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import Mathlib.GroupTheory.Commutator
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import Mathlib.GroupTheory.GroupAction.Basic
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import Rubin.Tactic
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import Rubin.Support
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import Rubin.RigidStabilizer
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namespace Rubin
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variable {G α: Type _}
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variable [Group G]
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variable [MulAction G α] [FaithfulSMul G α]
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theorem faithful_moves_point {g : G} (h2 : ∀ x : α, g • x = x) : g = 1 := by
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have h3 : ∀ x : α, g • x = (1 : G) • x := by group_action; apply h2
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exact eq_of_smul_eq_smul h3
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#align faithful_moves_point Rubin.faithful_moves_point
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theorem faithful_moves_point' {g : G} (α : Type _) [MulAction G α] [FaithfulSMul G α] :
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g ≠ 1 → ∃ x : α, g • x ≠ x := by
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intro k
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apply Classical.byContradiction; intro h
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rewrite [Classical.not_exists_not] at h
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exact k (faithful_moves_point h)
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#align faithful_moves_point' Rubin.faithful_moves_point'
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theorem faithful_rigid_stabilizer_moves_point {g : G} {U : Set α} :
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g ∈ RigidStabilizer G U → g ≠ 1 → ∃ x ∈ U, g • x ≠ x :=
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by
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intro g_rigid g_ne_one
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let ⟨x, xmoved⟩ := Rubin.faithful_moves_point' α g_ne_one
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exact ⟨x, rigidStabilizer_support.mp g_rigid xmoved, xmoved⟩
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#align faithful_rist_moves_point Rubin.faithful_rigid_stabilizer_moves_point
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theorem ne_one_support_nonempty {g : G} : g ≠ 1 → (Support α g).Nonempty :=
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by
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intro h1
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let ⟨x, h⟩ := Rubin.faithful_moves_point' α h1
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use x
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exact h
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#align ne_one_support_nempty Rubin.ne_one_support_nonempty
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theorem disjoint_commute {f g : G} :
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Disjoint (Support α f) (Support α g) → Commute f g :=
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by
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intro hdisjoint
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rw [← commutatorElement_eq_one_iff_commute]
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apply Rubin.faithful_moves_point (α := α)
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intro x
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rw [commutatorElement_def, mul_smul, mul_smul, mul_smul]
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by_cases hffixed: (x ∈ Support α f)
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{
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rename _ => hfmoved
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rw [
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smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp (Set.disjoint_left.mp hdisjoint hfmoved)),
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not_mem_support.mp
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(Set.disjoint_left.mp hdisjoint (inv_smul_mem_support hfmoved)),
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smul_inv_smul
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]
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}
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by_cases hgfixed: (x ∈ Support α g)
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· rename _ => hgmoved
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rw [smul_eq_iff_inv_smul_eq.mp
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(not_mem_support.mp <|
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Set.disjoint_right.mp hdisjoint (inv_smul_mem_support hgmoved)),
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smul_inv_smul, not_mem_support.mp hffixed]
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· rw [
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smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp hgfixed),
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smul_eq_iff_inv_smul_eq.mp (not_mem_support.mp hffixed),
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not_mem_support.mp hgfixed,
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not_mem_support.mp hffixed
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]
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#align disjoint_commute Rubin.disjoint_commute
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end Rubin
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