Loogle!

Result

Found 2744 declarations mentioning SMulCommClass. Of these, only the first 200 are shown.

SMulCommClass 📋 Mathlib.Algebra.Group.Action.Defs
(M : Type u_9) (N : Type u_10) (α : Type u_11) [SMul M α] [SMul N α] : Prop
SMulCommClass.symm 📋 Mathlib.Algebra.Group.Action.Defs
(M : Type u_9) (N : Type u_10) (α : Type u_11) [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass N M α
SMul.comp.smulCommClass 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul β α] [SMulCommClass M β α] (g : N → M) : SMulCommClass N β α
SMul.comp.smulCommClass' 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul β α] [SMulCommClass β M α] (g : N → M) : SMulCommClass β N α
SMulCommClass.op_left 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} [SMul M α] [SMul Mᵐᵒᵖ α] [IsCentralScalar M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass Mᵐᵒᵖ N α
SMulCommClass.op_right 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} [SMul M α] [SMul N α] [SMul Nᵐᵒᵖ α] [IsCentralScalar N α] [SMulCommClass M N α] : SMulCommClass M Nᵐᵒᵖ α
instSMulCommClassOfIsScalarTower 📋 Mathlib.Algebra.Group.Action.Defs
{R : Type u_9} {M : Type u_10} [CommMonoid M] [SMul R M] [IsScalarTower R M M] : SMulCommClass R M M
smulCommClass_self 📋 Mathlib.Algebra.Group.Action.Defs
(M : Type u_9) (α : Type u_10) [CommMonoid M] [MulAction M α] : SMulCommClass M M α
Commute.smul_left 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {α : Type u_5} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {a b : α} (h : Commute a b) (r : M) : Commute (r • a) b
Commute.smul_right 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {α : Type u_5} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {a b : α} (h : Commute a b) (r : M) : Commute a (r • b)
SemiconjBy.smul_left 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {α : Type u_5} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {x a b : α} (h : SemiconjBy x a b) (r : M) : SemiconjBy (r • x) a b
SMulCommClass.of_commMonoid 📋 Mathlib.Algebra.Group.Action.Defs
(A : Type u_9) (B : Type u_10) (G : Type u_11) [CommMonoid G] [SMul A G] [SMul B G] [IsScalarTower A G G] [IsScalarTower B G G] : SMulCommClass A B G
mul_smul_comm 📋 Mathlib.Algebra.Group.Action.Defs
{α : Type u_5} {β : Type u_6} [Mul β] [SMul α β] [SMulCommClass α β β] (s : α) (x y : β) : x * s • y = s • (x * y)
SMulCommClass.mk 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_9} {N : Type u_10} {α : Type u_11} [SMul M α] [SMul N α] (smul_comm : ∀ (m : M) (n : N) (a : α), m • n • a = n • m • a) : SMulCommClass M N α
SMulCommClass.smul_comm 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_9} {N : Type u_10} {α : Type u_11} {inst✝ : SMul M α} {inst✝¹ : SMul N α} [self : SMulCommClass M N α] (m : M) (n : N) (a : α) : m • n • a = n • m • a
SemiconjBy.smul_right 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {α : Type u_5} [SMul M α] [Mul α] [SMulCommClass M α α] [IsScalarTower M α α] {x a b : α} (h : SemiconjBy x a b) (r : M) : SemiconjBy x (r • a) (r • b)
mul_smul_one 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_9} {N : Type u_10} [MulOneClass N] [SMul M N] [SMulCommClass M N N] (x : M) (y : N) : y * x • 1 = x • y
SMulCommClass.of_mul_smul_one 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_9} {N : Type u_10} [Monoid N] [SMul M N] (H : ∀ (x : M) (y : N), y * x • 1 = x • y) : SMulCommClass M N N
IsScalarTower.of_commMonoid 📋 Mathlib.Algebra.Group.Action.Defs
(R₁ : Type u_9) (R : Type u_10) [Monoid R₁] [CommMonoid R] [MulAction R₁ R] [SMulCommClass R₁ R R] : IsScalarTower R₁ R R
isScalarTower_iff_smulCommClass_of_commMonoid 📋 Mathlib.Algebra.Group.Action.Defs
(R₁ : Type u_9) (R : Type u_10) [Monoid R₁] [CommMonoid R] [MulAction R₁ R] : SMulCommClass R₁ R R ↔ IsScalarTower R₁ R R
Function.Injective.smulCommClass 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul N α] [SMul M β] [SMul N β] [SMulCommClass M N β] {f : α → β} (hf : Function.Injective f) (h₁ : ∀ (c : M) (x : α), f (c • x) = c • f x) (h₂ : ∀ (c : N) (x : α), f (c • x) = c • f x) : SMulCommClass M N α
Function.Surjective.smulCommClass 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul N α] [SMul M β] [SMul N β] [SMulCommClass M N α] {f : α → β} (hf : Function.Surjective f) (h₁ : ∀ (c : M) (x : α), f (c • x) = c • f x) (h₂ : ∀ (c : N) (x : α), f (c • x) = c • f x) : SMulCommClass M N β
mul_smul_mul_comm 📋 Mathlib.Algebra.Group.Action.Defs
{α : Type u_5} {β : Type u_6} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a b : α) (c d : β) : (a * b) • (c * d) = a • c * b • d
smul_mul_smul 📋 Mathlib.Algebra.Group.Action.Defs
{α : Type u_5} {β : Type u_6} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a • b * c • d = (a * c) • (b * d)
smul_mul_smul_comm 📋 Mathlib.Algebra.Group.Action.Defs
{α : Type u_5} {β : Type u_6} [Mul α] [Mul β] [SMul α β] [IsScalarTower α β β] [IsScalarTower α α β] [SMulCommClass α β β] (a : α) (b : β) (c : α) (d : β) : a • b * c • d = (a * c) • (b * d)
smul_smul_smul_comm 📋 Mathlib.Algebra.Group.Action.Defs
{α : Type u_5} {β : Type u_6} {γ : Type u_7} {δ : Type u_8} [SMul α β] [SMul α γ] [SMul β δ] [SMul α δ] [SMul γ δ] [IsScalarTower α β δ] [IsScalarTower α γ δ] [SMulCommClass β γ δ] (a : α) (b : β) (c : γ) (d : δ) : (a • b) • c • d = (a • c) • b • d
Commute.smul_left_iff 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] {g : G} [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {a b : H} : Commute (g • a) b ↔ Commute a b
Commute.smul_right_iff 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] {g : G} [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {a b : H} : Commute a (g • b) ↔ Commute a b
SemiconjBy.smul_left_iff 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {a b x : H} {r : G} : SemiconjBy (r • x) a b ↔ SemiconjBy x a b
smul_pow 📋 Mathlib.Algebra.Group.Action.Defs
{M : Type u_1} {N : Type u_2} [Monoid M] [Monoid N] [MulAction M N] [IsScalarTower M N N] [SMulCommClass M N N] (r : M) (x : N) (n : ℕ) : (r • x) ^ n = r ^ n • x ^ n
SemiconjBy.smul_right_iff 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] [Mul H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] {a b x : H} {r : G} : SemiconjBy x (r • a) (r • b) ↔ SemiconjBy x a b
smul_inv 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] [Group H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] (g : G) (a : H) : (g • a)⁻¹ = g⁻¹ • a⁻¹
smul_zpow 📋 Mathlib.Algebra.Group.Action.Defs
{G : Type u_3} {H : Type u_4} [Group G] [Group H] [MulAction G H] [SMulCommClass G H H] [IsScalarTower G H H] (g : G) (a : H) (n : ℤ) : (g • a) ^ n = g ^ n • a ^ n
Function.smulCommClass 📋 Mathlib.Algebra.Group.Action.Pi
{ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : Type u_7} [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass M N (ι → α)
Pi.smulCommClass 📋 Mathlib.Algebra.Group.Action.Pi
{ι : Type u_1} {M : Type u_2} {N : Type u_3} {α : ι → Type u_4} [(i : ι) → SMul M (α i)] [(i : ι) → SMul N (α i)] [∀ (i : ι), SMulCommClass M N (α i)] : SMulCommClass M N ((i : ι) → α i)
Pi.smulCommClass' 📋 Mathlib.Algebra.Group.Action.Pi
{ι : Type u_1} {M : Type u_2} {α : ι → Type u_4} {β : ι → Type u_5} [(i : ι) → SMul M (β i)] [(i : ι) → SMul (α i) (β i)] [∀ (i : ι), SMulCommClass M (α i) (β i)] : SMulCommClass M ((i : ι) → α i) ((i : ι) → β i)
Pi.smulCommClass'' 📋 Mathlib.Algebra.Group.Action.Pi
{ι : Type u_1} {α : ι → Type u_4} {β : ι → Type u_5} {γ : ι → Type u_6} [(i : ι) → SMul (β i) (γ i)] [(i : ι) → SMul (α i) (γ i)] [∀ (i : ι), SMulCommClass (α i) (β i) (γ i)] : SMulCommClass ((i : ι) → α i) ((i : ι) → β i) ((i : ι) → γ i)
Semigroup.opposite_smulCommClass 📋 Mathlib.Algebra.Group.Action.Opposite
{α : Type u_3} [Semigroup α] : SMulCommClass αᵐᵒᵖ α α
Semigroup.opposite_smulCommClass' 📋 Mathlib.Algebra.Group.Action.Opposite
{α : Type u_3} [Semigroup α] : SMulCommClass α αᵐᵒᵖ α
SMulCommClass.opposite_mid 📋 Mathlib.Algebra.Group.Action.Opposite
{M : Type u_5} {N : Type u_6} [Mul N] [SMul M N] [IsScalarTower M N N] : SMulCommClass M Nᵐᵒᵖ N
IsScalarTower.opposite_mid 📋 Mathlib.Algebra.Group.Action.Opposite
{M : Type u_5} {N : Type u_6} [Mul N] [SMul M N] [SMulCommClass M N N] : IsScalarTower M Nᵐᵒᵖ N
MulOpposite.instSMulCommClass 📋 Mathlib.Algebra.Group.Action.Opposite
{M : Type u_1} {N : Type u_2} {α : Type u_3} [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass M N αᵐᵒᵖ
AddMonoid.nat_smulCommClass 📋 Mathlib.Algebra.GroupWithZero.Action.Defs
{M : Type u_12} {A : Type u_13} [AddMonoid A] [DistribSMul M A] : SMulCommClass ℕ M A
AddMonoid.nat_smulCommClass' 📋 Mathlib.Algebra.GroupWithZero.Action.Defs
{M : Type u_12} {A : Type u_13} [AddMonoid A] [DistribSMul M A] : SMulCommClass M ℕ A
AddGroup.int_smulCommClass 📋 Mathlib.Algebra.GroupWithZero.Action.Defs
{M : Type u_1} {A : Type u_7} [AddGroup A] [DistribSMul M A] : SMulCommClass ℤ M A
AddGroup.int_smulCommClass' 📋 Mathlib.Algebra.GroupWithZero.Action.Defs
{M : Type u_1} {A : Type u_7} [AddGroup A] [DistribSMul M A] : SMulCommClass M ℤ A
smul_inv₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Defs
{G₀ : Type u_4} {G₀' : Type u_5} [GroupWithZero G₀] [GroupWithZero G₀'] [MulActionWithZero G₀ G₀'] [SMulCommClass G₀ G₀' G₀'] [IsScalarTower G₀ G₀' G₀'] (c : G₀) (x : G₀') : (c • x)⁻¹ = c⁻¹ • x⁻¹
Units.smulCommClass_left 📋 Mathlib.Algebra.Group.Action.Units
{M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid M] [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass Mˣ N α
Units.smulCommClass_right 📋 Mathlib.Algebra.Group.Action.Units
{M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid N] [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass M Nˣ α
Units.mulAction' 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] : MulAction G Mˣ
IsUnit.smul 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] {m : M} (g : G) (h : IsUnit m) : IsUnit (g • m)
Units.isScalarTower'_left 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {M : Type u_3} {α : Type u_5} [Group G] [Monoid M] [MulAction G M] [SMul M α] [SMul G α] [SMulCommClass G M M] [IsScalarTower G M M] [IsScalarTower G M α] : IsScalarTower G Mˣ α
Units.val_smul 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] (g : G) (m : Mˣ) : ↑(g • m) = g • ↑m
Units.smul_inv 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {M : Type u_3} [Group G] [Monoid M] [MulAction G M] [SMulCommClass G M M] [IsScalarTower G M M] (g : G) (m : Mˣ) : (g • m)⁻¹ = g⁻¹ • m⁻¹
Units.smulCommClass' 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {H : Type u_2} {M : Type u_3} [Group G] [Group H] [Monoid M] [MulAction G M] [SMulCommClass G M M] [MulAction H M] [SMulCommClass H M M] [IsScalarTower G M M] [IsScalarTower H M M] [SMulCommClass G H M] : SMulCommClass G H Mˣ
Units.isScalarTower' 📋 Mathlib.Algebra.Group.Action.Units
{G : Type u_1} {H : Type u_2} {M : Type u_3} [SMul G H] [Group G] [Group H] [Monoid M] [MulAction G M] [SMulCommClass G M M] [MulAction H M] [SMulCommClass H M M] [IsScalarTower G M M] [IsScalarTower H M M] [IsScalarTower G H M] : IsScalarTower G H Mˣ
isUnit_smul_iff 📋 Mathlib.Algebra.Group.Action.Basic
{α : Type u_5} {β : Type u_6} [Group α] [Monoid β] [MulAction α β] [SMulCommClass α β β] [IsScalarTower α β β] (g : α) (m : β) : IsUnit (g • m) ↔ IsUnit m
Additive.vaddCommClass 📋 Mathlib.Algebra.Group.Action.TypeTags
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : VAddCommClass (Additive α) (Additive β) γ
Multiplicative.smulCommClass 📋 Mathlib.Algebra.Group.Action.TypeTags
{α : Type u_1} {β : Type u_2} {γ : Type u_3} [VAdd α γ] [VAdd β γ] [VAddCommClass α β γ] : SMulCommClass (Multiplicative α) (Multiplicative β) γ
Submonoid.instSMulCommClassSubtypeMem 📋 Mathlib.Algebra.Group.Submonoid.MulAction
{M' : Type u_1} {α : Type u_2} {β : Type u_3} {S' : Type u_4} [SetLike S' M'] (s : S') [SMul M' β] [SMul α β] [SMulCommClass M' α β] : SMulCommClass (↥s) α β
Submonoid.instSMulCommClassSubtypeMem_1 📋 Mathlib.Algebra.Group.Submonoid.MulAction
{M' : Type u_1} {α : Type u_2} {β : Type u_3} {S' : Type u_4} [SetLike S' M'] (s : S') [SMul α β] [SMul M' β] [SMulCommClass α M' β] : SMulCommClass α (↥s) β
Submonoid.smulCommClass_left 📋 Mathlib.Algebra.Group.Submonoid.MulAction
{M' : Type u_1} {α : Type u_2} {β : Type u_3} [MulOneClass M'] [SMul M' β] [SMul α β] [SMulCommClass M' α β] (S : Submonoid M') : SMulCommClass (↥S) α β
Submonoid.smulCommClass_right 📋 Mathlib.Algebra.Group.Submonoid.MulAction
{M' : Type u_1} {α : Type u_2} {β : Type u_3} [MulOneClass M'] [SMul α β] [SMul M' β] [SMulCommClass α M' β] (S : Submonoid M') : SMulCommClass α (↥S) β
Set.smulCommClass_set 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : SMulCommClass α β (Set γ)
Set.smulCommClass_set' 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : SMulCommClass α (Set β) (Set γ)
Set.smulCommClass_set'' 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : SMulCommClass (Set α) β (Set γ)
Set.smulCommClass 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
{α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α γ] [SMul β γ] [SMulCommClass α β γ] : SMulCommClass (Set α) (Set β) (Set γ)
SMulCommClass.of_mclosure_eq_top 📋 Mathlib.Algebra.Group.Submonoid.Membership
{M : Type u_1} {N : Type u_4} {α : Type u_5} [Monoid M] [SMul N α] [MulAction M α] {s : Set M} (htop : Submonoid.closure s = ⊤) (hs : ∀ x ∈ s, ∀ (y : N) (z : α), x • y • z = y • x • z) : SMulCommClass M N α
MulAction.le_stabilizer_smul_right 📋 Mathlib.GroupTheory.GroupAction.Defs
{α : Type u_2} {β : Type u_3} {G' : Type u_4} [Group G'] [SMul α β] [MulAction G' β] [SMulCommClass G' α β] (a : α) (b : β) : MulAction.stabilizer G' b ≤ MulAction.stabilizer G' (a • b)
instSMulCommClassSubtypeMemSubmonoidSubmonoid 📋 Mathlib.GroupTheory.GroupAction.Defs
(M : Type u) (α : Type v) [Monoid M] [Monoid α] [MulDistribMulAction M α] : SMulCommClass M (↥(FixedPoints.submonoid M α)) α
MulAction.stabilizer_mul_eq_right 📋 Mathlib.GroupTheory.GroupAction.Defs
{G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [Group α] [SMulCommClass G α α] (a b : α) : MulAction.stabilizer G (a * b) = MulAction.stabilizer G b
MulAction.stabilizer_smul_eq_right 📋 Mathlib.GroupTheory.GroupAction.Defs
{G : Type u_1} {β : Type u_3} [Group G] [MulAction G β] {α : Type u_4} [Group α] [MulAction α β] [SMulCommClass G α β] (a : α) (b : β) : MulAction.stabilizer G (a • b) = MulAction.stabilizer G b
FixedPoints.instSMulCommClassSubtypeMemSubgroupSubgroup 📋 Mathlib.GroupTheory.GroupAction.Defs
(M : Type u) (α : Type v) [Monoid M] [Group α] [MulDistribMulAction M α] : SMulCommClass M (↥(FixedPoints.subgroup M α)) α
Submonoid.center.smulCommClass_left 📋 Mathlib.GroupTheory.Submonoid.Center
{M : Type u_1} [Monoid M] : SMulCommClass (↥(Submonoid.center M)) M M
Submonoid.center.smulCommClass_right 📋 Mathlib.GroupTheory.Submonoid.Center
{M : Type u_1} [Monoid M] : SMulCommClass M (↥(Submonoid.center M)) M
Submonoid.instSMulCommClassSubtypeMemCenter 📋 Mathlib.GroupTheory.Submonoid.Center
{M : Type u_2} {α : Type u_3} [Monoid M] [MulAction M α] : SMulCommClass (↥(Submonoid.center M)) M α
Submonoid.instSMulCommClassSubtypeMemCenter_1 📋 Mathlib.GroupTheory.Submonoid.Center
{M : Type u_2} {α : Type u_3} [Monoid M] [MulAction M α] : SMulCommClass M (↥(Submonoid.center M)) α
ConjAct.unitsSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.ConjAct
(α : Type u_1) {M : Type u_2} [Monoid M] [SMul α M] [SMulCommClass α M M] [IsScalarTower α M M] : SMulCommClass α (ConjAct Mˣ) M
ConjAct.unitsSMulCommClass' 📋 Mathlib.GroupTheory.GroupAction.ConjAct
(α : Type u_1) {M : Type u_2} [Monoid M] [SMul α M] [SMulCommClass M α M] [IsScalarTower α M M] : SMulCommClass (ConjAct Mˣ) α M
ConjAct.smulCommClass 📋 Mathlib.GroupTheory.GroupAction.ConjAct
(α : Type u_1) {G : Type u_3} [Group G] [SMul α G] [SMulCommClass α G G] [IsScalarTower α G G] : SMulCommClass α (ConjAct G) G
ConjAct.smulCommClass' 📋 Mathlib.GroupTheory.GroupAction.ConjAct
(α : Type u_1) {G : Type u_3} [Group G] [SMul α G] [SMulCommClass G α G] [IsScalarTower α G G] : SMulCommClass (ConjAct G) α G
OreLocalization.instSMulCommClass_1 📋 Mathlib.GroupTheory.OreLocalization.Basic
{R : Type u_1} {M : Type u_3} {X : Type u_4} [Monoid M] {S : Submonoid M} [OreLocalization.OreSet S] [MulAction M X] [SMul R X] [SMul R M] [IsScalarTower R M M] [IsScalarTower R M X] [SMulCommClass R M M] : SMulCommClass R (OreLocalization S M) (OreLocalization S X)
OreLocalization.instSMulCommClass 📋 Mathlib.GroupTheory.OreLocalization.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_3} {X : Type u_4} [Monoid M] {S : Submonoid M} [OreLocalization.OreSet S] [MulAction M X] [SMul R X] [SMul R M] [IsScalarTower R M M] [IsScalarTower R M X] [SMul R' X] [SMul R' M] [IsScalarTower R' M M] [IsScalarTower R' M X] [SMulCommClass R R' M] : SMulCommClass R R' (OreLocalization S X)
Commute.smul_left_iff₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Units
{α : Type u_4} {β : Type u_5} [GroupWithZero α] [MulAction α β] {a : α} [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {x y : β} (ha : a ≠ 0) : Commute (a • x) y ↔ Commute x y
Commute.smul_right_iff₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Units
{α : Type u_4} {β : Type u_5} [GroupWithZero α] [MulAction α β] {a : α} [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {x y : β} (ha : a ≠ 0) : Commute x (a • y) ↔ Commute x y
SemiconjBy.smul_left_iff₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Units
{α : Type u_4} {β : Type u_5} [GroupWithZero α] [MulAction α β] {a : α} [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {x y z : β} (ha : a ≠ 0) : SemiconjBy (a • x) y z ↔ SemiconjBy x y z
SemiconjBy.smul_right_iff₀ 📋 Mathlib.Algebra.GroupWithZero.Action.Units
{α : Type u_4} {β : Type u_5} [GroupWithZero α] [MulAction α β] {a : α} [Mul β] [SMulCommClass α β β] [IsScalarTower α β β] {x y z : β} (ha : a ≠ 0) : SemiconjBy x (a • y) (a • z) ↔ SemiconjBy x y z
SMulCommClass.toMulActionHom 📋 Mathlib.GroupTheory.GroupAction.Hom
{M : Type u_11} (N : Type u_9) (α : Type u_10) [SMul M α] [SMul N α] [SMulCommClass M N α] (c : M) : α →ₑ[id] α
MulActionHom.instSMulOfSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [SMul M X] [SMul N Y] [SMul R Y] [SMulCommClass N R Y] : SMul R (X →ₑ[σ] Y)
MulActionHom.instMulActionOfSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [SMul M X] [SMul N Y] [Monoid R] [MulAction R Y] [SMulCommClass N R Y] : MulAction R (X →ₑ[σ] Y)
SMulCommClass.toMulActionHom_apply 📋 Mathlib.GroupTheory.GroupAction.Hom
{M : Type u_11} (N : Type u_9) (α : Type u_10) [SMul M α] [SMul N α] [SMulCommClass M N α] (c : M) (x✝ : α) : (SMulCommClass.toMulActionHom N α c) x✝ = c • x✝
SMulCommClass.toDistribMulActionHom 📋 Mathlib.GroupTheory.GroupAction.Hom
{M : Type u_13} (N : Type u_11) (A : Type u_12) [Monoid N] [AddMonoid A] [DistribSMul M A] [DistribMulAction N A] [SMulCommClass M N A] (c : M) : A →+[N] A
MulActionHom.instDistribSMulOfSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [AddZeroClass Y] [SMul M X] [DistribSMul N Y] [DistribSMul R Y] [SMulCommClass N R Y] : DistribSMul R (X →ₑ[σ] Y)
MulActionHom.instDistribMulActionOfSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [AddMonoid Y] [Monoid R] [SMul M X] [DistribSMul N Y] [DistribMulAction R Y] [SMulCommClass N R Y] : DistribMulAction R (X →ₑ[σ] Y)
MulActionHom.instModuleOfSMulCommClass 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [AddCommMonoid Y] [Semiring R] [SMul M X] [DistribSMul N Y] [Module R Y] [SMulCommClass N R Y] : Module R (X →ₑ[σ] Y)
MulActionHom.coe_smul 📋 Mathlib.GroupTheory.GroupAction.Hom
{R : Type u_1} {M : Type u_2} {N : Type u_3} {X : Type u_4} {Y : Type u_5} {σ : M → N} [SMul M X] [SMul N Y] [SMul R Y] [SMulCommClass N R Y] (f : X →ₑ[σ] Y) (r : R) : ⇑(r • f) = r • ⇑f
SMulCommClass.toDistribMulActionHom_toFun 📋 Mathlib.GroupTheory.GroupAction.Hom
{M : Type u_13} (N : Type u_11) (A : Type u_12) [Monoid N] [AddMonoid A] [DistribSMul M A] [DistribMulAction N A] [SMulCommClass M N A] (c : M) (x✝ : A) : (SMulCommClass.toDistribMulActionHom N A c) x✝ = c • x✝
LinearMap.mulLeft 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a : A) : A →ₗ[R] A
LinearMap.instSMul 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] : SMul S (M →ₛₗ[σ₁₂] M₂)
LinearMap.instDistribMulAction 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Monoid S] [DistribMulAction S M₂] [SMulCommClass R₂ S M₂] : DistribMulAction S (M →ₛₗ[σ₁₂] M₂)
LinearMap.mulLeft_apply 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a b : A) : (LinearMap.mulLeft R a) b = a * b
LinearMap.module 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [Semiring S] [Module S M₂] [SMulCommClass R₂ S M₂] : Module S (M →ₛₗ[σ₁₂] M₂)
LinearMap.mulLeft_zero_eq_zero 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) (A : Type u_15) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] : LinearMap.mulLeft R 0 = 0
LinearMap.mulLeftRight 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (ab : A × A) : A →ₗ[R] A
LinearMap.mulLeftRight_apply 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_14} {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] (a b x : A) : (LinearMap.mulLeftRight R (a, b)) x = a * x * b
LinearMap.instSMulCommClass 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] [DistribSMul T M₂] [SMulCommClass R₂ T M₂] [SMulCommClass S T M₂] : SMulCommClass S T (M →ₛₗ[σ₁₂] M₂)
LinearMap.smul_apply 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] (a : S) (f : M →ₛₗ[σ₁₂] M₂) (x : M) : (a • f) x = a • f x
LinearMap.instIsCentralScalar 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] [DistribSMul Sᵐᵒᵖ M₂] [SMulCommClass R₂ Sᵐᵒᵖ M₂] [IsCentralScalar S M₂] : IsCentralScalar S (M →ₛₗ[σ₁₂] M₂)
LinearMap.instIsScalarTower 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {T : Type u_7} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] [DistribSMul T M₂] [SMulCommClass R₂ T M₂] [SMul S T] [IsScalarTower S T M₂] : IsScalarTower S T (M →ₛₗ[σ₁₂] M₂)
LinearMap.coe_smul 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} [Semiring R] [Semiring R₂] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R₂ M₂] {σ₁₂ : R →+* R₂} [DistribSMul S M₂] [SMulCommClass R₂ S M₂] (a : S) (f : M →ₛₗ[σ₁₂] M₂) : ⇑(a • f) = a • ⇑f
LinearMap.toAddMonoidHom_mulLeft 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a : A) : ↑(LinearMap.mulLeft R a) = AddMonoidHom.mulLeft a
LinearMap.smul_comp 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {R₂ : Type u_3} {R₃ : Type u_4} {S₃ : Type u_6} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R] [Semiring R₂] [Semiring R₃] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Module R₂ M₂] [Module R₃ M₃] {σ₁₂ : R →+* R₂} {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} [RingHomCompTriple σ₁₂ σ₂₃ σ₁₃] [Monoid S₃] [DistribMulAction S₃ M₃] [SMulCommClass R₃ S₃ M₃] (a : S₃) (g : M₂ →ₛₗ[σ₂₃] M₃) (f : M →ₛₗ[σ₁₂] M₂) : (a • g) ∘ₛₗ f = a • g ∘ₛₗ f
LinearMap.restrictScalarsₗ 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) (S : Type u_15) (M : Type u_16) (N : Type u_17) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (R₁ : Type u_19) [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] : (M →ₗ[S] N) →ₗ[R₁] M →ₗ[R] N
LinearMap.restrictScalars_smul 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_14} {S : Type u_15} {M : Type u_16} {N : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] {R₁ : Type u_19} [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] (c : R₁) (f : M →ₗ[S] N) : ↑R (c • f) = c • ↑R f
LinearMap.comp_smul 📋 Mathlib.Algebra.Module.LinearMap.Defs
{R : Type u_1} {S : Type u_5} {M : Type u_8} {M₂ : Type u_10} {M₃ : Type u_11} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] [Module R M] [Monoid S] [DistribMulAction S M₂] [Module R M₂] [Module R M₃] [SMulCommClass R S M₂] [DistribMulAction S M₃] [SMulCommClass R S M₃] [LinearMap.CompatibleSMul M₃ M₂ S R] (g : M₃ →ₗ[R] M₂) (a : S) (f : M →ₗ[R] M₃) : g ∘ₗ (a • f) = a • g ∘ₗ f
LinearMap.restrictScalarsₗ_apply 📋 Mathlib.Algebra.Module.LinearMap.Defs
(R : Type u_14) (S : Type u_15) (M : Type u_16) (N : Type u_17) [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module S M] [Module S N] [LinearMap.CompatibleSMul M N R S] (R₁ : Type u_19) [Semiring R₁] [Module R₁ N] [SMulCommClass S R₁ N] [SMulCommClass R R₁ N] (fₗ : M →ₗ[S] N) : (LinearMap.restrictScalarsₗ R S M N R₁) fₗ = ↑R fₗ
smul_algebraMap 📋 Mathlib.Algebra.Algebra.Defs
{R : Type u} {A : Type w} [CommSemiring R] [Semiring A] [Algebra R A] {α : Type u_1} [Monoid α] [MulDistribMulAction α A] [SMulCommClass α R A] (a : α) (r : R) : a • (algebraMap R A) r = (algebraMap R A) r
IsUnital.toAlgebra 📋 Mathlib.Algebra.Algebra.Defs
{R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] [SMulCommClass R A A] [IsUnital A] : Algebra R A
Prod.smulCommClass 📋 Mathlib.Algebra.Group.Action.Prod
{M : Type u_1} {N : Type u_2} {α : Type u_4} {β : Type u_5} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMulCommClass M N α] [SMulCommClass M N β] : SMulCommClass M N (α × β)
MulAction.prodOfSMulCommClass 📋 Mathlib.Algebra.Group.Action.Prod
(M : Type u_1) (N : Type u_2) (α : Type u_4) [Monoid M] [Monoid N] [MulAction M α] [MulAction N α] [SMulCommClass M N α] : MulAction (M × N) α
Prod.smulCommClassBoth 📋 Mathlib.Algebra.Group.Action.Prod
{M : Type u_1} {N : Type u_2} {P : Type u_3} [Mul N] [Mul P] [SMul M N] [SMul M P] [SMulCommClass M N N] [SMulCommClass M P P] : SMulCommClass M (N × P) (N × P)
MulAction.prodEquiv 📋 Mathlib.Algebra.Group.Action.Prod
(M : Type u_1) (N : Type u_2) (α : Type u_4) [Monoid M] [Monoid N] : MulAction (M × N) α ≃ (x : MulAction M α) ×' (x_1 : MulAction N α) ×' SMulCommClass M N α
smulMulHom 📋 Mathlib.Algebra.Group.Action.Prod
{α : Type u_4} {β : Type u_5} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] : α × β →ₙ* β
smulMonoidHom 📋 Mathlib.Algebra.Group.Action.Prod
{α : Type u_4} {β : Type u_5} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] : α × β →* β
smulMulHom_apply 📋 Mathlib.Algebra.Group.Action.Prod
{α : Type u_4} {β : Type u_5} [Monoid α] [Mul β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (a : α × β) : smulMulHom a = a.1 • a.2
smulMonoidHom_apply 📋 Mathlib.Algebra.Group.Action.Prod
{α : Type u_4} {β : Type u_5} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (a✝ : α × β) : smulMonoidHom a✝ = smulMulHom.toFun a✝
smulMonoidWithZeroHom 📋 Mathlib.Algebra.GroupWithZero.Action.Basic
{M₀ : Type u_4} {N₀ : Type u_5} [MonoidWithZero M₀] [MulZeroOneClass N₀] [MulActionWithZero M₀ N₀] [IsScalarTower M₀ N₀ N₀] [SMulCommClass M₀ N₀ N₀] : M₀ × N₀ →*₀ N₀
div_smul_div_comm 📋 Mathlib.Algebra.GroupWithZero.Action.Basic
{G : Type u_1} {G₀ : Type u_2} [Group G] [GroupWithZero G₀] [MulAction G G₀] [IsScalarTower G G₀ G₀] [SMulCommClass G G₀ G₀] (g h : G) (a b : G₀) : (g / h) • (a / b) = g • a / h • b
smulMonoidWithZeroHom_apply 📋 Mathlib.Algebra.GroupWithZero.Action.Basic
{M₀ : Type u_4} {N₀ : Type u_5} [MonoidWithZero M₀] [MulZeroOneClass N₀] [MulActionWithZero M₀ N₀] [IsScalarTower M₀ N₀ N₀] [SMulCommClass M₀ N₀ N₀] (a✝ : M₀ × N₀) : smulMonoidWithZeroHom a✝ = (↑smulMonoidHom).toFun a✝
IsUnit.smul_sub_iff_sub_inv_smul 📋 Mathlib.Algebra.GroupWithZero.Action.Basic
{G : Type u_1} {R : Type u_6} [Group G] [Monoid R] [AddGroup R] [DistribMulAction G R] [IsScalarTower G R R] [SMulCommClass G R R] (r : G) (a : R) : IsUnit (r • 1 - a) ↔ IsUnit (1 - r⁻¹ • a)
LinearEquiv.instSMulUnitsId 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] : SMul Sˣ (V ≃ₗ[R] W)
LinearEquiv.toLinearMap_smul 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] (e : V ≃ₗ[R] W) (α : Sˣ) : ↑(α • e) = ↑α • ↑e
LinearEquiv.smul_apply 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] (α : Sˣ) (e : V ≃ₗ[R] W) (x : V) : (α • e) x = ↑α • e x
LinearEquiv.symm_smul 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] [SMulCommClass R S V] (e : V ≃ₗ[R] W) (α : Sˣ) : (α • e).symm = α⁻¹ • e.symm
LinearEquiv.symm_smul_apply 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] (e : V ≃ₗ[R] W) (α : Sˣ) (x : W) : (α • e).symm x = ↑α⁻¹ • e.symm x
LinearEquiv.trans_smul 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} {G : Type u_18} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [AddCommMonoid G] [Module R G] [Module S G] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] [IsScalarTower S R G] (α : Sˣ) (e : G ≃ₗ[R] V) (f : V ≃ₗ[R] W) : e ≪≫ₗ (α • f) = α • e ≪≫ₗ f
LinearEquiv.smul_trans 📋 Mathlib.Algebra.Module.Equiv.Defs
{S : Type u_14} {R : Type u_15} {V : Type u_16} {W : Type u_17} {G : Type u_18} [Semiring R] [Semiring S] [AddCommMonoid V] [Module R V] [Module S V] [AddCommMonoid W] [Module R W] [Module S W] [AddCommMonoid G] [Module R G] [Module S G] [SMulCommClass R S W] [SMul S R] [IsScalarTower S R V] [IsScalarTower S R W] [SMulCommClass R S V] [IsScalarTower S R G] (α : Sˣ) (e : G ≃ₗ[R] V) (f : V ≃ₗ[R] W) : (α • e) ≪≫ₗ f = α • e ≪≫ₗ f
ZeroHom.instSMulCommClass 📋 Mathlib.Algebra.GroupWithZero.Action.Hom
{M : Type u_1} {N : Type u_2} {A : Type u_3} {B : Type u_4} [Zero A] [Zero B] [SMulZeroClass M B] [SMulZeroClass N B] [SMulCommClass M N B] : SMulCommClass M N (ZeroHom A B)
AddMonoidHom.instSMulCommClass 📋 Mathlib.Algebra.GroupWithZero.Action.Hom
{M : Type u_1} {N : Type u_2} {A : Type u_3} {B : Type u_4} [AddZeroClass A] [AddZeroClass B] [DistribSMul M B] [DistribSMul N B] [SMulCommClass M N B] : SMulCommClass M N (A →+ B)
DomMulAct.instSMulCommClassForall 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{M : Type u_1} {β : Type u_2} {α : Type u_3} {N : Type u_4} [SMul M α] [SMul N β] : SMulCommClass Mᵈᵐᵃ N (α → β)
DomMulAct.instSMulCommClassForall_1 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{M : Type u_1} {β : Type u_2} {α : Type u_3} {N : Type u_4} [SMul M α] [SMul N β] : SMulCommClass N Mᵈᵐᵃ (α → β)
DomMulAct.instSMulCommClassForall_2 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{M : Type u_1} {β : Type u_2} {α : Type u_3} {N : Type u_4} [SMul M α] [SMul N α] [SMulCommClass M N α] : SMulCommClass Mᵈᵐᵃ Nᵈᵐᵃ (α → β)
DomMulAct.instSMulCommClassAddMonoidHom_1 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{A : Type u_5} {B : Type u_6} {M : Type u_7} {M' : Type u_8} [AddMonoid A] [DistribSMul M A] [AddZeroClass B] [DistribSMul M' B] : SMulCommClass Mᵈᵐᵃ M' (A →+ B)
DomMulAct.instSMulCommClassAddMonoidHom 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{A : Type u_5} {B : Type u_6} {M : Type u_7} {M' : Type u_8} [AddMonoid A] [DistribSMul M A] [AddZeroClass B] [DistribSMul M' A] [SMulCommClass M M' A] : SMulCommClass Mᵈᵐᵃ M'ᵈᵐᵃ (A →+ B)
DomMulAct.instSMulCommClassMonoidHom 📋 Mathlib.GroupTheory.GroupAction.DomAct.Basic
{M : Type u_5} {M' : Type u_6} {A : Type u_7} {B : Type u_8} [Monoid M] [Monoid A] [MulDistribMulAction M A] [MulOneClass B] [Monoid M'] [MulDistribMulAction M' A] [SMulCommClass M M' A] : SMulCommClass Mᵈᵐᵃ M'ᵈᵐᵃ (A →* B)
AddMonoid.End.smulCommClass 📋 Mathlib.Algebra.Module.Hom
{R : Type u_1} {S : Type u_2} {A : Type u_4} [Monoid R] [Monoid S] [AddCommMonoid A] [DistribMulAction R A] [DistribMulAction S A] [SMulCommClass R S A] : SMulCommClass R S (AddMonoid.End A)
Module.IsTorsionFree.trans 📋 Mathlib.Algebra.Module.Torsion.Free
(R : Type u_1) {S : Type u_2} {M : Type u_3} [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [Module.IsTorsionFree R M] [Module S R] [Module.IsTorsionFree S R] [IsScalarTower S R R] [SMulCommClass S R R] [IsScalarTower S R M] : Module.IsTorsionFree S M
LinearMap.instSMulDomMulAct 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] : SMul S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')
LinearMap.mulLeft_inj 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_6} {A : Type u_7} [Semiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] {a b : A} : LinearMap.mulLeft R a = LinearMap.mulLeft R b ↔ a = b
LinearMap.mulLeft_one 📋 Mathlib.Algebra.Module.LinearMap.Basic
(R : Type u_6) (A : Type u_7) [Semiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] : LinearMap.mulLeft R 1 = LinearMap.id
LinearMap.instDistribMulActionDomMulActOfSMulCommClass 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] : DistribMulAction S'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')
LinearMap.instModuleDomMulActOfSMulCommClass 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {S : Type u_3} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} [Semiring S] [Module S M] [SMulCommClass R S M] : Module Sᵈᵐᵃ (M →ₛₗ[σ₁₂] M')
LinearMap.instIsTorsionFree 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {S : Type u_3} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} [Semiring S] [Module S M'] [SMulCommClass R' S M'] [Module.IsTorsionFree S M'] : Module.IsTorsionFree S (M →ₛₗ[σ₁₂] M')
LinearMap.ltoFun 📋 Mathlib.Algebra.Module.LinearMap.Basic
(R : Type u_6) (M : Type u_7) (N : Type u_8) (A : Type u_9) [Semiring R] [Semiring A] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module A N] [SMulCommClass R A N] : (M →ₗ[R] N) →ₗ[A] M → N
LinearMap.mulLeft_mul 📋 Mathlib.Algebra.Module.LinearMap.Basic
(R : Type u_6) (A : Type u_7) [Semiring R] [NonUnitalSemiring A] [Module R A] [SMulCommClass R A A] (a b : A) : LinearMap.mulLeft R (a * b) = LinearMap.mulLeft R a ∘ₗ LinearMap.mulLeft R b
LinearMap.mulLeft_eq_zero_iff 📋 Mathlib.Algebra.Module.LinearMap.Basic
(R : Type u_6) (A : Type u_7) [Semiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] (a : A) : LinearMap.mulLeft R a = 0 ↔ a = 0
DomMulAct.coe_smul_linearMap 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') : ⇑(a • f) = a • ⇑f
LinearMap.instSMulCommClassDomMulAct 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} {T' : Type u_7} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] [Monoid T'] [DistribMulAction T' M] [SMulCommClass R T' M] [SMulCommClass S' T' M] : SMulCommClass S'ᵈᵐᵃ T'ᵈᵐᵃ (M →ₛₗ[σ₁₂] M')
DomMulAct.mk_smul_linearMap_apply 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S') (f : M →ₛₗ[σ₁₂] M') (x : M) : (DomMulAct.mk a • f) x = f (a • x)
DomMulAct.smul_linearMap_apply 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_1} {R' : Type u_2} {M : Type u_4} {M' : Type u_5} [Semiring R] [Semiring R'] [AddCommMonoid M] [AddCommMonoid M'] [Module R M] [Module R' M'] {σ₁₂ : R →+* R'} {S' : Type u_6} [Monoid S'] [DistribMulAction S' M] [SMulCommClass R S' M] (a : S'ᵈᵐᵃ) (f : M →ₛₗ[σ₁₂] M') (x : M) : (a • f) x = f (DomMulAct.mk.symm a • x)
LinearMap.ltoFun_apply 📋 Mathlib.Algebra.Module.LinearMap.Basic
{R : Type u_6} {M : Type u_7} {N : Type u_8} {A : Type u_9} [Semiring R] [Semiring A] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Module A N] [SMulCommClass R A N] {f : M →ₗ[R] N} : (LinearMap.ltoFun R M N A) f = ⇑f
DistribSMul.toLinearMap 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [DistribSMul S M] [SMulCommClass S R M] (s : S) : M →ₗ[R] M
Module.toModuleEnd 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S M] [SMulCommClass S R M] : S →+* Module.End R M
DistribMulAction.toModuleEnd 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass S R M] : S →* Module.End R M
Module.End.apply_smulCommClass 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [SMul S M] [IsScalarTower S R M] : SMulCommClass S (Module.End R M) M
Module.End.apply_smulCommClass' 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [SMul S M] [IsScalarTower S R M] : SMulCommClass (Module.End R M) S M
DistribSMul.toLinearMap_apply 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [DistribSMul S M] [SMulCommClass S R M] (s : S) (a✝ : M) : (DistribSMul.toLinearMap R M s) a✝ = s • a✝
Module.End.instIsScalarTower 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] : IsScalarTower S (Module.End R M) (Module.End R M)
Module.toModuleEnd_apply 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Module S M] [SMulCommClass S R M] (s : S) : (Module.toModuleEnd R M) s = DistribSMul.toLinearMap R M s
Module.End.instSMulCommClass 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMul S R] [IsScalarTower S R M] : SMulCommClass S (Module.End R M) (Module.End R M)
Module.End.instSMulCommClass' 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] [SMul S R] [IsScalarTower S R M] : SMulCommClass (Module.End R M) S (Module.End R M)
Module.End.apply_isScalarTower 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} {S : Type u_3} {M : Type u_4} [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass R S M] : IsScalarTower S (Module.End R M) M
DistribMulAction.toModuleEnd_apply 📋 Mathlib.Algebra.Module.LinearMap.End
(R : Type u_1) {S : Type u_3} (M : Type u_4) [Semiring R] [AddCommMonoid M] [Module R M] [Monoid S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) : (DistribMulAction.toModuleEnd R M) s = DistribSMul.toLinearMap R M s
LinearMap.applyₗ' 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} (S : Type u_3) {M : Type u_4} {M₂ : Type u_6} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M₂] [SMulCommClass R S M₂] : M →+ (M →ₗ[R] M₂) →ₗ[S] M₂
LinearMap.compRight 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} (S : Type u_3) {M : Type u_4} {M₁ : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M] [Module R M₁] [Module R M₂] [Module S M₁] [Module S M₂] [SMulCommClass R S M₁] [SMulCommClass R S M₂] [LinearMap.CompatibleSMul M₁ M₂ S R] (f : M₁ →ₗ[R] M₂) : (M →ₗ[R] M₁) →ₗ[S] M →ₗ[R] M₂
LinearMap.compRight_apply 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} (S : Type u_3) {M : Type u_4} {M₁ : Type u_5} {M₂ : Type u_6} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₁] [AddCommMonoid M₂] [Module R M] [Module R M₁] [Module R M₂] [Module S M₁] [Module S M₂] [SMulCommClass R S M₁] [SMulCommClass R S M₂] [LinearMap.CompatibleSMul M₁ M₂ S R] (f : M₁ →ₗ[R] M₂) (g : M →ₗ[R] M₁) : (LinearMap.compRight S f) g = f ∘ₗ g
LinearMap.applyₗ'_apply_apply 📋 Mathlib.Algebra.Module.LinearMap.End
{R : Type u_1} (S : Type u_3) {M : Type u_4} {M₂ : Type u_6} [Semiring R] [Semiring S] [AddCommMonoid M] [AddCommMonoid M₂] [Module R M] [Module R M₂] [Module S M₂] [SMulCommClass R S M₂] (v : M) (f : M →ₗ[R] M₂) : ((LinearMap.applyₗ' S) v) f = f v
DistribMulAction.prodOfSMulCommClass 📋 Mathlib.Algebra.GroupWithZero.Action.Prod
(M : Type u_1) (N : Type u_2) (α : Type u_3) [Monoid M] [Monoid N] [AddMonoid α] [DistribMulAction M α] [DistribMulAction N α] [SMulCommClass M N α] : DistribMulAction (M × N) α
DistribMulAction.prodEquiv 📋 Mathlib.Algebra.GroupWithZero.Action.Prod
(M : Type u_1) (N : Type u_2) (α : Type u_3) [Monoid M] [Monoid N] [AddMonoid α] : DistribMulAction (M × N) α ≃ (x : DistribMulAction M α) ×' (x_1 : DistribMulAction N α) ×' SMulCommClass M N α
DistribMulAction.toLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) : M ≃ₗ[R] M
LinearMap.ringLmapEquivSelf 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) (S : Type u_4) (M : Type u_5) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] : (R →ₗ[R] M) ≃ₗ[S] M
DistribMulAction.toLinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) (a✝ : M) : (DistribMulAction.toLinearEquiv R M s) a✝ = s • a✝
DistribMulAction.toModuleAut 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] : S →* M ≃ₗ[R] M
LinearEquiv.congrLeft 📋 Mathlib.Algebra.Module.Equiv.Basic
(M : Type u_5) {M₂ : Type u_7} {M₃ : Type u_8} [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] {R : Type u_9} (S : Type u_10) [Semiring R] [Semiring S] [Module R M₂] [Module R M₃] [Module R M] [Module S M] [SMulCommClass R S M] (e : M₂ ≃ₗ[R] M₃) : (M₂ →ₗ[R] M) ≃ₗ[S] M₃ →ₗ[R] M
LinearEquiv.apply_smulCommClass 📋 Mathlib.Algebra.Module.Equiv.Basic
{R : Type u_1} {S : Type u_4} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [SMul S M] [IsScalarTower S R M] : SMulCommClass S (M ≃ₗ[R] M) M
LinearEquiv.apply_smulCommClass' 📋 Mathlib.Algebra.Module.Equiv.Basic
{R : Type u_1} {S : Type u_4} {M : Type u_5} [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [SMul S M] [IsScalarTower S R M] : SMulCommClass (M ≃ₗ[R] M) S M
Units.mulLeftLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_9) (A : Type u_10) [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] : Aˣ →* A ≃ₗ[R] A
DistribMulAction.toLinearEquiv_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) (a✝ : M) : (DistribMulAction.toLinearEquiv R M s).symm a✝ = s⁻¹ • a✝
LinearEquiv.domMulActCongrRight 📋 Mathlib.Algebra.Module.Equiv.Basic
{S : Type u_4} {R₁ : Type u_9} {R₁' : Type u_11} {R₂' : Type u_12} {M₁ : Type u_13} {M₁' : Type u_15} {M₂' : Type u_16} [Semiring R₁] [Semiring R₁'] [Semiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [Semiring S] [Module S M₁] [SMulCommClass R₁ S M₁] [RingHomCompTriple σ₁₂' σ₂'₁' σ₁₁'] (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') : (M₁ →ₛₗ[σ₁₁'] M₁') ≃ₗ[Sᵈᵐᵃ] M₁ →ₛₗ[σ₁₂'] M₂'
DistribMulAction.toModuleAut_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) {S : Type u_4} (M : Type u_5) [Semiring R] [AddCommMonoid M] [Module R M] [Group S] [DistribMulAction S M] [SMulCommClass S R M] (s : S) : (DistribMulAction.toModuleAut R M) s = DistribMulAction.toLinearEquiv R M s
LinearMap.ringLmapEquivSelf_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) (S : Type u_4) (M : Type u_5) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] (f : R →ₗ[R] M) : (LinearMap.ringLmapEquivSelf R S M) f = f 1
LinearEquiv.smul_refl 📋 Mathlib.Algebra.Module.Equiv.Basic
{R : Type u_1} {S : Type u_4} {M : Type u_5} [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] [SMul S R] [IsScalarTower S R M] (α : Sˣ) : α • LinearEquiv.refl R M = DistribMulAction.toLinearEquiv R M α
Units.toEquiv_mulLeftLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
{R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (u : Aˣ) : ((Units.mulLeftLinearEquiv R A) u).toEquiv = u.mulLeft
Units.toLinearMap_mulLeftLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
{R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (u : Aˣ) : ↑((Units.mulLeftLinearEquiv R A) u) = LinearMap.mulLeft R ↑u
LinearMap.ringLmapEquivSelf_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_1) (S : Type u_4) (M : Type u_5) [Semiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass R S M] (x : M) : (LinearMap.ringLmapEquivSelf R S M).symm x = LinearMap.smulRight 1 x
Units.mulLeftLinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_9) {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (a : Aˣ) (x : A) : ((Units.mulLeftLinearEquiv R A) a) x = ↑a * x
Units.symm_mulLeftLinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(R : Type u_9) {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [SMulCommClass R A A] (a : Aˣ) (x : A) : ((Units.mulLeftLinearEquiv R A) a).symm x = ↑a⁻¹ * x
LinearEquiv.congrLeft_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(M : Type u_5) {M₂ : Type u_7} {M₃ : Type u_8} [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] {R : Type u_9} (S : Type u_10) [Semiring R] [Semiring S] [Module R M₂] [Module R M₃] [Module R M] [Module S M] [SMulCommClass R S M] (e : M₂ ≃ₗ[R] M₃) (a✝ : M₂ →ₗ[R] M) : (LinearEquiv.congrLeft M S e) a✝ = (e.arrowCongrAddEquiv (LinearEquiv.refl R M)).toFun a✝
LinearEquiv.domMulActCongrRight_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
{S : Type u_4} {R₁ : Type u_9} {R₁' : Type u_11} {R₂' : Type u_12} {M₁ : Type u_13} {M₁' : Type u_15} {M₂' : Type u_16} [Semiring R₁] [Semiring R₁'] [Semiring R₂'] [AddCommMonoid M₁] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [Module R₁ M₁] [Module R₁' M₁'] [Module R₂' M₂'] {σ₁'₂' : R₁' →+* R₂'} {σ₂'₁' : R₂' →+* R₁'} {σ₁₁' : R₁ →+* R₁'} {σ₁₂' : R₁ →+* R₂'} [RingHomInvPair σ₁'₂' σ₂'₁'] [RingHomInvPair σ₂'₁' σ₁'₂'] [RingHomCompTriple σ₁₁' σ₁'₂' σ₁₂'] [Semiring S] [Module S M₁] [SMulCommClass R₁ S M₁] [RingHomCompTriple σ₁₂' σ₂'₁' σ₁₁'] (e₂ : M₁' ≃ₛₗ[σ₁'₂'] M₂') (a✝ : M₁ →ₛₗ[σ₁₁'] M₁') : e₂.domMulActCongrRight a✝ = ((LinearEquiv.refl R₁ M₁).arrowCongrAddEquiv e₂).toFun a✝
LinearEquiv.congrLeft_symm_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
(M : Type u_5) {M₂ : Type u_7} {M₃ : Type u_8} [AddCommMonoid M] [AddCommMonoid M₂] [AddCommMonoid M₃] {R : Type u_9} (S : Type u_10) [Semiring R] [Semiring S] [Module R M₂] [Module R M₃] [Module R M] [Module S M] [SMulCommClass R S M] (e : M₂ ≃ₗ[R] M₃) (a✝ : M₃ →ₗ[R] M) : (LinearEquiv.congrLeft M S e).symm a✝ = (e.arrowCongrAddEquiv (LinearEquiv.refl R M)).invFun a✝

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c