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[Merged by Bors] - feat(GroupTheory/GroupAction/SubMulAction): two more orbit lemmas #11463

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[Merged by Bors] - feat(GroupTheory/GroupAction/SubMulAction): two more orbit lemmas #11463

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feat(GroupTheory/GroupAction/SubMulAction): two more orbit lemmas
jsm28 Mar 17, 2024
b8b35b4
intermediate lemma
eric-wieser Mar 17, 2024
260729a
Update Mathlib/GroupTheory/GroupAction/SubMulAction.lean
jsm28 Mar 17, 2024
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15 changes: 15 additions & 0 deletions Mathlib/GroupTheory/GroupAction/SubMulAction.lean
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Expand Up @@ -329,6 +329,15 @@ theorem val_image_orbit {p : SubMulAction R M} (m : p) :
lemma orbit_of_sub_mul {p : SubMulAction R M} (m : p) :
(mul_action.orbit R m : set M) = MulAction.orbit R (m : M) := rfl
-/

theorem val_preimage_orbit {p : SubMulAction R M} (m : p) :
Subtype.val ⁻¹' MulAction.orbit R (m : M) = MulAction.orbit R m := by
rw [← val_image_orbit, Subtype.val_injective.preimage_image]

lemma mem_orbit_subMul_iff {p : SubMulAction R M} {x m : p} :
x ∈ MulAction.orbit R m ↔ (x : M) ∈ MulAction.orbit R (m : M) := by
rw [← val_preimage_orbit, Set.mem_preimage]

/-- Stabilizers in monoid SubMulAction coincide with stabilizers in the ambient space -/
theorem stabilizer_of_subMul.submonoid {p : SubMulAction R M} (m : p) :
MulAction.stabilizerSubmonoid R m = MulAction.stabilizerSubmonoid R (m : M) := by
Expand All @@ -342,6 +351,12 @@ section MulActionGroup

variable [Group R] [MulAction R M]

lemma orbitRel_of_subMul (p : SubMulAction R M) :
MulAction.orbitRel R p = (MulAction.orbitRel R M).comap Subtype.val := by
refine Setoid.ext_iff.2 (fun x y ↦ ?_)
rw [Setoid.comap_rel]
exact mem_orbit_subMul_iff

/-- Stabilizers in group SubMulAction coincide with stabilizers in the ambient space -/
theorem stabilizer_of_subMul {p : SubMulAction R M} (m : p) :
MulAction.stabilizer R m = MulAction.stabilizer R (m : M) := by
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