import Mathlib.Data.Finset.Basic import Mathlib.GroupTheory.Subgroup.Basic import Mathlib.GroupTheory.GroupAction.Basic import Mathlib.Topology.Basic import Rubin.MulActionExt import Rubin.Topology 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 theorem mem_smulImage' {x : α} (g : G) {U : Set α} : x ∈ U ↔ g • x ∈ g •'' U := by rw [mem_smulImage] rw [<-mul_smul, mul_left_inv, one_smul] @[simp] theorem smulImage_mul (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.smulImage_mul @[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 smulImage_disjoint (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.smulImage_disjoint theorem SmulImage.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 theorem SmulImage.inv_congr (g: G) {U V : Set α} : g •'' U = g •'' V → U = V := by intro h rw [<-one_smulImage (G := G) U] rw [<-one_smulImage (G := G) V] rw [<-mul_left_inv g] repeat rw [<-smulImage_mul] exact SmulImage.congr g⁻¹ h theorem smulImage_inv (g: G) (U V : Set α) : g •'' U = V ↔ U = g⁻¹ •'' V := by nth_rw 2 [<-one_smulImage (G := G) U] rw [<-mul_left_inv g, <-smulImage_mul] constructor · intro h rw [SmulImage.congr] exact h · intro h apply SmulImage.inv_congr at h exact h theorem smulImage_mono (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_mono 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_sUnion (g : G) {S : Set (Set α)} : g •'' (⋃₀ S) = ⋃₀ {g •'' T | T ∈ S} := by ext x constructor · intro h rw [mem_smulImage, Set.mem_sUnion] at h rw [Set.mem_sUnion] let ⟨T, ⟨T_in_S, ginv_x_in_T⟩⟩ := h simp use T constructor; trivial rw [mem_smulImage] exact ginv_x_in_T · intro h rw [Set.mem_sUnion] at h rw [mem_smulImage, Set.mem_sUnion] simp at h let ⟨T, ⟨T_in_S, x_in_gT⟩⟩ := h use T constructor; trivial rw [<-mem_smulImage] exact x_in_gT theorem smulImage_sInter (g : G) {S : Set (Set α)} : g •'' (⋂₀ S) = ⋂₀ {g •'' T | T ∈ S} := by ext x constructor · intro h rw [mem_smulImage, Set.mem_sInter] at h rw [Set.mem_sInter] simp intro T T_in_S rw [mem_smulImage] exact h T T_in_S · intro h rw [Set.mem_sInter] at h rw [mem_smulImage, Set.mem_sInter] intro T T_in_S rw [<-mem_smulImage] simp at h exact h T T_in_S @[simp] theorem smulImage_compl (g : G) (U : Set α) : (g •'' U)ᶜ = g •'' Uᶜ := by ext x rw [Set.mem_compl_iff] repeat rw [mem_smulImage] rw [Set.mem_compl_iff] 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_mono f⁻¹ at h rw [smulImage_mul] at h rw [mul_left_inv, one_smulImage] at h exact h · intro h apply smulImage_mono f at h rw [smulImage_mul] 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 V : Set α) : Disjoint (f •'' U) (g •'' V) ↔ Disjoint U ((f⁻¹ * g) •'' V) := by constructor · intro h apply smulImage_disjoint f⁻¹ at h repeat rw [smulImage_mul] at h rw [mul_left_inv, one_smulImage] at h exact h · intro h apply smulImage_disjoint f at h rw [smulImage_mul] at h rw [<-mul_assoc] at h rw [mul_right_inv, one_mul] at h exact h theorem smulImage_disjoint_inv_pow {G α : Type _} [Group G] [MulAction G α] (g : G) (i j : ℤ) (U V : Set α) : Disjoint (g^i •'' U) (g^j •'' V) ↔ Disjoint (g^(-j) •'' U) (g^(-i) •'' V) := by rw [smulImage_disjoint_mul] rw [<-zpow_neg, <-zpow_add, add_comm, zpow_add, zpow_neg] rw [<-inv_inv (g^j)] rw [<-smulImage_disjoint_mul] simp theorem smulImage_disjoint_subset {G α : Type _} [Group G] [MulAction G α] {f g : G} {U V : Set α} (h_sub: U ⊆ V): Disjoint (f •'' V) (g •'' V) → Disjoint (f •'' U) (g •'' U) := Set.disjoint_of_subset (smulImage_mono _ h_sub) (smulImage_mono _ h_sub) -- States that if `g^i •'' V` and `g^j •'' V` are disjoint for any `i ≠ j` and `x ∈ V` -- then `g^i • x` will always lie outside of `V` if `x ∈ V`. lemma smulImage_distinct_of_disjoint_pow {G α : Type _} [Group G] [MulAction G α] {g : G} {V : Set α} {n : ℕ} (n_pos : 0 < n) (h_disj : ∀ (i j : Fin n), i ≠ j → Disjoint (g ^ (i : ℕ) •'' V) (g ^ (j : ℕ) •'' V)) : ∀ (x : α) (_hx : x ∈ V) (i : Fin n), 0 < (i : ℕ) → g ^ (i : ℕ) • (x : α) ∉ V := by intro x hx i i_pos have i_ne_zero : i ≠ (⟨ 0, n_pos ⟩ : Fin n) := by intro h rw [h] at i_pos simp at i_pos have h_contra : g ^ (i : ℕ) • (x : α) ∈ g ^ (i : ℕ) •'' V := by use x have h_notin_V := Set.disjoint_left.mp (h_disj i (⟨0, n_pos⟩ : Fin n) i_ne_zero) h_contra simp only [pow_zero, one_smulImage] at h_notin_V exact h_notin_V #align distinct_images_from_disjoint Rubin.smulImage_distinct_of_disjoint_pow theorem continuousMulAction_elem_continuous {G : Type _} (α : Type _) [Group G] [TopologicalSpace α] [MulAction G α] [hc : ContinuousMulAction G α] (g : G): ∀ (S : Set α), IsOpen S → IsOpen (g •'' S) ∧ IsOpen ((g⁻¹) •'' S) := by intro S S_open repeat rw [smulImage_eq_inv_preimage] rw [inv_inv] constructor · exact (hc.continuous g⁻¹).isOpen_preimage _ S_open · exact (hc.continuous g).isOpen_preimage _ S_open end Rubin