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Result

Found 4764 declarations mentioning IsScalarTower. Of these, 3864 match your pattern(s). Of these, only the first 200 are shown.

• IsScalarTower.op_right 📋 Mathlib.Algebra.Group.Action.Defs
  {M : Type u_1} {N : Type u_2} {α : Type u_5} [SMul M α] [SMul M N] [SMul N α] [SMul Nᵐᵒᵖ α] [IsCentralScalar N α] [IsScalarTower M N α] : IsScalarTower M Nᵐᵒᵖ α
• SMul.comp.isScalarTower 📋 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 M β] [SMul α β] [IsScalarTower M α β] (g : N → M) : IsScalarTower N α β
• 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
• IsScalarTower.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 M N] [SMul Mᵐᵒᵖ N] [IsCentralScalar M N] [SMul N α] [IsScalarTower M N α] : IsScalarTower Mᵐᵒᵖ N 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
• smul_mul_assoc 📋 Mathlib.Algebra.Group.Action.Defs
  {α : Type u_5} {β : Type u_6} [Mul β] [SMul α β] [IsScalarTower α β β] (r : α) (x y : β) : r • x * y = r • (x * y)
• 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)
• smul_assoc 📋 Mathlib.Algebra.Group.Action.Defs
  {α : Type u_5} {M : Type u_9} {N : Type u_10} [SMul M N] [SMul N α] [SMul M α] [IsScalarTower M N α] (x : M) (y : N) (z : α) : (x • y) • z = x • y • z
• IsScalarTower.smul_assoc 📋 Mathlib.Algebra.Group.Action.Defs
  {M : Type u_9} {N : Type u_10} {α : Type u_11} {inst✝ : SMul M N} {inst✝¹ : SMul N α} {inst✝² : SMul M α} [self : IsScalarTower M N α] (x : M) (y : N) (z : α) : (x • y) • z = x • y • z
• smul_one_mul 📋 Mathlib.Algebra.Group.Action.Defs
  {M : Type u_9} {N : Type u_10} [MulOneClass N] [SMul M N] [IsScalarTower M N N] (x : M) (y : N) : x • 1 * y = x • y
• smul_div_assoc 📋 Mathlib.Algebra.Group.Action.Defs
  {α : Type u_5} {β : Type u_6} [DivInvMonoid β] [SMul α β] [IsScalarTower α β β] (r : α) (x y : β) : r • x / y = r • (x / y)
• IsScalarTower.to₁₂₄ 📋 Mathlib.Algebra.Group.Action.Defs
  (M : Type u_9) (N : Type u_10) (P : Type u_11) (Q : Type u_12) [SMul M N] [SMul M P] [SMul M Q] [SMul N P] [SMul N Q] [Monoid P] [MulAction P Q] [IsScalarTower M N P] [IsScalarTower M P Q] [IsScalarTower N P Q] : IsScalarTower M N Q
• smul_one_smul 📋 Mathlib.Algebra.Group.Action.Defs
  {α : Type u_5} {M : Type u_9} (N : Type u_10) [Monoid N] [SMul M N] [MulAction N α] [SMul M α] [IsScalarTower M N α] (x : M) (y : α) : (x • 1) • y = x • y
• 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)
• IsScalarTower.to₁₃₄ 📋 Mathlib.Algebra.Group.Action.Defs
  (M : Type u_9) (N : Type u_10) (P : Type u_11) (Q : Type u_12) [SMul M N] [SMul M P] [SMul M Q] [SMul P Q] [Monoid N] [MulAction N P] [MulAction N Q] [IsScalarTower M N P] [IsScalarTower M N Q] [IsScalarTower N P Q] : IsScalarTower M P Q
• 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
• IsScalarTower.to₂₃₄ 📋 Mathlib.Algebra.Group.Action.Defs
  (M : Type u_9) (N : Type u_10) (P : Type u_11) (Q : Type u_12) [SMul M N] [SMul M P] [SMul M Q] [SMul P Q] [Monoid N] [MulAction N P] [MulAction N Q] [IsScalarTower M N P] [IsScalarTower M N Q] [IsScalarTower M P Q] (h : Function.Surjective fun m => m • 1) : IsScalarTower N P Q
• 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
• faithfulSMul_iff_injective_smul_one 📋 Mathlib.Algebra.Group.Action.Faithful
  (R : Type u_4) (A : Type u_5) [MulOneClass A] [SMul R A] [IsScalarTower R A A] : FaithfulSMul R A ↔ Function.Injective fun r => r • 1
• IsScalarTower.to₁₂₃ 📋 Mathlib.Algebra.Group.Action.Faithful
  (M : Type u_4) (N : Type u_5) (P : Type u_6) (Q : Type u_7) [SMul M N] [SMul M P] [SMul M Q] [SMul N P] [SMul N Q] [SMul P Q] [FaithfulSMul P Q] [IsScalarTower M N Q] [IsScalarTower M P Q] [IsScalarTower N P Q] : IsScalarTower M N P
• FaithfulSMul.tower_bot 📋 Mathlib.Algebra.Group.Action.Faithful
  (R : Type u_4) (S : Type u_5) (T : Type u_6) [Monoid S] [MulOneClass T] [SMul R S] [SMul R T] [MulAction S T] [IsScalarTower R S S] [IsScalarTower R T T] [IsScalarTower R S T] [FaithfulSMul R T] : FaithfulSMul R S
• FaithfulSMul.trans 📋 Mathlib.Algebra.Group.Action.Faithful
  (R : Type u_4) (S : Type u_5) (T : Type u_6) [Monoid S] [MulOneClass T] [SMul R S] [IsScalarTower R S S] [MulAction S T] [IsScalarTower S T T] [SMul R T] [IsScalarTower R T T] [IsScalarTower R S T] [FaithfulSMul R S] [FaithfulSMul S T] : FaithfulSMul R T
• 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
• MulOpposite.instIsScalarTower 📋 Mathlib.Algebra.Group.Action.Opposite
  {M : Type u_1} {N : Type u_2} {α : Type u_3} [SMul M N] [SMul M α] [SMul N α] [IsScalarTower M N α] : IsScalarTower M N αᵐᵒᵖ
• 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.instIsScalarTower 📋 Mathlib.Algebra.Group.Action.Units
  {M : Type u_3} {N : Type u_4} {α : Type u_5} [Monoid M] [SMul M N] [SMul M α] [SMul N α] [IsScalarTower M N α] : IsScalarTower Mˣ N α
• 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
• MulAction.IsPretransitive.of_isScalarTower 📋 Mathlib.Algebra.Group.Action.Pretransitive
  (M : Type u_5) {N : Type u_6} {α : Type u_7} [Monoid N] [SMul M N] [MulAction N α] [SMul M α] [IsScalarTower M N α] [MulAction.IsPretransitive M α] : MulAction.IsPretransitive N α
• MonoidHom.smulOneHom_apply 📋 Mathlib.Algebra.Group.Action.Hom
  {M : Type u_4} {N : Type u_5} [Monoid M] [MulOneClass N] [MulAction M N] [IsScalarTower M N N] (x : M) : MonoidHom.smulOneHom x = x • 1
• Submonoid.instIsScalarTowerSubtypeMem 📋 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' α] [SMul M' β] [IsScalarTower M' α β] : IsScalarTower (↥s) α β
• Submonoid.isScalarTower 📋 Mathlib.Algebra.Group.Submonoid.MulAction
  {M' : Type u_1} {α : Type u_2} {β : Type u_3} [MulOneClass M'] [SMul α β] [SMul M' α] [SMul M' β] [IsScalarTower M' α β] (S : Submonoid M') : IsScalarTower (↥S) α β
• Set.isScalarTower 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α β] [SMul α γ] [SMul β γ] [IsScalarTower α β γ] : IsScalarTower α β (Set γ)
• Set.isScalarTower' 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α β] [SMul α γ] [SMul β γ] [IsScalarTower α β γ] : IsScalarTower α (Set β) (Set γ)
• Set.isScalarTower'' 📋 Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
  {α : Type u_2} {β : Type u_3} {γ : Type u_4} [SMul α β] [SMul α γ] [SMul β γ] [IsScalarTower α β γ] : IsScalarTower (Set α) (Set β) (Set γ)
• Submonoid.pow_smul_mem_closure_smul 📋 Mathlib.Algebra.Group.Submonoid.Pointwise
  {M : Type u_3} [Monoid M] {N : Type u_7} [CommMonoid N] [MulAction M N] [IsScalarTower M N N] (r : M) (s : Set N) {x : N} (hx : x ∈ Submonoid.closure s) : ∃ n, r ^ n • x ∈ Submonoid.closure (r • s)
• MulAction.le_stabilizer_smul_left 📋 Mathlib.GroupTheory.GroupAction.Defs
  {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G α] [MulAction G β] [SMul α β] [IsScalarTower G α β] (a : α) (b : β) : MulAction.stabilizer G a ≤ MulAction.stabilizer G (a • b)
• MulAction.stabilizer_mul_eq_left 📋 Mathlib.GroupTheory.GroupAction.Defs
  {G : Type u_1} {α : Type u_2} [Group G] [MulAction G α] [Group α] [IsScalarTower G α α] (a b : α) : MulAction.stabilizer G (a * b) = MulAction.stabilizer G a
• MulAction.stabilizer_smul_eq_left 📋 Mathlib.GroupTheory.GroupAction.Defs
  {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [MulAction G α] [MulAction G β] [SMul α β] [IsScalarTower G α β] (a : α) (b : β) (h : Function.Injective fun x => x • b) : MulAction.stabilizer G (a • b) = MulAction.stabilizer G a
• 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.instIsScalarTower_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] : IsScalarTower R (OreLocalization S M) (OreLocalization S X)
• 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)
• OreLocalization.instIsCentralScalar 📋 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] [SMul Rᵐᵒᵖ M] [SMul Rᵐᵒᵖ X] [IsScalarTower Rᵐᵒᵖ M M] [IsScalarTower Rᵐᵒᵖ M X] [IsCentralScalar R M] : IsCentralScalar R (OreLocalization S X)
• OreLocalization.instIsScalarTower 📋 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] [SMul R R'] [IsScalarTower R R' M] : IsScalarTower R R' (OreLocalization S X)
• OreLocalization.smul_one_smul 📋 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] (r : R) (x : OreLocalization S X) : (r • 1) • x = r • x
• OreLocalization.smul_oreDiv 📋 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] (r : R) (x : X) (s : ↥S) : r • (x /ₒ s) = OreLocalization.oreNum (r • 1) s • x /ₒ OreLocalization.oreDenom (r • 1) s
• OreLocalization.smul_oreDiv_one 📋 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] (r : R) (x : X) : r • (x /ₒ 1) = r • x /ₒ 1
• OreLocalization.smul_one_oreDiv_one_smul 📋 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] (r : R) (x : OreLocalization S X) : (r • 1 /ₒ 1) • x = r • x
• Localization.smul_mk 📋 Mathlib.GroupTheory.MonoidLocalization.Basic
  {M : Type u_1} [CommMonoid M] {S : Submonoid M} {R : Type u_4} [SMul R M] [IsScalarTower R M M] (c : R) (a : M) (b : ↥S) : c • Localization.mk a b = Localization.mk (c • a) b
• 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
• RingHom.smulOneHom_apply 📋 Mathlib.Algebra.Module.RingHom
  {R : Type u_1} {S : Type u_2} [Semiring R] [NonAssocSemiring S] [Module R S] [IsScalarTower R S S] (x : R) : RingHom.smulOneHom x = x • 1
• IsSMulRegular.of_mul 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ab : IsSMulRegular M (a * b)) : IsSMulRegular M b
• IsSMulRegular.of_smul 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {s : S} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (a : R) (ab : IsSMulRegular M (a • s)) : IsSMulRegular M s
• IsSMulRegular.mul 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ra : IsSMulRegular M a) (rb : IsSMulRegular M b) : IsSMulRegular M (a * b)
• IsSMulRegular.mul_iff_right 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] (ha : IsSMulRegular M a) : IsSMulRegular M (a * b) ↔ IsSMulRegular M b
• IsSMulRegular.smul 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (ra : IsSMulRegular M a) (rs : IsSMulRegular M s) : IsSMulRegular M (a • s)
• IsSMulRegular.smul_iff 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} [SMul R M] [SMul R S] [SMul S M] [IsScalarTower R S M] (b : S) (ha : IsSMulRegular M a) : IsSMulRegular M (a • b) ↔ IsSMulRegular M b
• IsSMulRegular.mul_iff 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {M : Type u_3} {a b : R} [CommSemigroup R] [SMul R M] [IsScalarTower R R M] : IsSMulRegular M (a * b) ↔ IsSMulRegular M a ∧ IsSMulRegular M b
• IsSMulRegular.mul_and_mul_iff 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {M : Type u_3} {a b : R} [SMul R M] [Mul R] [IsScalarTower R R M] : IsSMulRegular M (a * b) ∧ IsSMulRegular M (b * a) ↔ IsSMulRegular M a ∧ IsSMulRegular M b
• IsSMulRegular.of_smul_eq_one 📋 Mathlib.Algebra.Regular.SMul
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {a : R} {s : S} [Monoid S] [SMul R M] [SMul R S] [MulAction S M] [IsScalarTower R S M] (h : a • s = 1) : IsSMulRegular M s
• IsScalarTower.smulHomClass 📋 Mathlib.GroupTheory.GroupAction.Hom
  (M' : Type u_1) (X : Type u_5) [SMul M' X] (Y : Type u_6) [SMul M' Y] (F : Type u_8) [FunLike F X Y] [MulOneClass X] [SMul X Y] [IsScalarTower M' X Y] [MulActionHomClass F X X Y] : MulActionHomClass F M' X Y
• LinearMap.IsScalarTower.compatibleSMul' 📋 Mathlib.Algebra.Module.LinearMap.Defs
  {M : Type u_8} [AddCommMonoid M] {R : Type u_14} {S : Type u_15} [Semiring S] [SMul R M] [Module S M] [SMul R S] [IsScalarTower R S M] : LinearMap.CompatibleSMul S M R S
• LinearMap.IsScalarTower.compatibleSMul 📋 Mathlib.Algebra.Module.LinearMap.Defs
  {M : Type u_8} {M₂ : Type u_10} [AddCommMonoid M] [AddCommMonoid M₂] {R : Type u_14} {S : Type u_15} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] [SMul R S] [IsScalarTower R S M] [IsScalarTower R S M₂] : LinearMap.CompatibleSMul M M₂ R S
• LinearMap.isScalarTower_of_injective 📋 Mathlib.Algebra.Module.LinearMap.Defs
  {M : Type u_8} {M₂ : Type u_10} [AddCommMonoid M] [AddCommMonoid M₂] (R : Type u_14) {S : Type u_15} [Semiring S] [SMul R M] [Module S M] [SMul R M₂] [Module S M₂] [SMul R S] [LinearMap.CompatibleSMul M M₂ R S] [IsScalarTower R S M₂] (f : M →ₗ[S] M₂) (hf : Function.Injective ⇑f) : IsScalarTower R S M
• LinearMap.mulRight_apply 📋 Mathlib.Algebra.Module.LinearMap.Defs
  (R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] (a b : A) : (LinearMap.mulRight R a) b = b * a
• LinearMap.mulRight_zero_eq_zero 📋 Mathlib.Algebra.Module.LinearMap.Defs
  (R : Type u_14) (A : Type u_15) [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] : LinearMap.mulRight R 0 = 0
• 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.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.toAddMonoidHom_mulRight 📋 Mathlib.Algebra.Module.LinearMap.Defs
  (R : Type u_14) {A : Type u_15} [Semiring R] [NonUnitalNonAssocSemiring A] [Module R A] [IsScalarTower R A A] (a : A) : ↑(LinearMap.mulRight R a) = AddMonoidHom.mulRight a
• algebraMap.coe_smul 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [SMul A C] [IsScalarTower A B C] : ↑(a • b) = a • ↑b
• algebraMap.smul 📋 Mathlib.Algebra.Algebra.Defs
  {A : Type u_1} {B : Type u_2} (a : A) (b : B) (C : Type u_3) [SMul A B] [CommSemiring B] [Semiring C] [Algebra B C] [SMul A C] [IsScalarTower A B C] : (algebraMap B C) (a • b) = a • (algebraMap B C) b
• Prod.isScalarTower 📋 Mathlib.Algebra.Group.Action.Prod
  {M : Type u_1} {N : Type u_2} {α : Type u_5} {β : Type u_6} [SMul M α] [SMul M β] [SMul N α] [SMul N β] [SMul M N] [IsScalarTower M N α] [IsScalarTower M N β] : IsScalarTower M N (α × β)
• Prod.isScalarTowerBoth 📋 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] [IsScalarTower M N N] [IsScalarTower M P P] : IsScalarTower M (N × P) (N × P)
• smulMulHom_apply 📋 Mathlib.Algebra.Group.Action.Prod
  {α : Type u_5} {β : Type u_6} [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_5} {β : Type u_6} [Monoid α] [MulOneClass β] [MulAction α β] [IsScalarTower α β β] [SMulCommClass α β β] (a✝ : α × β) : smulMonoidHom a✝ = smulMulHom.toFun a✝
• 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_5} {N₀ : Type u_6} [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_7} [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.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.instIsScalarTower 📋 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] [SMul M N] [SMulZeroClass M B] [SMulZeroClass N B] [IsScalarTower M N B] : IsScalarTower M N (ZeroHom A B)
• AddMonoidHom.instIsScalarTower 📋 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] [SMul M N] [DistribSMul M B] [DistribSMul N B] [IsScalarTower M N B] : IsScalarTower M N (A →+ B)
• AddMonoid.End.isScalarTower 📋 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] [SMul R S] [IsScalarTower R S A] : IsScalarTower 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.mulRight_inj 📋 Mathlib.Algebra.Module.LinearMap.Basic
  {R : Type u_6} {A : Type u_7} [Semiring R] [NonAssocSemiring A] [Module R A] [IsScalarTower R A A] {a b : A} : LinearMap.mulRight R a = LinearMap.mulRight R b ↔ a = b
• LinearMap.mulRight_one 📋 Mathlib.Algebra.Module.LinearMap.Basic
  (R : Type u_6) (A : Type u_7) [Semiring R] [NonAssocSemiring A] [Module R A] [IsScalarTower R A A] : LinearMap.mulRight R 1 = LinearMap.id
• LinearMap.mulRight_mul 📋 Mathlib.Algebra.Module.LinearMap.Basic
  (R : Type u_6) (A : Type u_7) [Semiring R] [NonUnitalSemiring A] [Module R A] [IsScalarTower R A A] (a b : A) : LinearMap.mulRight R (a * b) = LinearMap.mulRight R b ∘ₗ LinearMap.mulRight R a
• LinearMap.mulRight_eq_zero_iff 📋 Mathlib.Algebra.Module.LinearMap.Basic
  (R : Type u_6) (A : Type u_7) [Semiring R] [NonAssocSemiring A] [Module R A] [IsScalarTower R A A] (a : A) : LinearMap.mulRight R a = 0 ↔ a = 0
• 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
• 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)
• LinearMap.smulRight_zero 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {S : Type u_3} {M : Type u_4} {M₁ : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] [Semiring S] [Module R S] [Module S M] [IsScalarTower R S M] (f : M₁ →ₗ[R] S) : f.smulRight 0 = 0
• LinearMap.zero_smulRight 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {S : Type u_3} {M : Type u_4} {M₁ : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] [Semiring S] [Module R S] [Module S M] [IsScalarTower R S M] (x : M) : LinearMap.smulRight 0 x = 0
• LinearMap.coe_smulRight 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {S : Type u_3} {M : Type u_4} {M₁ : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] [Semiring S] [Module R S] [Module S M] [IsScalarTower R S M] (f : M₁ →ₗ[R] S) (x : M) : ⇑(f.smulRight x) = fun c => f c • x
• LinearMap.smulRight_apply 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {S : Type u_3} {M : Type u_4} {M₁ : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] [Semiring S] [Module R S] [Module S M] [IsScalarTower R S M] (f : M₁ →ₗ[R] S) (x : M) (c : M₁) : (f.smulRight x) c = f c • x
• LinearMap.smulRight_apply_eq_zero_iff 📋 Mathlib.Algebra.Module.LinearMap.End
  {R : Type u_1} {S : Type u_3} {M : Type u_4} {M₁ : Type u_5} [Semiring R] [AddCommMonoid M] [AddCommMonoid M₁] [Module R M] [Module R M₁] [Semiring S] [Module R S] [Module S M] [IsScalarTower R S M] [IsDomain S] {f : M₁ →ₗ[R] S} {x : M} [Module.IsTorsionFree S M] : f.smulRight x = 0 ↔ f = 0 ∨ x = 0
• Units.toEquiv_mulRightLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (u : Aˣ) : (Units.mulRightLinearEquiv R u).toEquiv = u.mulRight
• Units.toLinearMap_mulRightLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (u : Aˣ) : ↑(Units.mulRightLinearEquiv R u) = LinearMap.mulRight R ↑u
• Units.symm_mulRightLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a : Aˣ) : (Units.mulRightLinearEquiv R a).symm = Units.mulRightLinearEquiv R a⁻¹
• 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.mulRightLinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  (R : Type u_9) {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a : Aˣ) (x : A) : (Units.mulRightLinearEquiv R a) x = x * ↑a
• Units.symm_mulRightLinearEquiv_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  (R : Type u_9) {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a : Aˣ) (x : A) : (Units.mulRightLinearEquiv R a).symm x = x * ↑a⁻¹
• Units.mulRightLinearEquiv_trans_mulRightLinearEquiv 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (a b : Aˣ) : Units.mulRightLinearEquiv R a ≪≫ₗ Units.mulRightLinearEquiv R b = Units.mulRightLinearEquiv R (a * b)
• Units.mulRightLinearEquiv_mul_apply 📋 Mathlib.Algebra.Module.Equiv.Basic
  {R : Type u_9} {A : Type u_10} [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] (u v : Aˣ) (x : A) : (Units.mulRightLinearEquiv R (u * v)) x = (Units.mulRightLinearEquiv R v) ((Units.mulRightLinearEquiv R u) x)
• 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
• Subgroup.instIsScalarTowerSubtypeMem 📋 Mathlib.Algebra.Group.Subgroup.Actions
  {G : Type u_1} {α : Type u_2} {β : Type u_3} [Group G] [SMul α β] [MulAction G α] [MulAction G β] [IsScalarTower G α β] (S : Subgroup G) : IsScalarTower (↥S) α β
• SMulMemClass.ofIsScalarTower 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  (S : Type u_1) (M : Type u_2) (N : Type u_3) (α : Type u_4) [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] : SMulMemClass S M α
• SetLike.instIsScalarTower 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] [Mul M] [MulMemClass S M] [IsScalarTower R M M] (s : S) : IsScalarTower R ↥s ↥s
• SetLike.instIsScalarTowerSubtypeMem 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) : IsScalarTower M N ↥s
• SetLike.instSMulCommClassSubtypeMem_1 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) [SMulCommClass M N α] : SMulCommClass M N ↥s
• SetLike.instSMulCommClassSubtypeMem_2 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) [SMulCommClass N M α] : SMulCommClass N M ↥s
• SubMulAction.isScalarTower 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) : IsScalarTower S R ↥p
• SubMulAction.isCentralScalar 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] : IsCentralScalar S ↥p
• SubMulAction.smul_of_tower_mem 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) (s : S) {x : M} (h : x ∈ p) : s • x ∈ p
• SubMulAction.isScalarTower' 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) {S' : Type u_1} [SMul S' R] [SMul S' S] [SMul S' M] [IsScalarTower S' R M] [IsScalarTower S' S M] : IsScalarTower S' S ↥p
• SetLike.val_smul_of_tower 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : ↥s) : ↑(r • x) = r • ↑x
• SubMulAction.smul_mem_iff' 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {R : Type u} {M : Type v} [Monoid R] [MulAction R M] (p : SubMulAction R M) {G : Type u_1} [Group G] [SMul G R] [MulAction G M] [IsScalarTower G R M] (g : G) {x : M} : g • x ∈ p ↔ x ∈ p
• SubMulAction.smul_mem_iff 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [GroupWithZero S] [Monoid R] [MulAction R M] [SMul S R] [MulAction S M] [IsScalarTower S R M] (p : SubMulAction R M) {s : S} {x : M} (s0 : s ≠ 0) : s • x ∈ p ↔ x ∈ p
• SetLike.mk_smul_of_tower_mk 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : α) (hx : x ∈ s) : r • ⟨x, hx⟩ = ⟨r • x, ⋯⟩
• SubMulAction.val_smul_of_tower 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {R : Type u} {M : Type v} [Monoid R] [MulAction R M] [SMul S R] [SMul S M] [IsScalarTower S R M] (p : SubMulAction R M) (s : S) (x : ↥p) : ↑(s • x) = s • ↑x
• SetLike.smul_of_tower_def 📋 Mathlib.GroupTheory.GroupAction.SubMulAction
  {S : Type u'} {M : Type v} {N : Type u_1} {α : Type u_2} [SetLike S α] [SMul M N] [SMul M α] [Monoid N] [MulAction N α] [SMulMemClass S N α] [IsScalarTower M N α] (s : S) (r : M) (x : ↥s) : r • x = ⟨r • ↑x, ⋯⟩
• Submodule.smul_of_tower_mem 📋 Mathlib.Algebra.Module.Submodule.Defs
  {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (h : x ∈ p) : r • x ∈ p
• Submodule.smul_mem_iff' 📋 Mathlib.Algebra.Module.Submodule.Defs
  {G : Type u''} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {x : M} [Group G] [MulAction G M] [SMul G R] [IsScalarTower G R M] (g : G) : g • x ∈ p ↔ x ∈ p
• Submodule.isScalarTower' 📋 Mathlib.Algebra.Module.Submodule.Defs
  {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) {S' : Type u_1} [SMul S R] [SMul S M] [SMul S' R] [SMul S' M] [SMul S S'] [IsScalarTower S' R M] [IsScalarTower S S' M] [IsScalarTower S R M] : IsScalarTower S S' ↥p
• Submodule.isScalarTower 📋 Mathlib.Algebra.Module.Submodule.Defs
  {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] : IsScalarTower S R ↥p
• Submodule.coe_smul_of_tower 📋 Mathlib.Algebra.Module.Submodule.Defs
  {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} {p : Submodule R M} [SMul S R] [SMul S M] [IsScalarTower S R M] (r : S) (x : ↥p) : ↑(r • x) = r • ↑x
• Submodule.isCentralScalar 📋 Mathlib.Algebra.Module.Submodule.Basic
  {S : Type u'} {R : Type u} {M : Type v} [Semiring R] [AddCommMonoid M] {module_M : Module R M} (p : Submodule R M) [SMul S R] [SMul S M] [IsScalarTower S R M] [SMul Sᵐᵒᵖ R] [SMul Sᵐᵒᵖ M] [IsScalarTower Sᵐᵒᵖ R M] [IsCentralScalar S M] : IsCentralScalar S ↥p
• Submodule.smul_mem_iff 📋 Mathlib.Algebra.Module.Submodule.Basic
  {S : Type u'} {R : Type u} {M : Type v} [DivisionSemiring S] [Semiring R] [AddCommMonoid M] [Module R M] [SMul S R] [Module S M] [IsScalarTower S R M] (p : Submodule R M) {s : S} {x : M} (s0 : s ≠ 0) : s • x ∈ p ↔ x ∈ p
• Submodule.restrictScalars_injective 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] : Function.Injective (Submodule.restrictScalars S)
• Submodule.toAddSubmonoid_restrictScalars 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (V : Submodule R M) : (Submodule.restrictScalars S V).toAddSubmonoid = V.toAddSubmonoid
• Submodule.restrictScalars_bot 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] : Submodule.restrictScalars S ⊥ = ⊥
• Submodule.restrictScalars_top 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] : Submodule.restrictScalars S ⊤ = ⊤
• Submodule.coe_restrictScalars 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (V : Submodule R M) : ↑(Submodule.restrictScalars S V) = ↑V
• Submodule.restrictScalars_monotone 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] : Monotone (Submodule.restrictScalars S)
• Submodule.restrictScalars_inj 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {V₁ V₂ : Submodule R M} : Submodule.restrictScalars S V₁ = Submodule.restrictScalars S V₂ ↔ V₁ = V₂
• Submodule.restrictScalars_eq_bot_iff 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {p : Submodule R M} : Submodule.restrictScalars S p = ⊥ ↔ p = ⊥
• Submodule.restrictScalars_eq_top_iff 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {p : Submodule R M} : Submodule.restrictScalars S p = ⊤ ↔ p = ⊤
• Submodule.restrictScalars_mem 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (V : Submodule R M) (m : M) : m ∈ Submodule.restrictScalars S V ↔ m ∈ V
• Submodule.restrictScalars_iInf 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {ι : Sort u_4} (s : ι → Submodule R M) : Submodule.restrictScalars S (iInf s) = ⨅ i, Submodule.restrictScalars S (s i)
• Submodule.codisjoint_restrictScalars_iff 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} : Codisjoint (Submodule.restrictScalars S s) (Submodule.restrictScalars S t) ↔ Codisjoint s t
• Submodule.disjoint_restrictScalars_iff 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} : Disjoint (Submodule.restrictScalars S s) (Submodule.restrictScalars S t) ↔ Disjoint s t
• Submodule.restrictScalars_sInf 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (s : Set (Submodule R M)) : Submodule.restrictScalars S (sInf s) = sInf (Submodule.restrictScalars S '' s)
• Submodule.restrictScalars_inf 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (s t : Submodule R M) : Submodule.restrictScalars S (s ⊓ t) = Submodule.restrictScalars S s ⊓ Submodule.restrictScalars S t
• Submodule.isCompl_restrictScalars_iff 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} : IsCompl (Submodule.restrictScalars S s) (Submodule.restrictScalars S t) ↔ IsCompl s t
• Submodule.restrictScalars_mono 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} (hst : s ≤ t) : Submodule.restrictScalars S s ≤ Submodule.restrictScalars S t
• Submodule.restrictScalars_le 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} : Submodule.restrictScalars S s ≤ Submodule.restrictScalars S t ↔ s ≤ t
• Submodule.restrictScalars_lt 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {s t : Submodule R M} : Submodule.restrictScalars S s < Submodule.restrictScalars S t ↔ s < t
• Submodule.restrictScalars_iSup 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] {ι : Sort u_4} (s : ι → Submodule R M) : Submodule.restrictScalars S (iSup s) = ⨆ i, Submodule.restrictScalars S (s i)
• Submodule.restrictScalars_sSup 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (s : Set (Submodule R M)) : Submodule.restrictScalars S (sSup s) = sSup (Submodule.restrictScalars S '' s)
• Submodule.restrictScalars_sup 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (s t : Submodule R M) : Submodule.restrictScalars S (s ⊔ t) = Submodule.restrictScalars S s ⊔ Submodule.restrictScalars S t
• Submodule.restrictScalars_restrictScalars 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) {R : Type u_2} {M : Type u_3} [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (T : Type u_4) [Semiring T] [SMul T R] [SMul S T] [Module T M] [IsScalarTower S T M] [IsScalarTower T R M] (V : Submodule R M) : Submodule.restrictScalars S (Submodule.restrictScalars T V) = Submodule.restrictScalars S V
• Submodule.restrictScalarsEmbedding_apply 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (V : Submodule R M) : (Submodule.restrictScalarsEmbedding S R M) V = Submodule.restrictScalars S V
• Submodule.restrictScalarsEquiv_apply 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) (a✝ : ↥p) : (Submodule.restrictScalarsEquiv S R M p) a✝ = a✝
• Submodule.restrictScalars.isScalarTower 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) : IsScalarTower S R ↥(Submodule.restrictScalars S p)
• Submodule.restrictScalarsEquiv_symm_apply 📋 Mathlib.Algebra.Module.Submodule.RestrictScalars
  (S : Type u_1) (R : Type u_2) (M : Type u_3) [Semiring R] [AddCommMonoid M] [Semiring S] [Module S M] [Module R M] [SMul S R] [IsScalarTower S R M] (p : Submodule R M) (a✝ : ↥p) : (Submodule.restrictScalarsEquiv S R M p).symm a✝ = a✝
• LinearMap.ker_restrictScalars 📋 Mathlib.Algebra.Module.Submodule.Ker
  (R : Type u_11) {S : Type u_12} {M : Type u_13} {N : Type u_14} [Semiring R] [Semiring S] [SMul R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [AddCommMonoid N] [Module R N] [Module S N] [IsScalarTower R S N] (f : M →ₗ[S] N) : (↑R f).ker = Submodule.restrictScalars R f.ker
• ULift.isScalarTower 📋 Mathlib.Algebra.Module.ULift
  {R : Type u} {M : Type v} {N : Type w} [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] : IsScalarTower (ULift.{u_1, u} R) M N
• ULift.isScalarTower' 📋 Mathlib.Algebra.Module.ULift
  {R : Type u} {M : Type v} {N : Type w} [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] : IsScalarTower R (ULift.{u_1, v} M) N
• ULift.isScalarTower'' 📋 Mathlib.Algebra.Module.ULift
  {R : Type u} {M : Type v} {N : Type w} [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] : IsScalarTower R M (ULift.{u_1, w} N)
• Subsemiring.isScalarTower 📋 Mathlib.Algebra.Ring.Subsemiring.Basic
  {R' : Type u_1} {α : Type u_2} {β : Type u_3} [NonAssocSemiring R'] [SMul α β] [SMul R' α] [SMul R' β] [IsScalarTower R' α β] (S : Subsemiring R') : IsScalarTower (↥S) α β
• Module.IsTorsionFree.trans_faithfulSMul 📋 Mathlib.Algebra.Algebra.Basic
  (R : Type u_1) (A : Type u_3) (M : Type u_4) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] [Nontrivial R] [IsCancelMulZero A] [AddCommMonoid M] [Module A M] [Module R M] [Module.IsTorsionFree A M] [IsScalarTower R A M] : Module.IsTorsionFree R M
• Module.IsTorsionFree.of_faithfulSMul 📋 Mathlib.Algebra.Algebra.Basic
  (R : Type u_1) (S : Type u_2) (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [FaithfulSMul R A] [Semiring S] [Module S R] [Module S A] [IsScalarTower S R A] [Module.IsTorsionFree S A] : Module.IsTorsionFree S R
• IsScalarTower.to_smulCommClass 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : SMulCommClass R A M
• IsScalarTower.to_smulCommClass' 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] : SMulCommClass A R M
• isSMulRegular_algebraMap_iff 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] (A : Type u_2) [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] {r : R} : IsSMulRegular M ((algebraMap R A) r) ↔ IsSMulRegular M r
• NeZero.of_faithfulSMul 📋 Mathlib.Algebra.Algebra.Basic
  (R : Type u_1) (A : Type u_2) [Semiring R] [Semiring A] [Module R A] [IsScalarTower R A A] [FaithfulSMul R A] (n : ℕ) [NeZero ↑n] : NeZero ↑n
• algebraMap_smul 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] (A : Type u_2) [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (r : R) (m : M) : (algebraMap R A) r • m = r • m
• algebra_compatible_smul 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] (A : Type u_2) [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (r : R) (m : M) : r • m = (algebraMap R A) r • m
• smul_algebra_smul_comm 📋 Mathlib.Algebra.Algebra.Basic
  {R : Type u_1} [CommSemiring R] {A : Type u_2} [Semiring A] [Algebra R A] {M : Type u_3} [AddCommMonoid M] [Module A M] [Module R M] [IsScalarTower R A M] (r : R) (a : A) (m : M) : a • r • m = r • a • m
• Module.algebraMap_end_apply 📋 Mathlib.Algebra.Algebra.Basic
  (R : Type u) (S : Type v) (M : Type w) [CommSemiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass S R M] [SMul R S] [IsScalarTower R S M] (a : R) (m : M) : ((algebraMap R (Module.End S M)) a) m = a • m
• Module.algebraMap_end_eq_smul_id 📋 Mathlib.Algebra.Algebra.Basic
  (R : Type u) (S : Type v) (M : Type w) [CommSemiring R] [Semiring S] [AddCommMonoid M] [Module R M] [Module S M] [SMulCommClass S R M] [SMul R S] [IsScalarTower R S M] (a : R) : (algebraMap R (Module.End S M)) a = a • LinearMap.id

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:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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.

5. 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 c026161