group_theory.group_action.fixing_subgroup ⟷ Mathlib.GroupTheory.GroupAction.FixingSubgroup

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,8 +3,8 @@ Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir
 -/
-import Mathbin.GroupTheory.Subgroup.Actions
-import Mathbin.GroupTheory.GroupAction.Basic
+import GroupTheory.Subgroup.Actions
+import GroupTheory.GroupAction.Basic
 
 #align_import group_theory.group_action.fixing_subgroup from "leanprover-community/mathlib"@"fac369018417f980cec5fcdafc766a69f88d8cfe"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -55,7 +55,7 @@ def fixingSubmonoid (s : Set α) : Submonoid M
     where
   carrier := {ϕ : M | ∀ x : s, ϕ • (x : α) = x}
   one_mem' _ := one_smul _ _
-  mul_mem' x y hx hy z := by rw [mul_smul, hy z, hx z]
+  hMul_mem' x y hx hy z := by rw [mul_smul, hy z, hx z]
 #align fixing_submonoid fixingSubmonoid
 #align fixing_add_submonoid fixingAddSubmonoid
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir
-
-! This file was ported from Lean 3 source module group_theory.group_action.fixing_subgroup
-! leanprover-community/mathlib commit fac369018417f980cec5fcdafc766a69f88d8cfe
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.GroupTheory.Subgroup.Actions
 import Mathbin.GroupTheory.GroupAction.Basic
 
+#align_import group_theory.group_action.fixing_subgroup from "leanprover-community/mathlib"@"fac369018417f980cec5fcdafc766a69f88d8cfe"
+
 /-!
 
 # Fixing submonoid, fixing subgroup of an action

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -63,10 +63,12 @@ def fixingSubmonoid (s : Set α) : Submonoid M
 #align fixing_add_submonoid fixingAddSubmonoid
 -/
 
+#print mem_fixingSubmonoid_iff /-
 theorem mem_fixingSubmonoid_iff {s : Set α} {m : M} :
     m ∈ fixingSubmonoid M s ↔ ∀ y ∈ s, m • y = y :=
   ⟨fun hg y hy => hg ⟨y, hy⟩, fun h ⟨y, hy⟩ => h y hy⟩
 #align mem_fixing_submonoid_iff mem_fixingSubmonoid_iff
+-/
 
 variable (α)
 
@@ -79,19 +81,25 @@ theorem fixingSubmonoid_fixedPoints_gc :
 #align fixing_submonoid_fixed_points_gc fixingSubmonoid_fixedPoints_gc
 -/
 
+#print fixingSubmonoid_antitone /-
 theorem fixingSubmonoid_antitone : Antitone fun s : Set α => fixingSubmonoid M s :=
   (fixingSubmonoid_fixedPoints_gc M α).monotone_l
 #align fixing_submonoid_antitone fixingSubmonoid_antitone
+-/
 
+#print fixedPoints_antitone /-
 theorem fixedPoints_antitone : Antitone fun P : Submonoid M => fixedPoints P α :=
   (fixingSubmonoid_fixedPoints_gc M α).monotone_u.dual_left
 #align fixed_points_antitone fixedPoints_antitone
+-/
 
+#print fixingSubmonoid_union /-
 /-- Fixing submonoid of union is intersection -/
 theorem fixingSubmonoid_union {s t : Set α} :
     fixingSubmonoid M (s ∪ t) = fixingSubmonoid M s ⊓ fixingSubmonoid M t :=
   (fixingSubmonoid_fixedPoints_gc M α).l_sup
 #align fixing_submonoid_union fixingSubmonoid_union
+-/
 
 #print fixingSubmonoid_iUnion /-
 /-- Fixing submonoid of Union is intersection -/
@@ -101,17 +109,21 @@ theorem fixingSubmonoid_iUnion {ι : Sort _} {s : ι → Set α} :
 #align fixing_submonoid_Union fixingSubmonoid_iUnion
 -/
 
+#print fixedPoints_submonoid_sup /-
 /-- Fixed points of sup of submonoids is intersection -/
 theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
     fixedPoints (↥(P ⊔ Q)) α = fixedPoints P α ∩ fixedPoints Q α :=
   (fixingSubmonoid_fixedPoints_gc M α).u_inf
 #align fixed_points_submonoid_sup fixedPoints_submonoid_sup
+-/
 
+#print fixedPoints_submonoid_iSup /-
 /-- Fixed points of supr of submonoids is intersection -/
 theorem fixedPoints_submonoid_iSup {ι : Sort _} {P : ι → Submonoid M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
   (fixingSubmonoid_fixedPoints_gc M α).u_iInf
 #align fixed_points_submonoid_supr fixedPoints_submonoid_iSup
+-/
 
 end Monoid
 
@@ -130,9 +142,11 @@ def fixingSubgroup (s : Set α) : Subgroup M :=
 #align fixing_add_subgroup fixingAddSubgroup
 -/
 
+#print mem_fixingSubgroup_iff /-
 theorem mem_fixingSubgroup_iff {s : Set α} {m : M} : m ∈ fixingSubgroup M s ↔ ∀ y ∈ s, m • y = y :=
   ⟨fun hg y hy => hg ⟨y, hy⟩, fun h ⟨y, hy⟩ => h y hy⟩
 #align mem_fixing_subgroup_iff mem_fixingSubgroup_iff
+-/
 
 variable (α)
 
@@ -145,19 +159,25 @@ theorem fixingSubgroup_fixedPoints_gc :
 #align fixing_subgroup_fixed_points_gc fixingSubgroup_fixedPoints_gc
 -/
 
+#print fixingSubgroup_antitone /-
 theorem fixingSubgroup_antitone : Antitone (fixingSubgroup M : Set α → Subgroup M) :=
   (fixingSubgroup_fixedPoints_gc M α).monotone_l
 #align fixing_subgroup_antitone fixingSubgroup_antitone
+-/
 
+#print fixedPoints_subgroup_antitone /-
 theorem fixedPoints_subgroup_antitone : Antitone fun P : Subgroup M => fixedPoints P α :=
   (fixingSubgroup_fixedPoints_gc M α).monotone_u.dual_left
 #align fixed_points_subgroup_antitone fixedPoints_subgroup_antitone
+-/
 
+#print fixingSubgroup_union /-
 /-- Fixing subgroup of union is intersection -/
 theorem fixingSubgroup_union {s t : Set α} :
     fixingSubgroup M (s ∪ t) = fixingSubgroup M s ⊓ fixingSubgroup M t :=
   (fixingSubgroup_fixedPoints_gc M α).l_sup
 #align fixing_subgroup_union fixingSubgroup_union
+-/
 
 #print fixingSubgroup_iUnion /-
 /-- Fixing subgroup of Union is intersection -/
@@ -167,17 +187,21 @@ theorem fixingSubgroup_iUnion {ι : Sort _} {s : ι → Set α} :
 #align fixing_subgroup_Union fixingSubgroup_iUnion
 -/
 
+#print fixedPoints_subgroup_sup /-
 /-- Fixed points of sup of subgroups is intersection -/
 theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
     fixedPoints (↥(P ⊔ Q)) α = fixedPoints P α ∩ fixedPoints Q α :=
   (fixingSubgroup_fixedPoints_gc M α).u_inf
 #align fixed_points_subgroup_sup fixedPoints_subgroup_sup
+-/
 
+#print fixedPoints_subgroup_iSup /-
 /-- Fixed points of supr of subgroups is intersection -/
 theorem fixedPoints_subgroup_iSup {ι : Sort _} {P : ι → Subgroup M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
   (fixingSubgroup_fixedPoints_gc M α).u_iInf
 #align fixed_points_subgroup_supr fixedPoints_subgroup_iSup
+-/
 
 end Group
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -56,7 +56,7 @@ variable (M : Type _) {α : Type _} [Monoid M] [MulAction M α]
 @[to_additive " The additive submonoid fixing a set under an `add_action`. "]
 def fixingSubmonoid (s : Set α) : Submonoid M
     where
-  carrier := { ϕ : M | ∀ x : s, ϕ • (x : α) = x }
+  carrier := {ϕ : M | ∀ x : s, ϕ • (x : α) = x}
   one_mem' _ := one_smul _ _
   mul_mem' x y hx hy z := by rw [mul_smul, hy z, hx z]
 #align fixing_submonoid fixingSubmonoid

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -63,12 +63,6 @@ def fixingSubmonoid (s : Set α) : Submonoid M
 #align fixing_add_submonoid fixingAddSubmonoid
 -/
 
-/- warning: mem_fixing_submonoid_iff -> mem_fixingSubmonoid_iff is a dubious translation:
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-  forall (M : Type.{u1}) {α : Type.{u2}} [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {s : Set.{u2} α} {m : M}, Iff (Membership.mem.{u1, u1} M (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) m (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 s)) (forall (y : α), (Membership.mem.{u2, u2} α (Set.{u2} α) (Set.instMembershipSet.{u2} α) y s) -> (Eq.{succ u2} α (HSMul.hSMul.{u1, u2, u2} M α α (instHSMul.{u1, u2} M α (MulAction.toSMul.{u1, u2} M α _inst_1 _inst_2)) m y) y))
-Case conversion may be inaccurate. Consider using '#align mem_fixing_submonoid_iff mem_fixingSubmonoid_iffₓ'. -/
 theorem mem_fixingSubmonoid_iff {s : Set α} {m : M} :
     m ∈ fixingSubmonoid M s ↔ ∀ y ∈ s, m • y = y :=
   ⟨fun hg y hy => hg ⟨y, hy⟩, fun h ⟨y, hy⟩ => h y hy⟩
@@ -85,32 +79,14 @@ theorem fixingSubmonoid_fixedPoints_gc :
 #align fixing_submonoid_fixed_points_gc fixingSubmonoid_fixedPoints_gc
 -/
 
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-Case conversion may be inaccurate. Consider using '#align fixing_submonoid_antitone fixingSubmonoid_antitoneₓ'. -/
 theorem fixingSubmonoid_antitone : Antitone fun s : Set α => fixingSubmonoid M s :=
   (fixingSubmonoid_fixedPoints_gc M α).monotone_l
 #align fixing_submonoid_antitone fixingSubmonoid_antitone
 
-/- warning: fixed_points_antitone -> fixedPoints_antitone is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align fixed_points_antitone fixedPoints_antitoneₓ'. -/
 theorem fixedPoints_antitone : Antitone fun P : Submonoid M => fixedPoints P α :=
   (fixingSubmonoid_fixedPoints_gc M α).monotone_u.dual_left
 #align fixed_points_antitone fixedPoints_antitone
 
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-Case conversion may be inaccurate. Consider using '#align fixing_submonoid_union fixingSubmonoid_unionₓ'. -/
 /-- Fixing submonoid of union is intersection -/
 theorem fixingSubmonoid_union {s t : Set α} :
     fixingSubmonoid M (s ∪ t) = fixingSubmonoid M s ⊓ fixingSubmonoid M t :=
@@ -125,24 +101,12 @@ theorem fixingSubmonoid_iUnion {ι : Sort _} {s : ι → Set α} :
 #align fixing_submonoid_Union fixingSubmonoid_iUnion
 -/
 
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 /-- Fixed points of sup of submonoids is intersection -/
 theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
     fixedPoints (↥(P ⊔ Q)) α = fixedPoints P α ∩ fixedPoints Q α :=
   (fixingSubmonoid_fixedPoints_gc M α).u_inf
 #align fixed_points_submonoid_sup fixedPoints_submonoid_sup
 
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 /-- Fixed points of supr of submonoids is intersection -/
 theorem fixedPoints_submonoid_iSup {ι : Sort _} {P : ι → Submonoid M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
@@ -166,12 +130,6 @@ def fixingSubgroup (s : Set α) : Subgroup M :=
 #align fixing_add_subgroup fixingAddSubgroup
 -/
 
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 theorem mem_fixingSubgroup_iff {s : Set α} {m : M} : m ∈ fixingSubgroup M s ↔ ∀ y ∈ s, m • y = y :=
   ⟨fun hg y hy => hg ⟨y, hy⟩, fun h ⟨y, hy⟩ => h y hy⟩
 #align mem_fixing_subgroup_iff mem_fixingSubgroup_iff
@@ -187,32 +145,14 @@ theorem fixingSubgroup_fixedPoints_gc :
 #align fixing_subgroup_fixed_points_gc fixingSubgroup_fixedPoints_gc
 -/
 
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 theorem fixingSubgroup_antitone : Antitone (fixingSubgroup M : Set α → Subgroup M) :=
   (fixingSubgroup_fixedPoints_gc M α).monotone_l
 #align fixing_subgroup_antitone fixingSubgroup_antitone
 
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 theorem fixedPoints_subgroup_antitone : Antitone fun P : Subgroup M => fixedPoints P α :=
   (fixingSubgroup_fixedPoints_gc M α).monotone_u.dual_left
 #align fixed_points_subgroup_antitone fixedPoints_subgroup_antitone
 
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 /-- Fixing subgroup of union is intersection -/
 theorem fixingSubgroup_union {s t : Set α} :
     fixingSubgroup M (s ∪ t) = fixingSubgroup M s ⊓ fixingSubgroup M t :=
@@ -227,24 +167,12 @@ theorem fixingSubgroup_iUnion {ι : Sort _} {s : ι → Set α} :
 #align fixing_subgroup_Union fixingSubgroup_iUnion
 -/
 
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 /-- Fixed points of sup of subgroups is intersection -/
 theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
     fixedPoints (↥(P ⊔ Q)) α = fixedPoints P α ∩ fixedPoints Q α :=
   (fixingSubgroup_fixedPoints_gc M α).u_inf
 #align fixed_points_subgroup_sup fixedPoints_subgroup_sup
 
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 /-- Fixed points of supr of subgroups is intersection -/
 theorem fixedPoints_subgroup_iSup {ι : Sort _} {P : ι → Subgroup M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -117,12 +117,12 @@ theorem fixingSubmonoid_union {s t : Set α} :
   (fixingSubmonoid_fixedPoints_gc M α).l_sup
 #align fixing_submonoid_union fixingSubmonoid_union
 
-#print fixingSubmonoid_unionᵢ /-
+#print fixingSubmonoid_iUnion /-
 /-- Fixing submonoid of Union is intersection -/
-theorem fixingSubmonoid_unionᵢ {ι : Sort _} {s : ι → Set α} :
+theorem fixingSubmonoid_iUnion {ι : Sort _} {s : ι → Set α} :
     fixingSubmonoid M (⋃ i, s i) = ⨅ i, fixingSubmonoid M (s i) :=
-  (fixingSubmonoid_fixedPoints_gc M α).l_supᵢ
-#align fixing_submonoid_Union fixingSubmonoid_unionᵢ
+  (fixingSubmonoid_fixedPoints_gc M α).l_iSup
+#align fixing_submonoid_Union fixingSubmonoid_iUnion
 -/
 
 /- warning: fixed_points_submonoid_sup -> fixedPoints_submonoid_sup is a dubious translation:
@@ -137,17 +137,17 @@ theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
   (fixingSubmonoid_fixedPoints_gc M α).u_inf
 #align fixed_points_submonoid_sup fixedPoints_submonoid_sup
 
-/- warning: fixed_points_submonoid_supr -> fixedPoints_submonoid_supᵢ is a dubious translation:
+/- warning: fixed_points_submonoid_supr -> fixedPoints_submonoid_iSup is a dubious translation:
 lean 3 declaration is
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+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {ι : Sort.{u3}} {P : ι -> (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (iSup.{u1, u3} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteSemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)))) ι P)) α (Submonoid.toMonoid.{u1} M _inst_1 (iSup.{u1, u3} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteSemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)))) ι P)) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 (iSup.{u1, u3} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteSemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)))) ι P))) (Set.iInter.{u2, u3} α ι (fun (i : ι) => MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (P i)) α (Submonoid.toMonoid.{u1} M _inst_1 (P i)) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 (P i))))
 but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align fixed_points_submonoid_supr fixedPoints_submonoid_supᵢₓ'. -/
+  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Monoid.{u2} M] [_inst_2 : MulAction.{u2, u1} M α _inst_1] {ι : Sort.{u3}} {P : ι -> (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x (iSup.{u2, u3} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toSupSet.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) ι P))) α (Submonoid.toMonoid.{u2} M _inst_1 (iSup.{u2, u3} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toSupSet.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) ι P)) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 (iSup.{u2, u3} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toSupSet.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) ι P))) (Set.iInter.{u1, u3} α ι (fun (i : ι) => MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x (P i))) α (Submonoid.toMonoid.{u2} M _inst_1 (P i)) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 (P i))))
+Case conversion may be inaccurate. Consider using '#align fixed_points_submonoid_supr fixedPoints_submonoid_iSupₓ'. -/
 /-- Fixed points of supr of submonoids is intersection -/
-theorem fixedPoints_submonoid_supᵢ {ι : Sort _} {P : ι → Submonoid M} :
-    fixedPoints (↥(supᵢ P)) α = ⋂ i, fixedPoints (P i) α :=
-  (fixingSubmonoid_fixedPoints_gc M α).u_infᵢ
-#align fixed_points_submonoid_supr fixedPoints_submonoid_supᵢ
+theorem fixedPoints_submonoid_iSup {ι : Sort _} {P : ι → Submonoid M} :
+    fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
+  (fixingSubmonoid_fixedPoints_gc M α).u_iInf
+#align fixed_points_submonoid_supr fixedPoints_submonoid_iSup
 
 end Monoid
 
@@ -219,12 +219,12 @@ theorem fixingSubgroup_union {s t : Set α} :
   (fixingSubgroup_fixedPoints_gc M α).l_sup
 #align fixing_subgroup_union fixingSubgroup_union
 
-#print fixingSubgroup_unionᵢ /-
+#print fixingSubgroup_iUnion /-
 /-- Fixing subgroup of Union is intersection -/
-theorem fixingSubgroup_unionᵢ {ι : Sort _} {s : ι → Set α} :
+theorem fixingSubgroup_iUnion {ι : Sort _} {s : ι → Set α} :
     fixingSubgroup M (⋃ i, s i) = ⨅ i, fixingSubgroup M (s i) :=
-  (fixingSubgroup_fixedPoints_gc M α).l_supᵢ
-#align fixing_subgroup_Union fixingSubgroup_unionᵢ
+  (fixingSubgroup_fixedPoints_gc M α).l_iSup
+#align fixing_subgroup_Union fixingSubgroup_iUnion
 -/
 
 /- warning: fixed_points_subgroup_sup -> fixedPoints_subgroup_sup is a dubious translation:
@@ -239,17 +239,17 @@ theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
   (fixingSubgroup_fixedPoints_gc M α).u_inf
 #align fixed_points_subgroup_sup fixedPoints_subgroup_sup
 
-/- warning: fixed_points_subgroup_supr -> fixedPoints_subgroup_supᵢ is a dubious translation:
+/- warning: fixed_points_subgroup_supr -> fixedPoints_subgroup_iSup is a dubious translation:
 lean 3 declaration is
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+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {ι : Sort.{u3}} {P : ι -> (Subgroup.{u1} M _inst_1)}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (iSup.{u1, u3} (Subgroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1))) ι P)) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (iSup.{u1, u3} (Subgroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1))) ι P)) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (iSup.{u1, u3} (Subgroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1))) ι P)) (Subgroup.toGroup.{u1} M _inst_1 (iSup.{u1, u3} (Subgroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1))) ι P)))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 (iSup.{u1, u3} (Subgroup.{u1} M _inst_1) (CompleteSemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toCompleteSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1))) ι P))) (Set.iInter.{u2, u3} α ι (fun (i : ι) => MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (P i)) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (P i)) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (P i)) (Subgroup.toGroup.{u1} M _inst_1 (P i)))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 (P i))))
 but is expected to have type
-  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Group.{u2} M] [_inst_2 : MulAction.{u2, u1} M α (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1))] {ι : Sort.{u3}} {P : ι -> (Subgroup.{u2} M _inst_1)}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (supᵢ.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (supᵢ.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (supᵢ.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) (Set.interᵢ.{u1, u3} α ι (fun (i : ι) => MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (P i))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (P i))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (P i))))
-Case conversion may be inaccurate. Consider using '#align fixed_points_subgroup_supr fixedPoints_subgroup_supᵢₓ'. -/
+  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Group.{u2} M] [_inst_2 : MulAction.{u2, u1} M α (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1))] {ι : Sort.{u3}} {P : ι -> (Subgroup.{u2} M _inst_1)}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (iSup.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (iSup.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (iSup.{u2, u3} (Subgroup.{u2} M _inst_1) (CompleteLattice.toSupSet.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)) ι P))) (Set.iInter.{u1, u3} α ι (fun (i : ι) => MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (P i))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (P i))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (P i))))
+Case conversion may be inaccurate. Consider using '#align fixed_points_subgroup_supr fixedPoints_subgroup_iSupₓ'. -/
 /-- Fixed points of supr of subgroups is intersection -/
-theorem fixedPoints_subgroup_supᵢ {ι : Sort _} {P : ι → Subgroup M} :
-    fixedPoints (↥(supᵢ P)) α = ⋂ i, fixedPoints (P i) α :=
-  (fixingSubgroup_fixedPoints_gc M α).u_infᵢ
-#align fixed_points_subgroup_supr fixedPoints_subgroup_supᵢ
+theorem fixedPoints_subgroup_iSup {ι : Sort _} {P : ι → Subgroup M} :
+    fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
+  (fixingSubgroup_fixedPoints_gc M α).u_iInf
+#align fixed_points_subgroup_supr fixedPoints_subgroup_iSup
 
 end Group
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9da1b3534b65d9661eb8f42443598a92bbb49211

@@ -107,9 +107,9 @@ theorem fixedPoints_antitone : Antitone fun P : Submonoid M => fixedPoints P α
 
 /- warning: fixing_submonoid_union -> fixingSubmonoid_union is a dubious translation:
 lean 3 declaration is
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.hasUnion.{u2} α) s t)) (HasInf.inf.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.hasInf.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 t))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.hasUnion.{u2} α) s t)) (Inf.inf.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.hasInf.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 t))
 but is expected to have type
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t)) (HasInf.inf.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.instHasInfSubmonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 t))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t)) (Inf.inf.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.instInfSubmonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubmonoid.{u1, u2} M α _inst_1 _inst_2 t))
 Case conversion may be inaccurate. Consider using '#align fixing_submonoid_union fixingSubmonoid_unionₓ'. -/
 /-- Fixing submonoid of union is intersection -/
 theorem fixingSubmonoid_union {s t : Set α} :
@@ -127,9 +127,9 @@ theorem fixingSubmonoid_unionᵢ {ι : Sort _} {s : ι → Set α} :
 
 /- warning: fixed_points_submonoid_sup -> fixedPoints_submonoid_sup is a dubious translation:
 lean 3 declaration is
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {P : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)} {Q : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (HasSup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q)) α (Submonoid.toMonoid.{u1} M _inst_1 (HasSup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q)) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 (HasSup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q))) (Inter.inter.{u2} (Set.{u2} α) (Set.hasInter.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) P) α (Submonoid.toMonoid.{u1} M _inst_1 P) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 P)) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) Q) α (Submonoid.toMonoid.{u1} M _inst_1 Q) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 Q)))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Monoid.{u1} M] [_inst_2 : MulAction.{u1, u2} M α _inst_1] {P : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)} {Q : Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) (Sup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q)) α (Submonoid.toMonoid.{u1} M _inst_1 (Sup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q)) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 (Sup.sup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (SemilatticeSup.toHasSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Lattice.toSemilatticeSup.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (CompleteLattice.toLattice.{u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) (Submonoid.completeLattice.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))))) P Q))) (Inter.inter.{u2} (Set.{u2} α) (Set.hasInter.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) P) α (Submonoid.toMonoid.{u1} M _inst_1 P) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 P)) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1)) M (Submonoid.setLike.{u1} M (Monoid.toMulOneClass.{u1} M _inst_1))) Q) α (Submonoid.toMonoid.{u1} M _inst_1 Q) (Submonoid.mulAction.{u1, u2} M α _inst_1 _inst_2 Q)))
 but is expected to have type
-  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Monoid.{u2} M] [_inst_2 : MulAction.{u2, u1} M α _inst_1] {P : Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)} {Q : Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x (HasSup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toHasSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q))) α (Submonoid.toMonoid.{u2} M _inst_1 (HasSup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toHasSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q)) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 (HasSup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toHasSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x P)) α (Submonoid.toMonoid.{u2} M _inst_1 P) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 P)) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x Q)) α (Submonoid.toMonoid.{u2} M _inst_1 Q) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 Q)))
+  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Monoid.{u2} M] [_inst_2 : MulAction.{u2, u1} M α _inst_1] {P : Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)} {Q : Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x (Sup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q))) α (Submonoid.toMonoid.{u2} M _inst_1 (Sup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q)) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 (Sup.sup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SemilatticeSup.toSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Lattice.toSemilatticeSup.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (CompleteLattice.toLattice.{u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (Submonoid.instCompleteLatticeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))))) P Q))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x P)) α (Submonoid.toMonoid.{u2} M _inst_1 P) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 P)) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1)) M (Submonoid.instSetLikeSubmonoid.{u2} M (Monoid.toMulOneClass.{u2} M _inst_1))) x Q)) α (Submonoid.toMonoid.{u2} M _inst_1 Q) (Submonoid.mulAction.{u2, u1} M α _inst_1 _inst_2 Q)))
 Case conversion may be inaccurate. Consider using '#align fixed_points_submonoid_sup fixedPoints_submonoid_supₓ'. -/
 /-- Fixed points of sup of submonoids is intersection -/
 theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
@@ -209,9 +209,9 @@ theorem fixedPoints_subgroup_antitone : Antitone fun P : Subgroup M => fixedPoin
 
 /- warning: fixing_subgroup_union -> fixingSubgroup_union is a dubious translation:
 lean 3 declaration is
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Subgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.hasUnion.{u2} α) s t)) (HasInf.inf.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.hasInf.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 t))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Subgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.hasUnion.{u2} α) s t)) (Inf.inf.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.hasInf.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 t))
 but is expected to have type
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Subgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t)) (HasInf.inf.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.instHasInfSubgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 t))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {s : Set.{u2} α} {t : Set.{u2} α}, Eq.{succ u1} (Subgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t)) (Inf.inf.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.instInfSubgroup.{u1} M _inst_1) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 s) (fixingSubgroup.{u1, u2} M α _inst_1 _inst_2 t))
 Case conversion may be inaccurate. Consider using '#align fixing_subgroup_union fixingSubgroup_unionₓ'. -/
 /-- Fixing subgroup of union is intersection -/
 theorem fixingSubgroup_union {s t : Set α} :
@@ -229,9 +229,9 @@ theorem fixingSubgroup_unionᵢ {ι : Sort _} {s : ι → Set α} :
 
 /- warning: fixed_points_subgroup_sup -> fixedPoints_subgroup_sup is a dubious translation:
 lean 3 declaration is
-  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {P : Subgroup.{u1} M _inst_1} {Q : Subgroup.{u1} M _inst_1}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (HasSup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (HasSup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (HasSup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) (Subgroup.toGroup.{u1} M _inst_1 (HasSup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 (HasSup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q))) (Inter.inter.{u2} (Set.{u2} α) (Set.hasInter.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) (Subgroup.toGroup.{u1} M _inst_1 P))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 P)) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) (Subgroup.toGroup.{u1} M _inst_1 Q))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 Q)))
+  forall (M : Type.{u1}) (α : Type.{u2}) [_inst_1 : Group.{u1} M] [_inst_2 : MulAction.{u1, u2} M α (DivInvMonoid.toMonoid.{u1} M (Group.toDivInvMonoid.{u1} M _inst_1))] {P : Subgroup.{u1} M _inst_1} {Q : Subgroup.{u1} M _inst_1}, Eq.{succ u2} (Set.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (Sup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (Sup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) (Sup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)) (Subgroup.toGroup.{u1} M _inst_1 (Sup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q)))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 (Sup.sup.{u1} (Subgroup.{u1} M _inst_1) (SemilatticeSup.toHasSup.{u1} (Subgroup.{u1} M _inst_1) (Lattice.toSemilatticeSup.{u1} (Subgroup.{u1} M _inst_1) (CompleteLattice.toLattice.{u1} (Subgroup.{u1} M _inst_1) (Subgroup.completeLattice.{u1} M _inst_1)))) P Q))) (Inter.inter.{u2} (Set.{u2} α) (Set.hasInter.{u2} α) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) P) (Subgroup.toGroup.{u1} M _inst_1 P))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 P)) (MulAction.fixedPoints.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) α (DivInvMonoid.toMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) (Group.toDivInvMonoid.{u1} (coeSort.{succ u1, succ (succ u1)} (Subgroup.{u1} M _inst_1) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subgroup.{u1} M _inst_1) M (Subgroup.setLike.{u1} M _inst_1)) Q) (Subgroup.toGroup.{u1} M _inst_1 Q))) (Subgroup.mulAction.{u1, u2} M _inst_1 α _inst_2 Q)))
 but is expected to have type
-  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Group.{u2} M] [_inst_2 : MulAction.{u2, u1} M α (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1))] {P : Subgroup.{u2} M _inst_1} {Q : Subgroup.{u2} M _inst_1}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (HasSup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toHasSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (HasSup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toHasSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (HasSup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toHasSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x P)) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 P)) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 P)) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x Q)) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 Q)) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 Q)))
+  forall (M : Type.{u2}) (α : Type.{u1}) [_inst_1 : Group.{u2} M] [_inst_2 : MulAction.{u2, u1} M α (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1))] {P : Subgroup.{u2} M _inst_1} {Q : Subgroup.{u2} M _inst_1}, Eq.{succ u1} (Set.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x (Sup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 (Sup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 (Sup.sup.{u2} (Subgroup.{u2} M _inst_1) (SemilatticeSup.toSup.{u2} (Subgroup.{u2} M _inst_1) (Lattice.toSemilatticeSup.{u2} (Subgroup.{u2} M _inst_1) (CompleteLattice.toLattice.{u2} (Subgroup.{u2} M _inst_1) (Subgroup.instCompleteLatticeSubgroup.{u2} M _inst_1)))) P Q))) (Inter.inter.{u1} (Set.{u1} α) (Set.instInterSet.{u1} α) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x P)) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 P)) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 P)) (MulAction.fixedPoints.{u2, u1} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Subgroup.{u2} M _inst_1) (SetLike.instMembership.{u2, u2} (Subgroup.{u2} M _inst_1) M (Subgroup.instSetLikeSubgroup.{u2} M _inst_1)) x Q)) α (Submonoid.toMonoid.{u2} M (DivInvMonoid.toMonoid.{u2} M (Group.toDivInvMonoid.{u2} M _inst_1)) (Subgroup.toSubmonoid.{u2} M _inst_1 Q)) (Subgroup.instMulActionSubtypeMemSubgroupInstMembershipInstSetLikeSubgroupToMonoidToMonoidToDivInvMonoidToSubmonoid.{u2, u1} M _inst_1 α _inst_2 Q)))
 Case conversion may be inaccurate. Consider using '#align fixed_points_subgroup_sup fixedPoints_subgroup_supₓ'. -/
 /-- Fixed points of sup of subgroups is intersection -/
 theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

Introduces a new module, Mathlib.GroupTheory.GroupAction.FixedPoints, which contains some properties of MulAction.fixedBy that wouldn't quite belong to Mathlib.GroupTheory.GroupAction.Basic.

@@ -5,6 +5,7 @@ Authors: Antoine Chambert-Loir
 -/
 import Mathlib.GroupTheory.Subgroup.Actions
 import Mathlib.GroupTheory.GroupAction.Basic
+import Mathlib.GroupTheory.GroupAction.FixedPoints
 
 #align_import group_theory.group_action.fixing_subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
 
@@ -36,6 +37,8 @@ TODO :
 
 * Treat semigroups ?
 
+* add `to_additive` for the various lemmas
+
 -/
 
 
@@ -120,6 +123,14 @@ theorem mem_fixingSubgroup_iff {s : Set α} {m : M} : m ∈ fixingSubgroup M s 
   ⟨fun hg y hy => hg ⟨y, hy⟩, fun h ⟨y, hy⟩ => h y hy⟩
 #align mem_fixing_subgroup_iff mem_fixingSubgroup_iff
 
+theorem mem_fixingSubgroup_iff_subset_fixedBy {s : Set α} {m : M} :
+    m ∈ fixingSubgroup M s ↔ s ⊆ fixedBy α m := by
+  simp_rw [mem_fixingSubgroup_iff, Set.subset_def, mem_fixedBy]
+
+theorem mem_fixingSubgroup_compl_iff_movedBy_subset {s : Set α} {m : M} :
+    m ∈ fixingSubgroup M sᶜ ↔ (fixedBy α m)ᶜ ⊆ s := by
+  rw [mem_fixingSubgroup_iff_subset_fixedBy, Set.compl_subset_comm]
+
 variable (α)
 
 /-- The Galois connection between fixing subgroups and fixed points of a group action -/
@@ -161,4 +172,13 @@ theorem fixedPoints_subgroup_iSup {ι : Sort*} {P : ι → Subgroup M} :
   (fixingSubgroup_fixedPoints_gc M α).u_iInf
 #align fixed_points_subgroup_supr fixedPoints_subgroup_iSup
 
+/-- The orbit of the fixing subgroup of `sᶜ` (ie. the moving subgroup of `s`) is a subset of `s` -/
+theorem orbit_fixingSubgroup_compl_subset {s : Set α} {a : α} (a_in_s : a ∈ s) :
+    MulAction.orbit (fixingSubgroup M sᶜ) a ⊆ s := by
+  intro b b_in_orbit
+  let ⟨⟨g, g_fixing⟩, g_eq⟩ := MulAction.mem_orbit_iff.mp b_in_orbit
+  rw [Submonoid.mk_smul] at g_eq
+  rw [mem_fixingSubgroup_compl_iff_movedBy_subset] at g_fixing
+  rwa [← g_eq, smul_mem_of_set_mem_fixedBy (set_mem_fixedBy_of_movedBy_subset g_fixing)]
+
 end Group

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -43,7 +43,7 @@ section Monoid
 
 open MulAction
 
-variable (M : Type _) {α : Type _} [Monoid M] [MulAction M α]
+variable (M : Type*) {α : Type*} [Monoid M] [MulAction M α]
 
 /-- The submonoid fixing a set under a `MulAction`. -/
 @[to_additive " The additive submonoid fixing a set under an `AddAction`. "]
@@ -84,7 +84,7 @@ theorem fixingSubmonoid_union {s t : Set α} :
 #align fixing_submonoid_union fixingSubmonoid_union
 
 /-- Fixing submonoid of iUnion is intersection -/
-theorem fixingSubmonoid_iUnion {ι : Sort _} {s : ι → Set α} :
+theorem fixingSubmonoid_iUnion {ι : Sort*} {s : ι → Set α} :
     fixingSubmonoid M (⋃ i, s i) = ⨅ i, fixingSubmonoid M (s i) :=
   (fixingSubmonoid_fixedPoints_gc M α).l_iSup
 #align fixing_submonoid_Union fixingSubmonoid_iUnion
@@ -96,7 +96,7 @@ theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
 #align fixed_points_submonoid_sup fixedPoints_submonoid_sup
 
 /-- Fixed points of iSup of submonoids is intersection -/
-theorem fixedPoints_submonoid_iSup {ι : Sort _} {P : ι → Submonoid M} :
+theorem fixedPoints_submonoid_iSup {ι : Sort*} {P : ι → Submonoid M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
   (fixingSubmonoid_fixedPoints_gc M α).u_iInf
 #align fixed_points_submonoid_supr fixedPoints_submonoid_iSup
@@ -107,7 +107,7 @@ section Group
 
 open MulAction
 
-variable (M : Type _) {α : Type _} [Group M] [MulAction M α]
+variable (M : Type*) {α : Type*} [Group M] [MulAction M α]
 
 /-- The subgroup fixing a set under a `MulAction`. -/
 @[to_additive " The additive subgroup fixing a set under an `AddAction`. "]
@@ -144,7 +144,7 @@ theorem fixingSubgroup_union {s t : Set α} :
 #align fixing_subgroup_union fixingSubgroup_union
 
 /-- Fixing subgroup of iUnion is intersection -/
-theorem fixingSubgroup_iUnion {ι : Sort _} {s : ι → Set α} :
+theorem fixingSubgroup_iUnion {ι : Sort*} {s : ι → Set α} :
     fixingSubgroup M (⋃ i, s i) = ⨅ i, fixingSubgroup M (s i) :=
   (fixingSubgroup_fixedPoints_gc M α).l_iSup
 #align fixing_subgroup_Union fixingSubgroup_iUnion
@@ -156,7 +156,7 @@ theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
 #align fixed_points_subgroup_sup fixedPoints_subgroup_sup
 
 /-- Fixed points of iSup of subgroups is intersection -/
-theorem fixedPoints_subgroup_iSup {ι : Sort _} {P : ι → Subgroup M} :
+theorem fixedPoints_subgroup_iSup {ι : Sort*} {P : ι → Subgroup M} :
     fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
   (fixingSubgroup_fixedPoints_gc M α).u_iInf
 #align fixed_points_subgroup_supr fixedPoints_subgroup_iSup

@@ -12,7 +12,7 @@ import Mathlib.GroupTheory.GroupAction.Basic
 
 # Fixing submonoid, fixing subgroup of an action
 
-In the presence of of an action of a monoid or a group,
+In the presence of an action of a monoid or a group,
 this file defines the fixing submonoid or the fixing subgroup,
 and relates it to the set of fixed points via a Galois connection.
 

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,15 +2,12 @@
 Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Antoine Chambert-Loir
-
-! This file was ported from Lean 3 source module group_theory.group_action.fixing_subgroup
-! leanprover-community/mathlib commit f93c11933efbc3c2f0299e47b8ff83e9b539cbf6
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.GroupTheory.Subgroup.Actions
 import Mathlib.GroupTheory.GroupAction.Basic
 
+#align_import group_theory.group_action.fixing_subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
+
 /-!
 
 # Fixing submonoid, fixing subgroup of an action

• Run a non-interactive version of fix-comments.py on all files.
• Go through the diff and manually add/discard/edit chunks.

@@ -21,7 +21,7 @@ and relates it to the set of fixed points via a Galois connection.
 
 ## Main definitions
 
-* `fixingSubmonoid M s` : in the presence of `MulAction M α` (with `monoid M`)
+* `fixingSubmonoid M s` : in the presence of `MulAction M α` (with `Monoid M`)
   it is the `Submonoid M` consisting of elements which fix `s : Set α` pointwise.
 
 * `fixingSubmonoid_fixedPoints_gc M α` is the `GaloisConnection`

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -86,11 +86,11 @@ theorem fixingSubmonoid_union {s t : Set α} :
   (fixingSubmonoid_fixedPoints_gc M α).l_sup
 #align fixing_submonoid_union fixingSubmonoid_union
 
-/-- Fixing submonoid of unionᵢ is intersection -/
-theorem fixingSubmonoid_unionᵢ {ι : Sort _} {s : ι → Set α} :
+/-- Fixing submonoid of iUnion is intersection -/
+theorem fixingSubmonoid_iUnion {ι : Sort _} {s : ι → Set α} :
     fixingSubmonoid M (⋃ i, s i) = ⨅ i, fixingSubmonoid M (s i) :=
-  (fixingSubmonoid_fixedPoints_gc M α).l_supᵢ
-#align fixing_submonoid_Union fixingSubmonoid_unionᵢ
+  (fixingSubmonoid_fixedPoints_gc M α).l_iSup
+#align fixing_submonoid_Union fixingSubmonoid_iUnion
 
 /-- Fixed points of sup of submonoids is intersection -/
 theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
@@ -98,11 +98,11 @@ theorem fixedPoints_submonoid_sup {P Q : Submonoid M} :
   (fixingSubmonoid_fixedPoints_gc M α).u_inf
 #align fixed_points_submonoid_sup fixedPoints_submonoid_sup
 
-/-- Fixed points of supᵢ of submonoids is intersection -/
-theorem fixedPoints_submonoid_supᵢ {ι : Sort _} {P : ι → Submonoid M} :
-    fixedPoints (↥(supᵢ P)) α = ⋂ i, fixedPoints (P i) α :=
-  (fixingSubmonoid_fixedPoints_gc M α).u_infᵢ
-#align fixed_points_submonoid_supr fixedPoints_submonoid_supᵢ
+/-- Fixed points of iSup of submonoids is intersection -/
+theorem fixedPoints_submonoid_iSup {ι : Sort _} {P : ι → Submonoid M} :
+    fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
+  (fixingSubmonoid_fixedPoints_gc M α).u_iInf
+#align fixed_points_submonoid_supr fixedPoints_submonoid_iSup
 
 end Monoid
 
@@ -146,11 +146,11 @@ theorem fixingSubgroup_union {s t : Set α} :
   (fixingSubgroup_fixedPoints_gc M α).l_sup
 #align fixing_subgroup_union fixingSubgroup_union
 
-/-- Fixing subgroup of unionᵢ is intersection -/
-theorem fixingSubgroup_unionᵢ {ι : Sort _} {s : ι → Set α} :
+/-- Fixing subgroup of iUnion is intersection -/
+theorem fixingSubgroup_iUnion {ι : Sort _} {s : ι → Set α} :
     fixingSubgroup M (⋃ i, s i) = ⨅ i, fixingSubgroup M (s i) :=
-  (fixingSubgroup_fixedPoints_gc M α).l_supᵢ
-#align fixing_subgroup_Union fixingSubgroup_unionᵢ
+  (fixingSubgroup_fixedPoints_gc M α).l_iSup
+#align fixing_subgroup_Union fixingSubgroup_iUnion
 
 /-- Fixed points of sup of subgroups is intersection -/
 theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
@@ -158,10 +158,10 @@ theorem fixedPoints_subgroup_sup {P Q : Subgroup M} :
   (fixingSubgroup_fixedPoints_gc M α).u_inf
 #align fixed_points_subgroup_sup fixedPoints_subgroup_sup
 
-/-- Fixed points of supᵢ of subgroups is intersection -/
-theorem fixedPoints_subgroup_supᵢ {ι : Sort _} {P : ι → Subgroup M} :
-    fixedPoints (↥(supᵢ P)) α = ⋂ i, fixedPoints (P i) α :=
-  (fixingSubgroup_fixedPoints_gc M α).u_infᵢ
-#align fixed_points_subgroup_supr fixedPoints_subgroup_supᵢ
+/-- Fixed points of iSup of subgroups is intersection -/
+theorem fixedPoints_subgroup_iSup {ι : Sort _} {P : ι → Subgroup M} :
+    fixedPoints (↥(iSup P)) α = ⋂ i, fixedPoints (P i) α :=
+  (fixingSubgroup_fixedPoints_gc M α).u_iInf
+#align fixed_points_subgroup_supr fixedPoints_subgroup_iSup
 
 end Group

port of group_theory.group_action.fixing_subgroup

Co-authored-by: Johan Commelin <johan@commelin.net>