File: Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean

full_name: fixingAddSubmonoid
code: /-- The submonoid fixing a set under a `MulAction`. -/
@[to_additive " The additive submonoid fixing a set under an `AddAction`. "]
def fixingSubmonoid (s : Set α) : Submonoid M 
goal: AddSubmonoid M

full_name: fixingSubmonoid
code: /-- The submonoid fixing a set under a `MulAction`. -/
@[to_additive " The additive submonoid fixing a set under an `AddAction`. "]
def fixingSubmonoid (s : Set α) : Submonoid M 
goal: Submonoid M

full_name: mem_fixingSubmonoid_iff
code: theorem mem_fixingSubmonoid_iff {s : Set α} {m : M} :
    m ∈ fixingSubmonoid M s ↔ ∀ y ∈ s, m • y = y 
goal: m ∈ fixingSubmonoid M s ↔ ∀ y ∈ s, m • y = y

full_name: fixingSubmonoid_fixedPoints_gc
code: /-- The Galois connection between fixing submonoids and fixed points of a monoid action -/
theorem fixingSubmonoid_fixedPoints_gc :
    GaloisConnection (OrderDual.toDual ∘ fixingSubmonoid M)
      ((fun P : Submonoid M => fixedPoints P α) ∘ OrderDual.ofDual) 
goal: GaloisConnection (⇑OrderDual.toDual ∘ fixingSubmonoid M) ((fun P => MulAction.fixedPoints (↥P) α) ∘ ⇑OrderDual.ofDual)

full_name: fixingSubmonoid_antitone
code: theorem fixingSubmonoid_antitone : Antitone fun s : Set α => fixingSubmonoid M s 
goal: Antitone fun s => fixingSubmonoid M s

full_name: fixedPoints_antitone
code: theorem fixedPoints_antitone : Antitone fun P : Submonoid M => fixedPoints P α 
goal: Antitone fun P => MulAction.fixedPoints (↥P) α

full_name: fixingSubmonoid_union
code: /-- Fixing submonoid of union is intersection -/
theorem fixingSubmonoid_union {s t : Set α} :
    fixingSubmonoid M (s ∪ t) = fixingSubmonoid M s ⊓ fixingSubmonoid M t 
goal: fixingSubmonoid M (s ∪ t) = fixingSubmonoid M s ⊓ fixingSubmonoid M t

full_name: fixingSubmonoid_iUnion
code: /-- 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) 
goal: fixingSubmonoid M (⋃ i, s i) = ⨅ i, fixingSubmonoid M (s i)

full_name: fixedPoints_submonoid_sup
code: /-- 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 
goal: MulAction.fixedPoints (↥(P ⊔ Q)) α = MulAction.fixedPoints (↥P) α ∩ MulAction.fixedPoints (↥Q) α

full_name: fixedPoints_submonoid_iSup
code: /-- 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 
goal: MulAction.fixedPoints (↥(iSup P)) α = ⋂ i, MulAction.fixedPoints (↥(P i)) α

full_name: fixingSubgroup
code: /-- The subgroup fixing a set under a `MulAction`. -/
@[to_additive " The additive subgroup fixing a set under an `AddAction`. "]
def fixingSubgroup (s : Set α) : Subgroup M 
goal: Subgroup M

full_name: fixingAddSubgroup
code: /-- The subgroup fixing a set under a `MulAction`. -/
@[to_additive " The additive subgroup fixing a set under an `AddAction`. "]
def fixingSubgroup (s : Set α) : Subgroup M 
goal: AddSubgroup M

full_name: mem_fixingSubgroup_iff
code: theorem mem_fixingSubgroup_iff {s : Set α} {m : M} : m ∈ fixingSubgroup M s ↔ ∀ y ∈ s, m • y = y 
goal: m ∈ fixingSubgroup M s ↔ ∀ y ∈ s, m • y = y

full_name: mem_fixingSubgroup_iff_subset_fixedBy
code: theorem mem_fixingSubgroup_iff_subset_fixedBy {s : Set α} {m : M} :
    m ∈ fixingSubgroup M s ↔ s ⊆ fixedBy α m 
goal: m ∈ fixingSubgroup M s ↔ s ⊆ MulAction.fixedBy α m

full_name: mem_fixingSubgroup_compl_iff_movedBy_subset
code: theorem mem_fixingSubgroup_compl_iff_movedBy_subset {s : Set α} {m : M} :
    m ∈ fixingSubgroup M sᶜ ↔ (fixedBy α m)ᶜ ⊆ s 
goal: m ∈ fixingSubgroup M sᶜ ↔ (MulAction.fixedBy α m)ᶜ ⊆ s

full_name: fixingSubgroup_fixedPoints_gc
code: /-- The Galois connection between fixing subgroups and fixed points of a group action -/
theorem fixingSubgroup_fixedPoints_gc :
    GaloisConnection (OrderDual.toDual ∘ fixingSubgroup M)
      ((fun P : Subgroup M => fixedPoints P α) ∘ OrderDual.ofDual) 
goal: GaloisConnection (⇑OrderDual.toDual ∘ fixingSubgroup M) ((fun P => MulAction.fixedPoints (↥P) α) ∘ ⇑OrderDual.ofDual)

full_name: fixingSubgroup_antitone
code: theorem fixingSubgroup_antitone : Antitone (fixingSubgroup M : Set α → Subgroup M) 
goal: Antitone (fixingSubgroup M)

full_name: fixedPoints_subgroup_antitone
code: theorem fixedPoints_subgroup_antitone : Antitone fun P : Subgroup M => fixedPoints P α 
goal: Antitone fun P => MulAction.fixedPoints (↥P) α

full_name: fixingSubgroup_union
code: /-- Fixing subgroup of union is intersection -/
theorem fixingSubgroup_union {s t : Set α} :
    fixingSubgroup M (s ∪ t) = fixingSubgroup M s ⊓ fixingSubgroup M t 
goal: fixingSubgroup M (s ∪ t) = fixingSubgroup M s ⊓ fixingSubgroup M t

full_name: fixingSubgroup_iUnion
code: /-- 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) 
goal: fixingSubgroup M (⋃ i, s i) = ⨅ i, fixingSubgroup M (s i)

full_name: fixedPoints_subgroup_sup
code: /-- 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 
goal: MulAction.fixedPoints (↥(P ⊔ Q)) α = MulAction.fixedPoints (↥P) α ∩ MulAction.fixedPoints (↥Q) α

full_name: fixedPoints_subgroup_iSup
code: /-- 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 
goal: MulAction.fixedPoints (↥(iSup P)) α = ⋂ i, MulAction.fixedPoints (↥(P i)) α

full_name: orbit_fixingSubgroup_compl_subset
code: /-- 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 
goal: MulAction.orbit (↥(fixingSubgroup M sᶜ)) a ⊆ s