_left : SMulCommClass (center G) G G 
goal: SMulCommClass (↥(Subgroup.center G)) G G

full_name: Subgroup.center.smulCommClass_right
code: /-- The center of a group acts commutatively on that group. -/
instance center.smulCommClass_right : SMulCommClass G (center G) G 
goal: SMulCommClass G (↥(Subgroup.center G)) G