import Mathlib.Data.Finset.Basic import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.GroupAction.Basic import Rubin.MulActionExt namespace Rubin /-- The image of a group action (here generalized to any pair `(G, α)` implementing `SMul`) is the image of the elements of `U` under the `g • u` operation. An alternative definition (which is available through the [`mem_smulImage`] theorem and the [`smulImage_set`] equality) would be: `SmulImage g U = {x | g⁻¹ • x ∈ U}`. The notation used for this operator is `g •'' U`. -/ def SmulImage {G α : Type _} [SMul G α] (g : G) (U : Set α) := (g • ·) '' U #align subset_img Rubin.SmulImage infixl:60 " •'' " => Rubin.SmulImage /-- The pre-image of a group action (here generalized to any pair `(G, α)` implementing `SMul`) is the set of values `x: α` such that `g • x ∈ U`. Unlike [`SmulImage`], no notation is defined for this operator. --/ def SmulPreImage {G α : Type _} [SMul G α] (g : G) (U : Set α) := {x | g • x ∈ U} #align subset_preimg' Rubin.SmulPreImage variable {G α : Type _} variable [Group G] variable [MulAction G α] theorem smulImage_def {g : G} {U : Set α} : g •'' U = (· • ·) g '' U := rfl #align subset_img_def Rubin.smulImage_def theorem mem_smulImage {x : α} {g : G} {U : Set α} : x ∈ g •'' U ↔ g⁻¹ • x ∈ U := by rw [Rubin.smulImage_def, Set.mem_image (g • ·) U x] constructor · rintro ⟨y, yU, gyx⟩ let ygx : y = g⁻¹ • x := inv_smul_smul g y ▸ Rubin.smul_congr g⁻¹ gyx exact ygx ▸ yU · intro h exact ⟨g⁻¹ • x, ⟨Set.mem_preimage.mp h, smul_inv_smul g x⟩⟩ #align mem_smul'' Rubin.mem_smulImage -- Provides a way to express a [`SmulImage`] as a `Set`; -- this is simply [`mem_smulImage`] paired with set extensionality. theorem smulImage_set {g: G} {U: Set α} : g •'' U = {x | g⁻¹ • x ∈ U} := Set.ext (fun _x => mem_smulImage) theorem mem_inv_smulImage {x : α} {g : G} {U : Set α} : x ∈ g⁻¹ •'' U ↔ g • x ∈ U := by let msi := @Rubin.mem_smulImage _ _ _ _ x g⁻¹ U rw [inv_inv] at msi exact msi #align mem_inv_smul'' Rubin.mem_inv_smulImage -- TODO: rename to smulImage_mul @[simp] theorem mul_smulImage (g h : G) (U : Set α) : g •'' (h •'' U) = (g * h) •'' U := by ext rw [Rubin.mem_smulImage, Rubin.mem_smulImage, Rubin.mem_smulImage, ← mul_smul, mul_inv_rev] #align mul_smul'' Rubin.mul_smulImage @[simp] theorem one_smulImage (U : Set α) : (1 : G) •'' U = U := by ext rw [Rubin.mem_smulImage, inv_one, one_smul] #align one_smul'' Rubin.one_smulImage theorem disjoint_smulImage (g : G) {U V : Set α} : Disjoint U V → Disjoint (g •'' U) (g •'' V) := by intro disjoint_U_V rw [Set.disjoint_left] rw [Set.disjoint_left] at disjoint_U_V intro x x_in_gU by_contra h exact (disjoint_U_V (mem_smulImage.mp x_in_gU)) (mem_smulImage.mp h) #align disjoint_smul'' Rubin.disjoint_smulImage namespace SmulImage theorem congr (g : G) {U V : Set α} : U = V → g •'' U = g •'' V := congr_arg fun W : Set α => g •'' W #align smul''_congr Rubin.SmulImage.congr end SmulImage theorem smulImage_subset (g : G) {U V : Set α} : U ⊆ V → g •'' U ⊆ g •'' V := by intro h1 x rw [Rubin.mem_smulImage, Rubin.mem_smulImage] exact fun h2 => h1 h2 #align smul''_subset Rubin.smulImage_subset theorem smulImage_union (g : G) {U V : Set α} : g •'' U ∪ V = (g •'' U) ∪ (g •'' V) := by ext rw [Rubin.mem_smulImage, Set.mem_union, Set.mem_union, Rubin.mem_smulImage, Rubin.mem_smulImage] #align smul''_union Rubin.smulImage_union theorem smulImage_inter (g : G) {U V : Set α} : g •'' U ∩ V = (g •'' U) ∩ (g •'' V) := by ext rw [Set.mem_inter_iff, Rubin.mem_smulImage, Rubin.mem_smulImage, Rubin.mem_smulImage, Set.mem_inter_iff] #align smul''_inter Rubin.smulImage_inter theorem smulImage_eq_inv_preimage {g : G} {U : Set α} : g •'' U = (g⁻¹ • ·) ⁻¹' U := by ext constructor · intro h; rw [Set.mem_preimage]; exact mem_smulImage.mp h · intro h; rw [Rubin.mem_smulImage]; exact Set.mem_preimage.mp h #align smul''_eq_inv_preimage Rubin.smulImage_eq_inv_preimage theorem smulImage_eq_of_smul_eq {g h : G} {U : Set α} : (∀ x ∈ U, g • x = h • x) → g •'' U = h •'' U := by intro hU ext x rw [Rubin.mem_smulImage, Rubin.mem_smulImage] constructor · intro k; let a := congr_arg (h⁻¹ • ·) (hU (g⁻¹ • x) k); simp only [smul_inv_smul, inv_smul_smul] at a ; exact Set.mem_of_eq_of_mem a k · intro k; let a := congr_arg (g⁻¹ • ·) (hU (h⁻¹ • x) k); simp only [smul_inv_smul, inv_smul_smul] at a ; exact Set.mem_of_eq_of_mem a.symm k #align smul''_eq_of_smul_eq Rubin.smulImage_eq_of_smul_eq theorem smulImage_subset_inv {G α : Type _} [Group G] [MulAction G α] (f : G) (U V : Set α) : f •'' U ⊆ V ↔ U ⊆ f⁻¹ •'' V := by constructor · intro h apply smulImage_subset f⁻¹ at h rw [mul_smulImage] at h rw [mul_left_inv, one_smulImage] at h exact h · intro h apply smulImage_subset f at h rw [mul_smulImage] at h rw [mul_right_inv, one_smulImage] at h exact h theorem smulImage_subset_inv' {G α : Type _} [Group G] [MulAction G α] (f : G) (U V : Set α) : f⁻¹ •'' U ⊆ V ↔ U ⊆ f •'' V := by nth_rewrite 2 [<-inv_inv f] exact smulImage_subset_inv f⁻¹ U V theorem smulImage_disjoint_mul {G α : Type _} [Group G] [MulAction G α] (f g : G) (U : Set α) : Disjoint (f •'' U) (g •'' U) ↔ Disjoint U ((f⁻¹ * g) •'' U) := by constructor · intro h apply disjoint_smulImage f⁻¹ at h repeat rw [mul_smulImage] at h rw [mul_left_inv, one_smulImage] at h exact h · intro h apply disjoint_smulImage f at h rw [mul_smulImage] at h rw [<-mul_assoc] at h rw [mul_right_inv, one_mul] at h exact h end Rubin