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import Mathlib.Data.Finset.Basic
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import Mathlib.GroupTheory.GroupAction.Basic
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import Mathlib.GroupTheory.GroupAction.FixingSubgroup
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import Rubin.Support
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import Rubin.MulActionExt
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namespace Rubin
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-- comment by Cedric: would be nicer to define just a subset, and then show it is a subgroup
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def rigidStabilizer' (G : Type _) [Group G] [MulAction G α] (U : Set α) : Set G :=
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{g : G | ∀ x : α, g • x = x ∨ x ∈ U}
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#align rigid_stabilizer' Rubin.rigidStabilizer'
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-- TODO: rename to something else? Also check the literature on what this is called
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/--
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A "rigid stabilizer" is a subgroup of `G` associated with a set `U` for which `Support α g ⊆ U` is true for all of its elements.
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In other words, a rigid stabilizer for a set `U` contains all elements of `G` that don't move points outside of `U`.
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The notation for this subgroup is `G•[U]`.
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You might sometimes find an expression written as `↑G•[U]` when `G•[U]` is used as a set.
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--/
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def RigidStabilizer (G : Type _) [Group G] [MulAction G α] (U : Set α) : Subgroup G
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where
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carrier := {g : G | ∀ (x) (_ : x ∉ U), g • x = x}
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mul_mem' ha hb x x_notin_U := by rw [mul_smul, hb x x_notin_U, ha x x_notin_U]
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inv_mem' hg x x_notin_U := smul_eq_iff_inv_smul_eq.mp (hg x x_notin_U)
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one_mem' x _ := one_smul G x
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#align rigid_stabilizer Rubin.RigidStabilizer
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notation:max G "•[" U "]" => RigidStabilizer G U
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variable {G α: Type _}
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variable [Group G]
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variable [MulAction G α]
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theorem rigidStabilizer_eq_fixingSubgroup_compl (U : Set α) :
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G•[U] = fixingSubgroup G Uᶜ :=
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by
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ext g
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rw [mem_fixingSubgroup_iff, <-Subgroup.mem_carrier]
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unfold RigidStabilizer
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simp
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theorem rigidStabilizer_support {g : G} {U : Set α} :
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g ∈ RigidStabilizer G U ↔ Support α g ⊆ U :=
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⟨
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fun h x x_in_support =>
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by_contradiction (x_in_support ∘ h x),
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by
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intro support_sub
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rw [<-Subgroup.mem_carrier]
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unfold RigidStabilizer; simp
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intro x x_notin_U
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by_contra h
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exact x_notin_U (support_sub h)
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⟩
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#align rist_supported_in_set Rubin.rigidStabilizer_support
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theorem rigidStabilizer_mono {U V : Set α} (V_ss_U : V ⊆ U) :
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(RigidStabilizer G V : Set G) ⊆ (RigidStabilizer G U : Set G) :=
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by
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intro g g_in_ristV x x_notin_U
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have x_notin_V : x ∉ V := by intro x_in_V; exact x_notin_U (V_ss_U x_in_V)
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exact g_in_ristV x x_notin_V
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#align rist_ss_rist Rubin.rigidStabilizer_mono
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theorem monotone_rigidStabilizer : Monotone (RigidStabilizer (α := α) G) := fun _ _ => rigidStabilizer_mono
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theorem rigidStabilizer_compl [FaithfulSMul G α] {U : Set α} {f : G} (f_ne_one : f ≠ 1) :
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f ∈ G•[Uᶜ] → f ∉ G•[U] :=
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by
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intro f_in_rist_compl
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intro f_in_rist
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rw [rigidStabilizer_support] at f_in_rist_compl
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rw [rigidStabilizer_support] at f_in_rist
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rw [Set.subset_compl_iff_disjoint_left] at f_in_rist_compl
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have supp_empty : Support α f = ∅ := empty_of_subset_disjoint f_in_rist_compl.symm f_in_rist
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exact f_ne_one ((support_empty_iff f).mp supp_empty)
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theorem commute_if_rigidStabilizer_and_disjoint {g h : G} {U : Set α} [FaithfulSMul G α] :
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g ∈ RigidStabilizer G U → Disjoint U (Support α h) → Commute g h :=
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by
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intro g_in_rist U_disj
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unfold Commute
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unfold SemiconjBy
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apply eq_of_smul_eq_smul (α := α)
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intro x
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by_cases x_in_U?: x ∈ U
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{
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rw [rigidStabilizer_support] at g_in_rist
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have x_notin_support : x ∉ Support α h := disjoint_not_mem U_disj x_in_U?
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rw [mul_smul]
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rw [not_mem_support.mp x_notin_support]
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rw [mul_smul]
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by_cases gx_in_U?: g • x ∈ U
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{
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symm
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apply not_mem_support.mp
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apply disjoint_not_mem U_disj
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exact gx_in_U?
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}
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{
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have gx_notin_support : g • x ∉ Support α g := by
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intro h
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exact gx_in_U? (g_in_rist h)
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rw [<-support_inv] at gx_notin_support
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rw [not_mem_support] at gx_notin_support
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simp at gx_notin_support
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symm at gx_notin_support
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rw [fixes_inv] at gx_notin_support
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rw [<-gx_notin_support]
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symm
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group_action
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rw [not_mem_support.mp x_notin_support]
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}
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}
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{
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have x_fixed : g • x = x := g_in_rist _ x_in_U?
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repeat rw [mul_smul]
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rw [x_fixed]
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by_cases hx_in_U?: h • x ∈ U
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{
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have hx_notin_support := disjoint_not_mem U_disj hx_in_U?
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rw [<-support_inv] at hx_notin_support
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rw [not_mem_support] at hx_notin_support
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symm at hx_notin_support
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group_action at hx_notin_support
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rw [hx_notin_support]
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exact x_fixed
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}
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{
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rw [g_in_rist _ hx_in_U?]
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}
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}
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theorem rigidStabilizer_inter (U V : Set α) :
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G•[U ∩ V] = G•[U] ⊓ G•[V] :=
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by
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ext x
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simp
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repeat rw [rigidStabilizer_support]
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rw [Set.subset_inter_iff]
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theorem rigidStabilizer_empty (G α: Type _) [Group G] [MulAction G α] [FaithfulSMul G α]:
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G•[(∅ : Set α)] = ⊥ :=
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by
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rw [Subgroup.eq_bot_iff_forall]
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intro f f_in_rist
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rw [<-Subgroup.mem_carrier] at f_in_rist
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apply eq_of_smul_eq_smul (α := α)
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intro x
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rw [f_in_rist x (Set.not_mem_empty x)]
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simp
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theorem rigidStabilizer_sInter (S : Set (Set α)) :
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G•[⋂₀ S] = ⨅ T ∈ S, G•[T] :=
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by
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ext x
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rw [rigidStabilizer_support]
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constructor
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· intro supp_ss_sInter
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rw [Subgroup.mem_iInf]
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intro T
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rw [Subgroup.mem_iInf]
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intro T_in_S
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rw [rigidStabilizer_support]
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rw [Set.subset_sInter_iff] at supp_ss_sInter
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exact supp_ss_sInter T T_in_S
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· intro x_in_rist
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rw [Set.subset_sInter_iff]
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intro T T_in_S
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rw [<-rigidStabilizer_support]
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rw [Subgroup.mem_iInf] at x_in_rist
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specialize x_in_rist T
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rw [Subgroup.mem_iInf] at x_in_rist
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exact x_in_rist T_in_S
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theorem rigidStabilizer_smulImage (f g : G) (S : Set α) :
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g ∈ G•[f •'' S] ↔ f⁻¹ * g * f ∈ G•[S] :=
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by
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repeat rw [rigidStabilizer_support]
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nth_rw 3 [<-inv_inv f]
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rw [support_conjugate]
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rw [smulImage_subset_inv]
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simp
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theorem orbit_rigidStabilizer_subset {p : α} {U : Set α} (p_in_U : p ∈ U):
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MulAction.orbit G•[U] p ⊆ U :=
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by
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intro q q_in_orbit
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have ⟨⟨h, h_in_rist⟩, hp_eq_q⟩ := MulAction.mem_orbit_iff.mp q_in_orbit
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simp at hp_eq_q
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rw [<-hp_eq_q]
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rw [rigidStabilizer_support] at h_in_rist
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rw [<-elem_moved_in_support' p h_in_rist]
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assumption
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-- TODO: remov ethe need for FaithfulSMul?
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theorem rigidStabilizer_neBot [FaithfulSMul G α] {U : Set α}:
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G•[U] ≠ ⊥ → Set.Nonempty U :=
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by
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intro ne_bot
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by_contra empty
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apply ne_bot
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rw [Set.not_nonempty_iff_eq_empty] at empty
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rw [empty]
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exact rigidStabilizer_empty G α
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end Rubin
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