File: Mathlib/Algebra/Pointwise/Stabilizer.lean

full_name: AddAction.stabilizer_empty
code: @[to_additive (attr := simp)]
lemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ 
goal: AddAction.stabilizer G ∅ = ⊤

full_name: MulAction.stabilizer_empty
code: @[to_additive (attr := simp)]
lemma stabilizer_empty : stabilizer G (∅ : Set α) = ⊤ 
goal: MulAction.stabilizer G ∅ = ⊤

full_name: MulAction.stabilizer_univ
code: @[to_additive (attr := simp)]
lemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ 
goal: MulAction.stabilizer G Set.univ = ⊤

full_name: AddAction.stabilizer_univ
code: @[to_additive (attr := simp)]
lemma stabilizer_univ : stabilizer G (Set.univ : Set α) = ⊤ 
goal: AddAction.stabilizer G Set.univ = ⊤

full_name: AddAction.stabilizer_singleton
code: @[to_additive (attr := simp)]
lemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b 
goal: AddAction.stabilizer G {b} = AddAction.stabilizer G b

full_name: MulAction.stabilizer_singleton
code: @[to_additive (attr := simp)]
lemma stabilizer_singleton (b : α) : stabilizer G ({b} : Set α) = stabilizer G b 
goal: MulAction.stabilizer G {b} = MulAction.stabilizer G b

full_name: AddAction.mem_stabilizer_set
code: @[to_additive]
lemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
goal: a ∈ AddAction.stabilizer G s ↔ ∀ (b : α), a +ᵥ b ∈ s ↔ b ∈ s

full_name: MulAction.mem_stabilizer_set
code: @[to_additive]
lemma mem_stabilizer_set {s : Set α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
goal: a ∈ MulAction.stabilizer G s ↔ ∀ (b : α), a • b ∈ s ↔ b ∈ s

full_name: AddAction.map_stabilizer_le
code: @[to_additive]
lemma map_stabilizer_le (f : G →* H) (s : Set G) :
    (stabilizer G s).map f ≤ stabilizer H (f '' s) 
goal: AddSubgroup.map f (AddAction.stabilizer G s) ≤ AddAction.stabilizer H (⇑f '' s)

full_name: MulAction.map_stabilizer_le
code: @[to_additive]
lemma map_stabilizer_le (f : G →* H) (s : Set G) :
    (stabilizer G s).map f ≤ stabilizer H (f '' s) 
goal: Subgroup.map f (MulAction.stabilizer G s) ≤ MulAction.stabilizer H (⇑f '' s)

full_name: AddAction.stabilizer_add_self
code: @[to_additive (attr := simp)]
lemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s 
goal: ↑(AddAction.stabilizer G s) + s = s

full_name: MulAction.stabilizer_mul_self
code: @[to_additive (attr := simp)]
lemma stabilizer_mul_self (s : Set G) : (stabilizer G s : Set G) * s = s 
goal: ↑(MulAction.stabilizer G s) * s = s

full_name: MulAction.stabilizer_subgroup
code: @[to_additive (attr := simp)]
lemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s 
goal: MulAction.stabilizer G ↑s = s

full_name: AddAction.stabilizer_addSubgroup
code: @[to_additive (attr := simp)]
lemma stabilizer_subgroup (s : Subgroup G) : stabilizer G (s : Set G) = s 
goal: AddAction.stabilizer G ↑s = s

full_name: MulAction.stabilizer_op_subgroup
code: @[to_additive (attr := simp)]
lemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op 
goal: MulAction.stabilizer Gᵐᵒᵖ ↑s = s.op

full_name: AddAction.stabilizer_op_addSubgroup
code: @[to_additive (attr := simp)]
lemma stabilizer_op_subgroup (s : Subgroup G) : stabilizer Gᵐᵒᵖ (s : Set G) = s.op 
goal: AddAction.stabilizer Gᵃᵒᵖ ↑s = s.op

full_name: AddAction.stabilizer_addSubgroup_op
code: @[to_additive (attr := simp)]
lemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop 
goal: AddAction.stabilizer G ↑s = s.unop

full_name: MulAction.stabilizer_subgroup_op
code: @[to_additive (attr := simp)]
lemma stabilizer_subgroup_op (s : Subgroup Gᵐᵒᵖ) : stabilizer G (s : Set Gᵐᵒᵖ) = s.unop 
goal: MulAction.stabilizer G ↑s = s.unop

full_name: MulAction.stabilizer_coe_finset
code: @[to_additive (attr := simp)]
lemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s 
goal: MulAction.stabilizer G ↑s = MulAction.stabilizer G s

full_name: AddAction.stabilizer_coe_finset
code: @[to_additive (attr := simp)]
lemma stabilizer_coe_finset (s : Finset α) : stabilizer G (s : Set α) = stabilizer G s 
goal: AddAction.stabilizer G ↑s = AddAction.stabilizer G s

full_name: AddAction.stabilizer_finset_empty
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ 
goal: AddAction.stabilizer G ∅ = ⊤

full_name: MulAction.stabilizer_finset_empty
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_empty : stabilizer G (∅ : Finset α) = ⊤ 
goal: MulAction.stabilizer G ∅ = ⊤

full_name: AddAction.stabilizer_finset_univ
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ 
goal: AddAction.stabilizer G Finset.univ = ⊤

full_name: MulAction.stabilizer_finset_univ
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_univ [Fintype α] : stabilizer G (Finset.univ : Finset α) = ⊤ 
goal: MulAction.stabilizer G Finset.univ = ⊤

full_name: AddAction.stabilizer_finset_singleton
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b 
goal: AddAction.stabilizer G {b} = AddAction.stabilizer G b

full_name: MulAction.stabilizer_finset_singleton
code: @[to_additive (attr := simp)]
lemma stabilizer_finset_singleton (b : α) : stabilizer G ({b} : Finset α) = stabilizer G b 
goal: MulAction.stabilizer G {b} = MulAction.stabilizer G b

full_name: MulAction.mem_stabilizer_finset
code: @[to_additive]
lemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
goal: a ∈ MulAction.stabilizer G s ↔ ∀ (b : α), a • b ∈ s ↔ b ∈ s

full_name: AddAction.mem_stabilizer_finset
code: @[to_additive]
lemma mem_stabilizer_finset {s : Finset α} : a ∈ stabilizer G s ↔ ∀ b, a • b ∈ s ↔ b ∈ s 
goal: a ∈ AddAction.stabilizer G s ↔ ∀ (b : α), a +ᵥ b ∈ s ↔ b ∈ s

full_name: MulAction.mem_stabilizer_finset_iff_subset_smul_finset
code: @[to_additive]
lemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
goal: a ∈ MulAction.stabilizer G s ↔ s ⊆ a • s

full_name: AddAction.mem_stabilizer_finset_iff_subset_vadd_finset
code: @[to_additive]
lemma mem_stabilizer_finset_iff_subset_smul_finset {s : Finset α} :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
goal: a ∈ AddAction.stabilizer G s ↔ s ⊆ a +ᵥ s

full_name: AddAction.mem_stabilizer_finset_iff_vadd_finset_subset
code: @[to_additive]
lemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
goal: a ∈ AddAction.stabilizer G s ↔ a +ᵥ s ⊆ s

full_name: MulAction.mem_stabilizer_finset_iff_smul_finset_subset
code: @[to_additive]
lemma mem_stabilizer_finset_iff_smul_finset_subset {s : Finset α} :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
goal: a ∈ MulAction.stabilizer G s ↔ a • s ⊆ s

full_name: MulAction.mem_stabilizer_finset'
code: @[to_additive]
lemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
goal: a ∈ MulAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a • b ∈ s

full_name: AddAction.mem_stabilizer_finset'
code: @[to_additive]
lemma mem_stabilizer_finset' {s : Finset α} : a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
goal: a ∈ AddAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a +ᵥ b ∈ s

full_name: AddAction.mem_stabilizer_set_iff_subset_vadd_set
code: @[to_additive]
lemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
goal: a ∈ AddAction.stabilizer G s ↔ s ⊆ a +ᵥ s

full_name: MulAction.mem_stabilizer_set_iff_subset_smul_set
code: @[to_additive]
lemma mem_stabilizer_set_iff_subset_smul_set {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ s ⊆ a • s 
goal: a ∈ MulAction.stabilizer G s ↔ s ⊆ a • s

full_name: AddAction.mem_stabilizer_set_iff_vadd_set_subset
code: @[to_additive]
lemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
goal: a ∈ AddAction.stabilizer G s ↔ a +ᵥ s ⊆ s

full_name: MulAction.mem_stabilizer_set_iff_smul_set_subset
code: @[to_additive]
lemma mem_stabilizer_set_iff_smul_set_subset {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ a • s ⊆ s 
goal: a ∈ MulAction.stabilizer G s ↔ a • s ⊆ s

full_name: AddAction.mem_stabilizer_set'
code: @[to_additive]
lemma mem_stabilizer_set' {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
goal: a ∈ AddAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a +ᵥ b ∈ s

full_name: MulAction.mem_stabilizer_set'
code: @[to_additive]
lemma mem_stabilizer_set' {s : Set α} (hs : s.Finite) :
    a ∈ stabilizer G s ↔ ∀ ⦃b⦄, b ∈ s → a • b ∈ s 
goal: a ∈ MulAction.stabilizer G s ↔ ∀ ⦃b : α⦄, b ∈ s → a • b ∈ s

full_name: MulAction.mul_stabilizer_self
code: @[to_additive (attr := simp)]
lemma mul_stabilizer_self : s * stabilizer G s = s 
goal: s * ↑(MulAction.stabilizer G s) = s

full_name: AddAction.add_stabilizer_self
code: @[to_additive (attr := simp)]
lemma mul_stabilizer_self : s * stabilizer G s = s 
goal: s + ↑(AddAction.stabilizer G s) = s

full_name: MulAction._aux_Mathlib_Algebra_Pointwise_Stabilizer___macroRules_MulAction_termQ_1
code: local notation " Q " => G ⧸ stabilizer G s
goal: Lean.Macro

full_name: MulAction._aux_Mathlib_Algebra_Pointwise_Stabilizer___macroRules_MulAction_termQ_2
code: local notation " q " => ((↑) : G → Q)
goal: Lean.Macro

full_name: MulAction.stabilizer_image_coe_quotient
code: @[to_additive]
lemma stabilizer_image_coe_quotient : stabilizer Q (q '' s) = ⊥ 
goal: MulAction.stabilizer (G ⧸ MulAction.stabilizer G s) (QuotientGroup.mk '' s) = ⊥

full_name: AddAction.stabilizer_image_coe_quotient
code: @[to_additive]
lemma stabilizer_image_coe_quotient : stabilizer Q (q '' s) = ⊥ 
goal: AddAction.stabilizer (G ⧸ AddAction.stabilizer G s) (QuotientAddGroup.mk '' s) = ⊥