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Mathlib.GroupTheory.GroupAction.Embedding

Group actions on embeddings #

This file provides a MulAction G (α ↪ β) instance that agrees with the MulAction G (α → β) instances defined by Pi.mulAction.

Note that unlike the Pi instance, this requires G to be a group.

instance Function.Embedding.vadd {G : Type u_1} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddAction G β] :
VAdd G (α β)
Equations
instance Function.Embedding.smul {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] :
SMul G (α β)
Equations
theorem Function.Embedding.vadd_def {G : Type u_1} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddAction G β] (g : G) (f : α β) :
theorem Function.Embedding.smul_def {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] (g : G) (f : α β) :
@[simp]
theorem Function.Embedding.vadd_apply {G : Type u_1} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddAction G β] (g : G) (f : α β) (a : α) :
(g +ᵥ f) a = g +ᵥ f a
@[simp]
theorem Function.Embedding.smul_apply {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] (g : G) (f : α β) (a : α) :
(g f) a = g f a
theorem Function.Embedding.coe_vadd {G : Type u_1} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddAction G β] (g : G) (f : α β) :
(g +ᵥ f) = g +ᵥ f
theorem Function.Embedding.coe_smul {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] (g : G) (f : α β) :
(g f) = g f
instance Function.Embedding.instIsScalarTower {G : Type u_1} {G' : Type u_2} {α : Type u_3} {β : Type u_4} [Group G] [Group G'] [SMul G G'] [MulAction G β] [MulAction G' β] [IsScalarTower G G' β] :
IsScalarTower G G' (α β)
Equations
  • =
instance Function.Embedding.instVAddCommClass {G : Type u_1} {G' : Type u_2} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddGroup G'] [AddAction G β] [AddAction G' β] [VAddCommClass G G' β] :
VAddCommClass G G' (α β)
Equations
  • =
instance Function.Embedding.instSMulCommClass {G : Type u_1} {G' : Type u_2} {α : Type u_3} {β : Type u_4} [Group G] [Group G'] [MulAction G β] [MulAction G' β] [SMulCommClass G G' β] :
SMulCommClass G G' (α β)
Equations
  • =
instance Function.Embedding.instIsCentralScalar {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] [MulAction Gᵐᵒᵖ β] [IsCentralScalar G β] :
Equations
  • =
instance Function.Embedding.instAddAction {G : Type u_1} {α : Type u_3} {β : Type u_4} [AddGroup G] [AddAction G β] :
AddAction G (α β)
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theorem Function.Embedding.instAddAction.proof_1 {α : Type u_1} {β : Type u_2} :
Function.Injective fun (f : α β) => f
instance Function.Embedding.instMulAction {G : Type u_1} {α : Type u_3} {β : Type u_4} [Group G] [MulAction G β] :
MulAction G (α β)
Equations