File: Mathlib/GroupTheory/Submonoid/Centralizer.lean

full_name: AddSubmonoid.centralizer
code: /-- The centralizer of a subset of a monoid `M`. -/
@[to_additive "The centralizer of a subset of an additive monoid."]
def centralizer : Submonoid M 
goal: AddSubmonoid M

full_name: Submonoid.centralizer
code: /-- The centralizer of a subset of a monoid `M`. -/
@[to_additive "The centralizer of a subset of an additive monoid."]
def centralizer : Submonoid M 
goal: Submonoid M

full_name: Submonoid.coe_centralizer
code: @[to_additive (attr := simp, norm_cast)]
theorem coe_centralizer : ↑(centralizer S) = S.centralizer 
goal: ↑(Submonoid.centralizer S) = S.centralizer

full_name: AddSubmonoid.coe_centralizer
code: @[to_additive (attr := simp, norm_cast)]
theorem coe_centralizer : ↑(centralizer S) = S.centralizer 
goal: ↑(AddSubmonoid.centralizer S) = S.addCentralizer

full_name: Submonoid.centralizer_toSubsemigroup
code: theorem centralizer_toSubsemigroup : (centralizer S).toSubsemigroup = Subsemigroup.centralizer S 
goal: (Submonoid.centralizer S).toSubsemigroup = Subsemigroup.centralizer S

full_name: AddSubmonoid.centralizer_toAddSubsemigroup
code: theorem _root_.AddSubmonoid.centralizer_toAddSubsemigroup {M} [AddMonoid M] (S : Set M) :
    (AddSubmonoid.centralizer S).toAddSubsemigroup = AddSubsemigroup.centralizer S 
goal: (AddSubmonoid.centralizer S).toAddSubsemigroup = AddSubsemigroup.centralizer S

full_name: Submonoid.mem_centralizer_iff
code: @[to_additive]
theorem mem_centralizer_iff {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g 
goal: z ∈ Submonoid.centralizer S ↔ ∀ g ∈ S, g * z = z * g

full_name: AddSubmonoid.mem_centralizer_iff
code: @[to_additive]
theorem mem_centralizer_iff {z : M} : z ∈ centralizer S ↔ ∀ g ∈ S, g * z = z * g 
goal: z ∈ AddSubmonoid.centralizer S ↔ ∀ g ∈ S, g + z = z + g

full_name: Submonoid.center_le_centralizer
code: @[to_additive]
theorem center_le_centralizer (s) : center M ≤ centralizer s 
goal: Submonoid.center M ≤ Submonoid.centralizer s

full_name: AddSubmonoid.center_le_centralizer
code: @[to_additive]
theorem center_le_centralizer (s) : center M ≤ centralizer s 
goal: AddSubmonoid.center M ≤ AddSubmonoid.centralizer s

full_name: AddSubmonoid.decidableMemCentralizer
code: @[to_additive]
instance decidableMemCentralizer (a) [Decidable <| ∀ b ∈ S, b * a = a * b] :
    Decidable (a ∈ centralizer S) 
goal: Decidable (a ∈ AddSubmonoid.centralizer S)

full_name: Submonoid.decidableMemCentralizer
code: @[to_additive]
instance decidableMemCentralizer (a) [Decidable <| ∀ b ∈ S, b * a = a * b] :
    Decidable (a ∈ centralizer S) 
goal: Decidable (a ∈ Submonoid.centralizer S)

full_name: AddSubmonoid.centralizer_le
code: @[to_additive]
theorem centralizer_le (h : S ⊆ T) : centralizer T ≤ centralizer S 
goal: AddSubmonoid.centralizer T ≤ AddSubmonoid.centralizer S

full_name: Submonoid.centralizer_le
code: @[to_additive]
theorem centralizer_le (h : S ⊆ T) : centralizer T ≤ centralizer S 
goal: Submonoid.centralizer T ≤ Submonoid.centralizer S

full_name: AddSubmonoid.centralizer_eq_top_iff_subset
code: @[to_additive (attr := simp)]
theorem centralizer_eq_top_iff_subset {s : Set M} : centralizer s = ⊤ ↔ s ⊆ center M 
goal: AddSubmonoid.centralizer s = ⊤ ↔ s ⊆ ↑(AddSubmonoid.center M)

full_name: Submonoid.centralizer_eq_top_iff_subset
code: @[to_additive (attr := simp)]
theorem centralizer_eq_top_iff_subset {s : Set M} : centralizer s = ⊤ ↔ s ⊆ center M 
goal: Submonoid.centralizer s = ⊤ ↔ s ⊆ ↑(Submonoid.center M)

full_name: Submonoid.centralizer_univ
code: @[to_additive (attr := simp)]
theorem centralizer_univ : centralizer Set.univ = center M 
goal: Submonoid.centralizer Set.univ = Submonoid.center M

full_name: AddSubmonoid.centralizer_univ
code: @[to_additive (attr := simp)]
theorem centralizer_univ : centralizer Set.univ = center M 
goal: AddSubmonoid.centralizer Set.univ = AddSubmonoid.center M

full_name: AddSubmonoid.le_centralizer_centralizer
code: @[to_additive]
lemma le_centralizer_centralizer {s : Submonoid M} : s ≤ centralizer (centralizer (s : Set M)) 
goal: s ≤ AddSubmonoid.centralizer ↑(AddSubmonoid.centralizer ↑s)

full_name: Submonoid.le_centralizer_centralizer
code: @[to_additive]
lemma le_centralizer_centralizer {s : Submonoid M} : s ≤ centralizer (centralizer (s : Set M)) 
goal: s ≤ Submonoid.centralizer ↑(Submonoid.centralizer ↑s)

full_name: AddSubmonoid.centralizer_centralizer_centralizer
code: @[to_additive (attr := simp)]
lemma centralizer_centralizer_centralizer {s : Set M} :
    centralizer s.centralizer.centralizer = centralizer s 
goal: AddSubmonoid.centralizer s.addCentralizer.addCentralizer = AddSubmonoid.centralizer s

full_name: Submonoid.centralizer_centralizer_centralizer
code: @[to_additive (attr := simp)]
lemma centralizer_centralizer_centralizer {s : Set M} :
    centralizer s.centralizer.centralizer = centralizer s 
goal: Submonoid.centralizer s.centralizer.centralizer = Submonoid.centralizer s