Loogle!

Result

Found 778 declarations mentioning TensorProduct.tmul. Of these, only the first 200 are shown.

• TensorProduct.tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  (R : Type u_1) [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n : N) : TensorProduct R M N
• IsAddUnit.tmul_left 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  (R : Type u_1) [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {n : N} (hn : IsAddUnit n) (m : M) : IsAddUnit (m ⊗ₜ[R] n)
• IsAddUnit.tmul_right 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  (R : Type u_1) [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {m : M} (hm : IsAddUnit m) (n : N) : IsAddUnit (m ⊗ₜ[R] n)
• TensorProduct.tmul_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} (N : Type u_6) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) : m ⊗ₜ[R] 0 = 0
• TensorProduct.zero_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] (M : Type u_5) {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (n : N) : 0 ⊗ₜ[R] n = 0
• TensorProduct.sum_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {α : Type u_9} (s : Finset α) (m : α → M) (n : N) : (∑ a ∈ s, m a) ⊗ₜ[R] n = ∑ a ∈ s, m a ⊗ₜ[R] n
• TensorProduct.tmul_sum 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) {α : Type u_9} (s : Finset α) (n : α → N) : m ⊗ₜ[R] ∑ a ∈ s, n a = ∑ a ∈ s, m ⊗ₜ[R] n a
• TensorProduct.ite_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x₁ : M) (x₂ : N) (P : Prop) [Decidable P] : (if P then x₁ else 0) ⊗ₜ[R] x₂ = if P then x₁ ⊗ₜ[R] x₂ else 0
• TensorProduct.tmul_ite 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x₁ : M) (x₂ : N) (P : Prop) [Decidable P] : (x₁ ⊗ₜ[R] if P then x₂ else 0) = if P then x₁ ⊗ₜ[R] x₂ else 0
• TensorProduct.add_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m₁ m₂ : M) (n : N) : (m₁ + m₂) ⊗ₜ[R] n = m₁ ⊗ₜ[R] n + m₂ ⊗ₜ[R] n
• TensorProduct.single_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {N : Type u_6} [AddCommMonoid N] [Module R N] {ι : Type u_9} [DecidableEq ι] {M : ι → Type u_10} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (x : N) (m : M i) (j : ι) : Pi.single i m j ⊗ₜ[R] x = Pi.single i (m ⊗ₜ[R] x) j
• TensorProduct.tmul_add 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n₁ n₂ : N) : m ⊗ₜ[R] (n₁ + n₂) = m ⊗ₜ[R] n₁ + m ⊗ₜ[R] n₂
• TensorProduct.tmul_single 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {N : Type u_6} [AddCommMonoid N] [Module R N] {ι : Type u_9} [DecidableEq ι] {M : ι → Type u_10} [(i : ι) → AddCommMonoid (M i)] [(i : ι) → Module R (M i)] (i : ι) (x : N) (m : M i) (j : ι) : x ⊗ₜ[R] Pi.single i m j = Pi.single i (x ⊗ₜ[R] m) j
• TensorProduct.exists_eq_tmul_of_forall 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  (R : Type u_1) [CommSemiring R] (M : Type u_5) (N : Type u_6) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R M N) (h : ∀ (m₁ m₂ : M) (n₁ n₂ : N), ∃ m n, m₁ ⊗ₜ[R] n₁ + m₂ ⊗ₜ[R] n₂ = m ⊗ₜ[R] n) : ∃ m n, x = m ⊗ₜ[R] n
• TensorProduct.smul_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} {R' : Type u_2} [CommSemiring R] [Monoid R'] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [DistribMulAction R' M] [Module R M] [Module R N] [DistribMulAction R' N] [TensorProduct.CompatibleSMul R R' M N] (r : R') (m : M) (n : N) : (r • m) ⊗ₜ[R] n = m ⊗ₜ[R] (r • n)
• TensorProduct.CompatibleSMul.mk 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} {R' : Type u_2} [CommSemiring R] [Monoid R'] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [DistribMulAction R' M] [Module R M] [Module R N] [DistribMulAction R' N] (smul_tmul : ∀ (r : R') (m : M) (n : N), (r • m) ⊗ₜ[R] n = m ⊗ₜ[R] (r • n)) : TensorProduct.CompatibleSMul R R' M N
• TensorProduct.CompatibleSMul.smul_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} {R' : Type u_2} {inst✝ : CommSemiring R} {inst✝¹ : Monoid R'} {M : Type u_5} {N : Type u_6} {inst✝² : AddCommMonoid M} {inst✝³ : AddCommMonoid N} {inst✝⁴ : DistribMulAction R' M} {inst✝⁵ : Module R M} {inst✝⁶ : Module R N} {inst✝⁷ : DistribMulAction R' N} [self : TensorProduct.CompatibleSMul R R' M N] (r : R') (m : M) (n : N) : (r • m) ⊗ₜ[R] n = m ⊗ₜ[R] (r • n)
• TensorProduct.induction_on 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {motive : TensorProduct R M N → Prop} (z : TensorProduct R M N) (zero : motive 0) (tmul : ∀ (x : M) (y : N), motive (x ⊗ₜ[R] y)) (add : ∀ (x y : TensorProduct R M N), motive x → motive y → motive (x + y)) : motive z
• TensorProduct.span_tmul_eq_top 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  (R : Type u_1) [CommSemiring R] (M : Type u_5) (N : Type u_6) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] : Submodule.span R {t | ∃ m n, m ⊗ₜ[R] n = t} = ⊤
• TensorProduct.smul_tmul_smul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (r s : R) (m : M) (n : N) : (r • m) ⊗ₜ[R] (s • n) = (r * s) • m ⊗ₜ[R] n
• TensorProduct.tmul_eq_smul_one_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} [AddCommMonoid M] [Module R M] {S : Type u_9} [Semiring S] [Module R S] [SMulCommClass R S S] (s : S) (m : M) : s ⊗ₜ[R] m = s • 1 ⊗ₜ[R] m
• TensorProduct.SMul.aux_of 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {R' : Type u_9} [SMul R' M] (r : R') (m : M) (n : N) : (TensorProduct.SMul.aux r) (FreeAddMonoid.of (m, n)) = (r • m) ⊗ₜ[R] n
• TensorProduct.smul_tmul' 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} {R' : Type u_2} [CommSemiring R] [Monoid R'] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [DistribMulAction R' M] [Module R M] [Module R N] [SMulCommClass R R' M] (r : R') (m : M) (n : N) : r • m ⊗ₜ[R] n = (r • m) ⊗ₜ[R] n
• TensorProduct.tmul_smul 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} {R' : Type u_2} [CommSemiring R] [Monoid R'] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [DistribMulAction R' M] [Module R M] [Module R N] [SMulCommClass R R' M] [DistribMulAction R' N] [TensorProduct.CompatibleSMul R R' M N] (r : R') (x : M) (y : N) : x ⊗ₜ[R] (r • y) = r • x ⊗ₜ[R] y
• TensorProduct.mk_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Defs
  {R : Type u_1} [CommSemiring R] {M : Type u_5} {N : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n : N) : ((TensorProduct.mk R M N) m) n = m ⊗ₜ[R] n
• TensorProduct.neg_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} [CommSemiring R] {M : Type u_2} {N : Type u_3} [AddCommGroup M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n : N) : (-m) ⊗ₜ[R] n = -m ⊗ₜ[R] n
• TensorProduct.tmul_neg 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} [CommSemiring R] {M : Type u_2} {P : Type u_4} [AddCommGroup M] [AddCommGroup P] [Module R M] [Module R P] (m : M) (p : P) : m ⊗ₜ[R] (-p) = -m ⊗ₜ[R] p
• TensorProduct.sub_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} [CommSemiring R] {M : Type u_2} {N : Type u_3} [AddCommGroup M] [AddCommMonoid N] [Module R M] [Module R N] (m₁ m₂ : M) (n : N) : (m₁ - m₂) ⊗ₜ[R] n = m₁ ⊗ₜ[R] n - m₂ ⊗ₜ[R] n
• TensorProduct.tmul_sub 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} [CommSemiring R] {M : Type u_2} {P : Type u_4} [AddCommGroup M] [AddCommGroup P] [Module R M] [Module R P] (m : M) (p₁ p₂ : P) : m ⊗ₜ[R] (p₁ - p₂) = m ⊗ₜ[R] p₁ - m ⊗ₜ[R] p₂
• TensorProduct.comm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  (R : Type u_1) [CommSemiring R] (M : Type u_6) (N : Type u_7) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n : N) : (TensorProduct.comm R M N) (m ⊗ₜ[R] n) = n ⊗ₜ[R] m
• TensorProduct.ext' 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] {g h : TensorProduct R M N →ₛₗ[σ₁₂] P₂} (H : ∀ (x : M) (y : N), g (x ⊗ₜ[R] y) = h (x ⊗ₜ[R] y)) : g = h
• TensorProduct.curry_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f : TensorProduct R M N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : ((TensorProduct.curry f) m) n = f (m ⊗ₜ[R] n)
• TensorProduct.comm_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  (R : Type u_1) [CommSemiring R] (M : Type u_6) (N : Type u_7) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (m : M) (n : N) : (TensorProduct.comm R M N).symm (n ⊗ₜ[R] m) = m ⊗ₜ[R] n
• TensorProduct.liftAux_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f' : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : (TensorProduct.liftAux f') (m ⊗ₜ[R] n) = (f' m) n
• TensorProduct.lift.tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] {f' : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂} (x : M) (y : N) : (TensorProduct.lift f') (x ⊗ₜ[R] y) = (f' x) y
• TensorProduct.mapOfCompatibleSMul_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  (R : Type u_1) [CommSemiring R] (A : Type u_13) (S : Type u_14) (M : Type u_15) (N : Type u_16) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [CommSemiring A] [Module A M] [Module A N] [SMulCommClass R A M] [CommSemiring S] [Module S M] [SMulCommClass R S M] [SMulCommClass A S M] [TensorProduct.CompatibleSMul R A M N] (m : M) (n : N) : (TensorProduct.mapOfCompatibleSMul R A S M N) (m ⊗ₜ[A] n) = m ⊗ₜ[R] n
• TensorProduct.lift.tmul' 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] {f' : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂} (x : M) (y : N) : (TensorProduct.lift f').toAddHom (x ⊗ₜ[R] y) = (f' x) y
• TensorProduct.lift.unique 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] {f' : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂} {g : TensorProduct R M N →ₛₗ[σ₁₂] P₂} (H : ∀ (x : M) (y : N), g (x ⊗ₜ[R] y) = (f' x) y) : g = TensorProduct.lift f'
• TensorProduct.liftAddHom_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} [CommSemiring R] {M : Type u_6} {N : Type u_7} {P : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] (f : M →+ N →+ P) (hf : ∀ (r : R) (m : M) (n : N), (f (r • m)) n = (f m) (r • n)) (m : M) (n : N) : (TensorProduct.liftAddHom f hf) (m ⊗ₜ[R] n) = (f m) n
• TensorProduct.ext_threefold 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P : Type u_8} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] [Module R P] {g h : TensorProduct R (TensorProduct R M N) P →ₛₗ[σ₁₂] P₂} (H : ∀ (x : M) (y : N) (z : P), g (x ⊗ₜ[R] y ⊗ₜ[R] z) = h (x ⊗ₜ[R] y ⊗ₜ[R] z)) : g = h
• TensorProduct.ext_threefold' 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P : Type u_8} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] [Module R P] {g h : TensorProduct R M (TensorProduct R N P) →ₛₗ[σ₁₂] P₂} (H : ∀ (x : M) (y : N) (z : P), g (x ⊗ₜ[R] (y ⊗ₜ[R] z)) = h (x ⊗ₜ[R] (y ⊗ₜ[R] z))) : g = h
• TensorProduct.ext_fourfold' 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P : Type u_8} {Q : Type u_9} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] [Module R P] [Module R Q] {φ ψ : TensorProduct R (TensorProduct R M N) (TensorProduct R P Q) →ₛₗ[σ₁₂] P₂} (H : ∀ (w : M) (x : N) (y : P) (z : Q), φ (w ⊗ₜ[R] x ⊗ₜ[R] (y ⊗ₜ[R] z)) = ψ (w ⊗ₜ[R] x ⊗ₜ[R] (y ⊗ₜ[R] z))) : φ = ψ
• TensorProduct.lcurry_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f : TensorProduct R M N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : (((TensorProduct.lcurry σ₁₂ M N P₂) f) m) n = f (m ⊗ₜ[R] n)
• TensorProduct.uncurry_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : ((TensorProduct.uncurry σ₁₂ M N P₂) f) (m ⊗ₜ[R] n) = (f m) n
• TensorProduct.lift.equiv_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] (σ₁₂ : R →+* R₂) (M : Type u_6) (N : Type u_7) (P₂ : Type u_11) [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f : M →ₛₗ[σ₁₂] N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : ((TensorProduct.lift.equiv σ₁₂ M N P₂) f) (m ⊗ₜ[R] n) = (f m) n
• TensorProduct.ext_fourfold 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P : Type u_8} {Q : Type u_9} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] [Module R P] [Module R Q] {g h : TensorProduct R (TensorProduct R (TensorProduct R M N) P) Q →ₛₗ[σ₁₂] P₂} (H : ∀ (w : M) (x : N) (y : P) (z : Q), g (w ⊗ₜ[R] x ⊗ₜ[R] y ⊗ₜ[R] z) = h (w ⊗ₜ[R] x ⊗ₜ[R] y ⊗ₜ[R] z)) : g = h
• TensorProduct.ext_fourfold'' 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_6} {N : Type u_7} {P : Type u_8} {Q : Type u_9} {P₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] [Module R P] [Module R Q] {φ ψ : TensorProduct R (TensorProduct R M (TensorProduct R N P)) Q →ₛₗ[σ₁₂] P₂} (H : ∀ (w : M) (x : N) (y : P) (z : Q), φ (w ⊗ₜ[R] (x ⊗ₜ[R] y) ⊗ₜ[R] z) = ψ (w ⊗ₜ[R] (x ⊗ₜ[R] y) ⊗ₜ[R] z)) : φ = ψ
• TensorProduct.lift.equiv_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basic
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] (σ₁₂ : R →+* R₂) (M : Type u_6) (N : Type u_7) (P₂ : Type u_11) [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P₂] [Module R M] [Module R N] [Module R₂ P₂] (f : TensorProduct R M N →ₛₗ[σ₁₂] P₂) (m : M) (n : N) : (((TensorProduct.lift.equiv σ₁₂ M N P₂).symm f) m) n = f (m ⊗ₜ[R] n)
• LinearMap.lTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) (m : M) (n : N) : (LinearMap.lTensor M f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n
• LinearMap.rTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) (m : M) (n : N) : (LinearMap.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m
• TensorProduct.map_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_4} {N : Type u_5} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R M] [Module R N] [Module R₂ M₂] [Module R₂ N₂] (f : M →ₛₗ[σ₁₂] M₂) (g : N →ₛₗ[σ₁₂] N₂) (m : M) (n : N) : (TensorProduct.map f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R₂] g n
• TensorProduct.range_map_eq_span_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : (TensorProduct.map f g).range = Submodule.span R {t | ∃ m n, f m ⊗ₜ[R] g n = t}
• LinearEquiv.lTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (n : N) : (LinearEquiv.lTensor M f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n
• LinearEquiv.rTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (n : N) : (LinearEquiv.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m
• TensorProduct.congr_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_4} {N : Type u_5} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R M] [Module R N] [Module R₂ M₂] [Module R₂ N₂] {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : M ≃ₛₗ[σ₁₂] M₂) (g : N ≃ₛₗ[σ₁₂] N₂) (m : M) (n : N) : (TensorProduct.congr f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R₂] g n
• TensorProduct.congr_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_4} {N : Type u_5} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R M] [Module R N] [Module R₂ M₂] [Module R₂ N₂] {σ₂₁ : R₂ →+* R} [RingHomInvPair σ₁₂ σ₂₁] [RingHomInvPair σ₂₁ σ₁₂] (f : M ≃ₛₗ[σ₁₂] M₂) (g : N ≃ₛₗ[σ₁₂] N₂) (p : M₂) (q : N₂) : (TensorProduct.congr f g).symm (p ⊗ₜ[R₂] q) = f.symm p ⊗ₜ[R] g.symm q
• LinearEquiv.lTensor_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (p : P) : (LinearEquiv.lTensor M f).symm (m ⊗ₜ[R] p) = m ⊗ₜ[R] f.symm p
• LinearEquiv.rTensor_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) (m : M) (p : P) : (LinearEquiv.rTensor M f).symm (p ⊗ₜ[R] m) = f.symm p ⊗ₜ[R] m
• TensorProduct.lTensorHomToHomLTensor_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {P : Type u_6} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid P] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R₂ M₂] [Module R₂ N₂] [Module R P] (m₂ : M₂) (f : P →ₛₗ[σ₁₂] N₂) (p : P) : ((TensorProduct.lTensorHomToHomLTensor σ₁₂ P M₂ N₂) (m₂ ⊗ₜ[R₂] f)) p = m₂ ⊗ₜ[R₂] f p
• TensorProduct.rTensorHomToHomRTensor_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {P : Type u_6} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid P] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R₂ M₂] [Module R₂ N₂] [Module R P] (f : P →ₛₗ[σ₁₂] M₂) (n₂ : N₂) (p : P) : ((TensorProduct.rTensorHomToHomRTensor σ₁₂ P M₂ N₂) (f ⊗ₜ[R₂] n₂)) p = f p ⊗ₜ[R₂] n₂
• TensorProduct.map₂_apply_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} {R₃ : Type u_3} [CommSemiring R] [CommSemiring R₂] [CommSemiring R₃] {σ₂₃ : R₂ →+* R₃} {σ₁₃ : R →+* R₃} {M : Type u_4} {N : Type u_5} {M₂ : Type u_9} {M₃ : Type u_10} {N₂ : Type u_11} {N₃ : Type u_12} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid M₂] [AddCommMonoid N₂] [AddCommMonoid M₃] [AddCommMonoid N₃] [Module R M] [Module R N] [Module R₂ M₂] [Module R₂ N₂] [Module R₃ M₃] [Module R₃ N₃] (f : M →ₛₗ[σ₁₃] M₂ →ₛₗ[σ₂₃] M₃) (g : N →ₛₗ[σ₁₃] N₂ →ₛₗ[σ₂₃] N₃) (m : M) (n : N) : (TensorProduct.map₂ f g) (m ⊗ₜ[R] n) = TensorProduct.map (f m) (g n)
• TensorProduct.homTensorHomMap_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Map
  {R : Type u_1} {R₂ : Type u_2} [CommSemiring R] [CommSemiring R₂] {σ₁₂ : R →+* R₂} {M : Type u_4} {N : Type u_5} {M₂ : Type u_9} {N₂ : Type u_11} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid M₂] [AddCommMonoid N₂] [Module R M] [Module R N] [Module R₂ M₂] [Module R₂ N₂] (f : M →ₛₗ[σ₁₂] M₂) (g : N →ₛₗ[σ₁₂] N₂) : (TensorProduct.homTensorHomMap σ₁₂ M N M₂ N₂) (f ⊗ₜ[R₂] g) = TensorProduct.map f g
• LinearMap.mul'_apply 📋 Mathlib.Algebra.Algebra.Bilinear
  {R : Type u_1} {A : Type u_2} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] {a b : A} : (LinearMap.mul' R A) (a ⊗ₜ[R] b) = a * b
• TensorProduct.lid_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] (m : M) (r : R) : (TensorProduct.lid R M) (r ⊗ₜ[R] m) = r • m
• TensorProduct.rid_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] (m : M) (r : R) : (TensorProduct.rid R M) (m ⊗ₜ[R] r) = r • m
• TensorProduct.lid_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] (m : M) : (TensorProduct.lid R M).symm m = 1 ⊗ₜ[R] m
• TensorProduct.rid_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] (m : M) : (TensorProduct.rid R M).symm m = m ⊗ₜ[R] 1
• TensorProduct.lidOfCompatibleSMul_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] (A : Type u_3) (M : Type u_4) [AddCommMonoid M] [Module R M] [CommSemiring A] [Module A M] [Module R A] [SMulCommClass R A A] [TensorProduct.CompatibleSMul R A A M] [TensorProduct.CompatibleSMul A R A M] (a : A) (m : M) : (TensorProduct.lidOfCompatibleSMul R A M) (a ⊗ₜ[R] m) = a • m
• TensorProduct.includeRight_lid 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] {S : Type u_10} [Semiring S] [Algebra R S] (m : TensorProduct R R M) : 1 ⊗ₜ[R] (TensorProduct.lid R M) m = (LinearMap.rTensor M (Algebra.algHom R R S).toLinearMap) m
• TensorProduct.assoc_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (m : M) (n : N) (p : P) : (TensorProduct.assoc R M N P) (m ⊗ₜ[R] n ⊗ₜ[R] p) = m ⊗ₜ[R] (n ⊗ₜ[R] p)
• TensorProduct.leftComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (m : M) (n : N) (p : P) : (TensorProduct.leftComm R M N P) (m ⊗ₜ[R] (n ⊗ₜ[R] p)) = n ⊗ₜ[R] (m ⊗ₜ[R] p)
• TensorProduct.rightComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (m : M) (n : N) (p : P) : (TensorProduct.rightComm R M N P) (m ⊗ₜ[R] n ⊗ₜ[R] p) = m ⊗ₜ[R] p ⊗ₜ[R] n
• TensorProduct.assoc_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  {R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (m : M) (n : N) (p : P) : (TensorProduct.assoc R M N P).symm (m ⊗ₜ[R] (n ⊗ₜ[R] p)) = m ⊗ₜ[R] n ⊗ₜ[R] p
• TensorProduct.leftComm_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (m : M) (n : N) (p : P) : (TensorProduct.leftComm R M N P).symm (n ⊗ₜ[R] (m ⊗ₜ[R] p)) = m ⊗ₜ[R] (n ⊗ₜ[R] p)
• TensorProduct.tensorTensorTensorComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (m : M) (n : N) (p : P) (q : Q) : (TensorProduct.tensorTensorTensorComm R M N P Q) (m ⊗ₜ[R] n ⊗ₜ[R] (p ⊗ₜ[R] q)) = m ⊗ₜ[R] p ⊗ₜ[R] (n ⊗ₜ[R] q)
• TensorProduct.tensorTensorTensorAssoc_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (m : M) (n : N) (p : P) (q : Q) : (TensorProduct.tensorTensorTensorAssoc R M N P Q) (m ⊗ₜ[R] n ⊗ₜ[R] (p ⊗ₜ[R] q)) = m ⊗ₜ[R] (n ⊗ₜ[R] p) ⊗ₜ[R] q
• TensorProduct.tensorTensorTensorAssoc_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
  (R : Type u_1) [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (m : M) (n : N) (p : P) (q : Q) : (TensorProduct.tensorTensorTensorAssoc R M N P Q).symm (m ⊗ₜ[R] (n ⊗ₜ[R] p) ⊗ₜ[R] q) = m ⊗ₜ[R] n ⊗ₜ[R] (p ⊗ₜ[R] q)
• Submodule.tmul_mem_baseChange_of_mem 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type u_1} {M : Type u_2} {A : Type u_3} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] {p : Submodule R M} (a : A) {m : M} (hm : m ∈ p) : a ⊗ₜ[R] m ∈ Submodule.baseChange A p
• LinearMap.baseChange_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type u_1} {A : Type u_2} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] N) (a : A) (x : M) : (LinearMap.baseChange A f) (a ⊗ₜ[R] x) = a ⊗ₜ[R] f x
• TensorProduct.AlgebraTensorModule.rid_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} (A : Type uA) {M : Type uM} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (r : R) (m : M) : (TensorProduct.AlgebraTensorModule.rid R A M) (m ⊗ₜ[R] r) = r • m
• TensorProduct.AlgebraTensorModule.rid_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {M : Type uM} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] (m : M) : (TensorProduct.AlgebraTensorModule.rid R A M).symm m = m ⊗ₜ[R] 1
• TensorProduct.AlgebraTensorModule.map_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) (m : M) (n : N) : (TensorProduct.AlgebraTensorModule.map f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n
• TensorProduct.AlgebraTensorModule.ext 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] {g h : TensorProduct R M N →ₗ[A] P} (H : ∀ (x : M) (y : N), g (x ⊗ₜ[R] y) = h (x ⊗ₜ[R] y)) : g = h
• TensorProduct.AlgebraTensorModule.lift_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] N →ₗ[R] P) (x : M) (y : N) : (TensorProduct.AlgebraTensorModule.lift f) (x ⊗ₜ[R] y) = (f x) y
• LinearEquiv.baseChange_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type u_1) (A : Type u_2) (M : Type u_4) (N : Type u_5) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {e : M ≃ₗ[R] N} (a : A) (m : M) : (LinearEquiv.baseChange R A M N e) (a ⊗ₜ[R] m) = a ⊗ₜ[R] e m
• TensorProduct.AlgebraTensorModule.mk_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) [CommSemiring R] (A : Type u_1) (M : Type u_2) (N : Type u_3) [Semiring A] [AddCommMonoid M] [Module R M] [Module A M] [SMulCommClass R A M] [AddCommMonoid N] [Module R N] (m : M) : (TensorProduct.AlgebraTensorModule.mk R A M N) m = { toFun := fun x2 => m ⊗ₜ[R] x2, map_add' := ⋯, map_smul' := ⋯ }
• TensorProduct.AlgebraTensorModule.congr_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) (m : M) (n : N) : (TensorProduct.AlgebraTensorModule.congr f g) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] g n
• LinearEquiv.baseChange_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type u_1) (A : Type u_2) (M : Type u_4) (N : Type u_5) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {e : M ≃ₗ[R] N} (a : A) (n : N) : (LinearEquiv.baseChange R A M N e).symm (a ⊗ₜ[R] n) = a ⊗ₜ[R] e.symm n
• TensorProduct.AlgebraTensorModule.congr_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] (f : M ≃ₗ[A] P) (g : N ≃ₗ[R] Q) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.congr f g).symm (p ⊗ₜ[R] q) = f.symm p ⊗ₜ[R] g.symm q
• TensorProduct.AlgebraTensorModule.rTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) (m : M) (n : N) : ((TensorProduct.AlgebraTensorModule.rTensor R N) f) (m ⊗ₜ[R] n) = f m ⊗ₜ[R] n
• TensorProduct.AlgebraTensorModule.lTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {Q : Type uQ} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (f : N →ₗ[R] Q) (m : M) (n : N) : ((TensorProduct.AlgebraTensorModule.lTensor A M) f) (m ⊗ₜ[R] n) = m ⊗ₜ[R] f n
• TensorProduct.AlgebraTensorModule.leftComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.leftComm R A M P Q) (m ⊗ₜ[A] (p ⊗ₜ[R] q)) = p ⊗ₜ[A] (m ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.cancelBaseChange_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [Algebra A B] [IsScalarTower A B M] (m : M) (n : N) (a : A) : (TensorProduct.AlgebraTensorModule.cancelBaseChange R A B M N) (m ⊗ₜ[A] (a ⊗ₜ[R] n)) = (a • m) ⊗ₜ[R] n
• TensorProduct.AlgebraTensorModule.rightComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) {S : Type uS} (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.rightComm R S B M P Q) (m ⊗ₜ[S] p ⊗ₜ[R] q) = m ⊗ₜ[R] q ⊗ₜ[S] p
• TensorProduct.AlgebraTensorModule.leftComm_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.leftComm R A M P Q).symm (p ⊗ₜ[A] (m ⊗ₜ[R] q)) = m ⊗ₜ[A] (p ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.cancelBaseChange_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [Algebra A B] [IsScalarTower A B M] (m : M) (n : N) : (TensorProduct.AlgebraTensorModule.cancelBaseChange R A B M N).symm (m ⊗ₜ[R] n) = m ⊗ₜ[A] (1 ⊗ₜ[R] n)
• Submodule.coe_toBaseChange_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type u_1} {M : Type u_2} (A : Type u_3) [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] (p : Submodule R M) (a : A) (x : ↥p) : ↑((Submodule.toBaseChange A p) (a ⊗ₜ[R] x)) = a ⊗ₜ[R] ↑x
• TensorProduct.AlgebraTensorModule.assoc_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.assoc R A B M P Q) (m ⊗ₜ[A] p ⊗ₜ[R] q) = m ⊗ₜ[A] (p ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.rightComm_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) {S : Type uS} (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring B] [Algebra R B] [AddCommMonoid M] [Module R M] [Module B M] [IsScalarTower R B M] [AddCommMonoid P] [AddCommMonoid Q] [Module R Q] [CommSemiring S] [Module S M] [Module S P] [Algebra S B] [IsScalarTower S B M] [SMulCommClass R S M] [SMulCommClass S R M] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.rightComm R S B M P Q).symm (m ⊗ₜ[R] q ⊗ₜ[S] p) = m ⊗ₜ[S] p ⊗ₜ[R] q
• TensorProduct.AlgebraTensorModule.assoc_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) (B : Type uB) {M : Type uM} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] (m : M) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.assoc R A B M P Q).symm (m ⊗ₜ[A] (p ⊗ₜ[R] q)) = m ⊗ₜ[A] p ⊗ₜ[R] q
• TensorProduct.AlgebraTensorModule.uliftEquiv_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (m : M) (n : N) : (TensorProduct.AlgebraTensorModule.uliftEquiv R A M N) { down := m ⊗ₜ[R] n } = { down := m } ⊗ₜ[ULift.{u_3, uR} R] { down := n }
• TensorProduct.AlgebraTensorModule.distribBaseChange_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {N : Type uN} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (n : N) (q : Q) (a : A) : (TensorProduct.AlgebraTensorModule.distribBaseChange R A N Q) (a ⊗ₜ[R] (n ⊗ₜ[R] q)) = a ⊗ₜ[R] n ⊗ₜ[A] (1 ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.homTensorHomMap_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  {R : Type uR} {A : Type uA} {B : Type uB} {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] [AddCommMonoid Q] [Module R Q] [Module B P] [IsScalarTower R B P] [SMulCommClass A B P] (f : M →ₗ[A] P) (g : N →ₗ[R] Q) : (TensorProduct.AlgebraTensorModule.homTensorHomMap R A B M N P Q) (f ⊗ₜ[R] g) = TensorProduct.AlgebraTensorModule.map f g
• TensorProduct.AlgebraTensorModule.distribBaseChange_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {N : Type uN} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid N] [Module R N] [AddCommMonoid Q] [Module R Q] (n : N) (q : Q) (a b : A) : (TensorProduct.AlgebraTensorModule.distribBaseChange R A N Q).symm (a ⊗ₜ[R] n ⊗ₜ[A] (b ⊗ₜ[R] q)) = (a * b) ⊗ₜ[R] (n ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.down_uliftEquiv_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (m : ULift.{u_1, uM} M) (n : ULift.{u_2, uN} N) : ((TensorProduct.AlgebraTensorModule.uliftEquiv R A M N).symm (m ⊗ₜ[ULift.{u_4, uR} R] n)).down = m.down ⊗ₜ[R] n.down
• TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] (m : M) (n : N) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R S A B M N P Q) (m ⊗ₜ[S] n ⊗ₜ[A] (p ⊗ₜ[R] q)) = m ⊗ₜ[A] p ⊗ₜ[S] (n ⊗ₜ[R] q)
• TensorProduct.AlgebraTensorModule.tensorTensorTensorComm_symm_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
  (R : Type uR) (S : Type uS) (A : Type uA) (B : Type uB) {M : Type uM} {N : Type uN} {P : Type uP} {Q : Type uQ} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] [AddCommMonoid M] [Module R M] [Module A M] [Module B M] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid Q] [Module R Q] [Module R P] [IsScalarTower R A P] [Algebra A B] [IsScalarTower A B M] [CommSemiring S] [Algebra R S] [Algebra S B] [Module S M] [Module S N] [IsScalarTower R S M] [SMulCommClass A S M] [SMulCommClass S A M] [IsScalarTower S B M] [IsScalarTower R S N] (m : M) (n : N) (p : P) (q : Q) : (TensorProduct.AlgebraTensorModule.tensorTensorTensorComm R S A B M N P Q).symm (m ⊗ₜ[A] p ⊗ₜ[S] (n ⊗ₜ[R] q)) = m ⊗ₜ[S] n ⊗ₜ[A] (p ⊗ₜ[R] q)
• TensorProduct.directSumRight_tmul 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₂ : Type v₂} [DecidableEq ι₂] {M₁' : Type w₁'} {M₂ : ι₂ → Type w₂} [AddCommMonoid M₁'] [(i₂ : ι₂) → AddCommMonoid (M₂ i₂)] [Module R M₁'] [(i₂ : ι₂) → Module R (M₂ i₂)] [Module S M₁'] [IsScalarTower R S M₁'] (m : M₁') (n : DirectSum ι₂ fun i => M₂ i) (i : ι₂) : ((TensorProduct.directSumRight R S M₁' M₂) (m ⊗ₜ[R] n)) i = m ⊗ₜ[R] n i
• TensorProduct.directSumLeft_tmul 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} [DecidableEq ι₁] {M₁ : ι₁ → Type w₁} {M₂' : Type w₂'} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [AddCommMonoid M₂'] [(i₁ : ι₁) → Module R (M₁ i₁)] [Module R M₂'] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] (m : DirectSum ι₁ fun i => M₁ i) (n : M₂') (i : ι₁) : ((TensorProduct.directSumLeft R S M₁ M₂') (m ⊗ₜ[R] n)) i = m i ⊗ₜ[R] n
• TensorProduct.directSumRight_tmul_lof 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₂ : Type v₂} [DecidableEq ι₂] {M₁' : Type w₁'} {M₂ : ι₂ → Type w₂} [AddCommMonoid M₁'] [(i₂ : ι₂) → AddCommMonoid (M₂ i₂)] [Module R M₁'] [(i₂ : ι₂) → Module R (M₂ i₂)] [Module S M₁'] [IsScalarTower R S M₁'] (x : M₁') (i : ι₂) (y : M₂ i) : (TensorProduct.directSumRight R S M₁' M₂) (x ⊗ₜ[R] (DirectSum.lof R ι₂ M₂ i) y) = (DirectSum.lof S ι₂ (fun i => TensorProduct R M₁' (M₂ i)) i) (x ⊗ₜ[R] y)
• TensorProduct.directSumLeft_tmul_lof 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} [DecidableEq ι₁] {M₁ : ι₁ → Type w₁} {M₂' : Type w₂'} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [AddCommMonoid M₂'] [(i₁ : ι₁) → Module R (M₁ i₁)] [Module R M₂'] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] (i : ι₁) (x : M₁ i) (y : M₂') : (TensorProduct.directSumLeft R S M₁ M₂') ((DirectSum.lof S ι₁ M₁ i) x ⊗ₜ[R] y) = (DirectSum.lof S ι₁ (fun i => TensorProduct R (M₁ i) M₂') i) (x ⊗ₜ[R] y)
• TensorProduct.directSumRight_symm_lof_tmul 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₂ : Type v₂} [DecidableEq ι₂] {M₁' : Type w₁'} {M₂ : ι₂ → Type w₂} [AddCommMonoid M₁'] [(i₂ : ι₂) → AddCommMonoid (M₂ i₂)] [Module R M₁'] [(i₂ : ι₂) → Module R (M₂ i₂)] [Module S M₁'] [IsScalarTower R S M₁'] (x : M₁') (i : ι₂) (y : M₂ i) : (TensorProduct.directSumRight R S M₁' M₂).symm ((DirectSum.lof S ι₂ (fun i => TensorProduct R M₁' (M₂ i)) i) (x ⊗ₜ[R] y)) = x ⊗ₜ[R] (DirectSum.lof R ι₂ M₂ i) y
• TensorProduct.directSumLeft_symm_lof_tmul 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} [DecidableEq ι₁] {M₁ : ι₁ → Type w₁} {M₂' : Type w₂'} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [AddCommMonoid M₂'] [(i₁ : ι₁) → Module R (M₁ i₁)] [Module R M₂'] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] (i : ι₁) (x : M₁ i) (y : M₂') : (TensorProduct.directSumLeft R S M₁ M₂').symm ((DirectSum.lof S ι₁ (fun i => TensorProduct R (M₁ i) M₂') i) (x ⊗ₜ[R] y)) = (DirectSum.lof S ι₁ M₁ i) x ⊗ₜ[R] y
• TensorProduct.directSum_lof_tmul_lof 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} {ι₂ : Type v₂} [DecidableEq ι₁] [DecidableEq ι₂] {M₁ : ι₁ → Type w₁} {M₂ : ι₂ → Type w₂} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [(i₂ : ι₂) → AddCommMonoid (M₂ i₂)] [(i₁ : ι₁) → Module R (M₁ i₁)] [(i₂ : ι₂) → Module R (M₂ i₂)] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) : (TensorProduct.directSum R S M₁ M₂) ((DirectSum.lof S ι₁ M₁ i₁) m₁ ⊗ₜ[R] (DirectSum.lof R ι₂ M₂ i₂) m₂) = (DirectSum.lof S (ι₁ × ι₂) (fun i => TensorProduct R (M₁ i.1) (M₂ i.2)) (i₁, i₂)) (m₁ ⊗ₜ[R] m₂)
• TensorProduct.directSum_symm_lof_tmul 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
  (R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} {ι₂ : Type v₂} [DecidableEq ι₁] [DecidableEq ι₂] {M₁ : ι₁ → Type w₁} {M₂ : ι₂ → Type w₂} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [(i₂ : ι₂) → AddCommMonoid (M₂ i₂)] [(i₁ : ι₁) → Module R (M₁ i₁)] [(i₂ : ι₂) → Module R (M₂ i₂)] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] (i₁ : ι₁) (m₁ : M₁ i₁) (i₂ : ι₂) (m₂ : M₂ i₂) : (TensorProduct.directSum R S M₁ M₂).symm ((DirectSum.lof S (ι₁ × ι₂) (fun i => TensorProduct R (M₁ i.1) (M₂ i.2)) (i₁, i₂)) (m₁ ⊗ₜ[R] m₂)) = (DirectSum.lof S ι₁ M₁ i₁) m₁ ⊗ₜ[R] (DirectSum.lof R ι₂ M₂ i₂) m₂
• TensorProduct.finsuppScalarLeft_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} [CommSemiring R] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ R) (n : N) : (TensorProduct.finsuppScalarLeft R N ι) (p ⊗ₜ[R] n) = p.sum fun i m => fun₀ | i => m • n
• TensorProduct.finsuppScalarLeft_apply_tmul_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} [CommSemiring R] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ R) (n : N) (i : ι) : ((TensorProduct.finsuppScalarLeft R N ι) (p ⊗ₜ[R] n)) i = p i • n
• TensorProduct.finsuppScalarRight_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ R) : (TensorProduct.finsuppScalarRight R S M ι) (m ⊗ₜ[R] p) = p.sum fun i n => fun₀ | i => n • m
• TensorProduct.finsuppScalarRight_apply_tmul_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ R) (i : ι) : ((TensorProduct.finsuppScalarRight R S M ι) (m ⊗ₜ[R] p)) i = p i • m
• TensorProduct.finsuppScalarLeft_symm_apply_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} [CommSemiring R] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (i : ι) (n : N) : ((TensorProduct.finsuppScalarLeft R N ι).symm fun₀ | i => n) = (fun₀ | i => 1) ⊗ₜ[R] n
• TensorProduct.finsuppRight_tmul_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (i : ι) (m : M) (n : N) : (TensorProduct.finsuppRight R S M N ι) (m ⊗ₜ[R] fun₀ | i => n) = fun₀ | i => m ⊗ₜ[R] n
• TensorProduct.finsuppRight_apply_tmul_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ N) (i : ι) : ((TensorProduct.finsuppRight R S M N ι) (m ⊗ₜ[R] p)) i = m ⊗ₜ[R] p i
• TensorProduct.finsuppRight_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (m : M) (p : ι →₀ N) : (TensorProduct.finsuppRight R S M N ι) (m ⊗ₜ[R] p) = p.sum fun i n => fun₀ | i => m ⊗ₜ[R] n
• TensorProduct.finsuppScalarRight_symm_apply_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {ι : Type u_5} [DecidableEq ι] (i : ι) (m : M) : ((TensorProduct.finsuppScalarRight R S M ι).symm fun₀ | i => m) = m ⊗ₜ[R] fun₀ | i => 1
• finsuppTensorFinsuppLid_single_tmul_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (a : ι) (b : κ) (r : R) (n : N) : (finsuppTensorFinsuppLid R N ι κ) ((fun₀ | a => r) ⊗ₜ[R] fun₀ | b => n) = fun₀ | (a, b) => r • n
• finsuppTensorFinsuppRid_single_tmul_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (a : ι) (b : κ) (m : M) (r : R) : (finsuppTensorFinsuppRid R M ι κ) ((fun₀ | a => m) ⊗ₜ[R] fun₀ | b => r) = fun₀ | (a, b) => r • m
• TensorProduct.finsuppRight_symm_apply_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (i : ι) (m : M) (n : N) : ((TensorProduct.finsuppRight R S M N ι).symm fun₀ | i => m ⊗ₜ[R] n) = m ⊗ₜ[R] fun₀ | i => n
• finsuppTensorFinsuppLid_apply_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (f : ι →₀ R) (g : κ →₀ N) (a : ι) (b : κ) : ((finsuppTensorFinsuppLid R N ι κ) (f ⊗ₜ[R] g)) (a, b) = f a • g b
• finsuppTensorFinsuppRid_apply_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (f : ι →₀ M) (g : κ →₀ R) (a : ι) (b : κ) : ((finsuppTensorFinsuppRid R M ι κ) (f ⊗ₜ[R] g)) (a, b) = g b • f a
• TensorProduct.finsuppLeft_apply_tmul_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ M) (n : N) (i : ι) : ((TensorProduct.finsuppLeft R S M N ι) (p ⊗ₜ[R] n)) i = p i ⊗ₜ[R] n
• TensorProduct.finsuppLeft_apply_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (p : ι →₀ M) (n : N) : (TensorProduct.finsuppLeft R S M N ι) (p ⊗ₜ[R] n) = p.sum fun i m => fun₀ | i => m ⊗ₜ[R] n
• finsuppTensorFinsupp'_single_tmul_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (a : ι) (b : κ) (r₁ r₂ : R) : (finsuppTensorFinsupp' R ι κ) ((fun₀ | a => r₁) ⊗ₜ[R] fun₀ | b => r₂) = fun₀ | (a, b) => r₁ * r₂
• finsuppTensorFinsupp'_apply_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (f : ι →₀ R) (g : κ →₀ R) (a : ι) (b : κ) : ((finsuppTensorFinsupp' R ι κ) (f ⊗ₜ[R] g)) (a, b) = f a * g b
• finsuppTensorFinsuppLid_symm_single_smul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid N] [Module R N] (i : ι × κ) (r : R) (n : N) : ((finsuppTensorFinsuppLid R N ι κ).symm fun₀ | i => r • n) = (fun₀ | i.1 => r) ⊗ₜ[R] fun₀ | i.2 => n
• finsuppTensorFinsuppRid_symm_single_smul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (M : Type u_3) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] (i : ι × κ) (m : M) (r : R) : ((finsuppTensorFinsuppRid R M ι κ).symm fun₀ | i => r • m) = (fun₀ | i.1 => m) ⊗ₜ[R] fun₀ | i.2 => r
• TensorProduct.finsuppLeft_symm_apply_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  {R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (i : ι) (m : M) (n : N) : ((TensorProduct.finsuppLeft R S M N ι).symm fun₀ | i => m ⊗ₜ[R] n) = (fun₀ | i => m) ⊗ₜ[R] n
• finsuppTensorFinsupp_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (i : ι) (m : M) (k : κ) (n : N) : (finsuppTensorFinsupp R S M N ι κ) ((fun₀ | i => m) ⊗ₜ[R] fun₀ | k => n) = fun₀ | (i, k) => m ⊗ₜ[R] n
• finsuppTensorFinsupp'_symm_single_eq_single_one_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r : R) : ((finsuppTensorFinsupp' R ι κ).symm fun₀ | i => r) = (fun₀ | i.1 => 1) ⊗ₜ[R] fun₀ | i.2 => r
• finsuppTensorFinsupp'_symm_single_eq_tmul_single_one 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r : R) : ((finsuppTensorFinsupp' R ι κ).symm fun₀ | i => r) = (fun₀ | i.1 => r) ⊗ₜ[R] fun₀ | i.2 => 1
• finsuppTensorFinsupp'_symm_single_mul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] (i : ι × κ) (r₁ r₂ : R) : ((finsuppTensorFinsupp' R ι κ).symm fun₀ | i => r₁ * r₂) = (fun₀ | i.1 => r₁) ⊗ₜ[R] fun₀ | i.2 => r₂
• finsuppTensorFinsupp_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) : ((finsuppTensorFinsupp R S M N ι κ) (f ⊗ₜ[R] g)) (i, k) = f i ⊗ₜ[R] g k
• finsuppTensorFinsupp_symm_single 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (S : Type u_2) (M : Type u_3) (N : Type u_4) (ι : Type u_5) (κ : Type u_6) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [Semiring S] [Algebra R S] [Module S M] [IsScalarTower R S M] (i : ι × κ) (m : M) (n : N) : ((finsuppTensorFinsupp R S M N ι κ).symm fun₀ | i => m ⊗ₜ[R] n) = (fun₀ | i.1 => m) ⊗ₜ[R] fun₀ | i.2 => n
• Finsupp.linearCombination_one_tmul 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
  (R : Type u_1) (S : Type u_2) (M : Type u_3) (ι : Type u_5) [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring S] [Algebra R S] [DecidableEq ι] {v : ι → M} : ↑R (Finsupp.linearCombination S fun x => 1 ⊗ₜ[R] v x) = LinearMap.lTensor S (Finsupp.linearCombination R v) ∘ₗ ↑(TensorProduct.finsuppScalarRight R R S ι).symm
• TensorProduct.eq_repr_basis_left 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (ℬ : Module.Basis ι R M) (x : TensorProduct R M N) : ∃ c, (c.sum fun i n => ℬ i ⊗ₜ[R] n) = x
• TensorProduct.eq_repr_basis_right 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (𝒞 : Module.Basis κ R N) (x : TensorProduct R M N) : ∃ b, (b.sum fun i m => m ⊗ₜ[R] 𝒞 i) = x
• TensorProduct.sum_tmul_basis_left_eq_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (ℬ : Module.Basis ι R M) (b : ι →₀ N) (h : (b.sum fun i n => ℬ i ⊗ₜ[R] n) = 0) : b = 0
• TensorProduct.sum_tmul_basis_right_eq_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (𝒞 : Module.Basis κ R N) (b : κ →₀ M) (h : (b.sum fun i m => m ⊗ₜ[R] 𝒞 i) = 0) : b = 0
• Module.Basis.baseChange_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (S : Type u_7) [Semiring S] [Algebra R S] (b : Module.Basis ι R M) (i : ι) : (Module.Basis.baseChange S b) i = 1 ⊗ₜ[R] b i
• Module.Basis.tensorProduct_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [AddCommMonoid N] [Module R N] (b : Module.Basis ι S M) (c : Module.Basis κ R N) (i : ι) (j : κ) : (b.tensorProduct c) (i, j) = b i ⊗ₜ[R] c j
• Module.Basis.tensorProduct_apply' 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [AddCommMonoid N] [Module R N] (b : Module.Basis ι S M) (c : Module.Basis κ R N) (i : ι × κ) : (b.tensorProduct c) i = b i.1 ⊗ₜ[R] c i.2
• TensorProduct.equivFinsuppOfBasisLeft_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq ι] (ℬ : Module.Basis ι R M) (b : ι →₀ N) : (TensorProduct.equivFinsuppOfBasisLeft ℬ).symm b = b.sum fun i n => ℬ i ⊗ₜ[R] n
• TensorProduct.equivFinsuppOfBasisRight_symm_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq κ] (𝒞 : Module.Basis κ R N) (b : κ →₀ M) : (TensorProduct.equivFinsuppOfBasisRight 𝒞).symm b = b.sum fun i m => m ⊗ₜ[R] 𝒞 i
• TensorProduct.equivFinsuppOfBasisLeft_apply_tmul_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq ι] (ℬ : Module.Basis ι R M) (m : M) (n : N) (i : ι) : ((TensorProduct.equivFinsuppOfBasisLeft ℬ) (m ⊗ₜ[R] n)) i = (ℬ.repr m) i • n
• TensorProduct.equivFinsuppOfBasisRight_apply_tmul_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq κ] (𝒞 : Module.Basis κ R N) (m : M) (n : N) (i : κ) : ((TensorProduct.equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ[R] n)) i = (𝒞.repr n) i • m
• TensorProduct.equivFinsuppOfBasisLeft_apply_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq ι] (ℬ : Module.Basis ι R M) (m : M) (n : N) : (TensorProduct.equivFinsuppOfBasisLeft ℬ) (m ⊗ₜ[R] n) = Finsupp.mapRange (fun x => x • n) ⋯ (ℬ.repr m)
• TensorProduct.equivFinsuppOfBasisRight_apply_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {N : Type u_4} {κ : Type u_6} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq κ] (𝒞 : Module.Basis κ R N) (m : M) (n : N) : (TensorProduct.equivFinsuppOfBasisRight 𝒞) (m ⊗ₜ[R] n) = Finsupp.mapRange (fun x => x • m) ⋯ (𝒞.repr n)
• Module.Basis.baseChange_repr_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {M : Type u_3} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] (S : Type u_7) [Semiring S] [Algebra R S] (b : Module.Basis ι R M) (x : S) (y : M) (i : ι) : ((Module.Basis.baseChange S b).repr (x ⊗ₜ[R] y)) i = (b.repr y) i • x
• Module.Basis.tensorProduct_repr_tmul_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
  {R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} {ι : Type u_5} {κ : Type u_6} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] [AddCommMonoid N] [Module R N] (b : Module.Basis ι S M) (c : Module.Basis κ R N) (m : M) (n : N) (i : ι) (j : κ) : ((b.tensorProduct c).repr (m ⊗ₜ[R] n)) (i, j) = (c.repr n) j • (b.repr m) i
• TensorProduct.star_tmul 📋 Mathlib.Algebra.Star.TensorProduct
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [StarRing R] [AddCommMonoid A] [StarAddMonoid A] [Module R A] [StarModule R A] [AddCommMonoid B] [StarAddMonoid B] [Module R B] [StarModule R B] (x : A) (y : B) : star (x ⊗ₜ[R] y) = star x ⊗ₜ[R] star y
• Algebra.TensorProduct.one_def 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommMonoidWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] : 1 = 1 ⊗ₜ[R] 1
• Algebra.TensorProduct.natCast_def 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommMonoidWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] (n : ℕ) : ↑n = ↑n ⊗ₜ[R] 1
• Algebra.TensorProduct.natCast_def' 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommMonoidWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] (n : ℕ) : ↑n = 1 ⊗ₜ[R] ↑n
• Algebra.TensorProduct.intCast_def 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [AddCommGroupWithOne A] [Module R A] [AddCommMonoidWithOne B] [Module R B] (z : ℤ) : ↑z = ↑z ⊗ₜ[R] 1
• Algebra.TensorProduct.includeRight_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (b : B) : Algebra.TensorProduct.includeRight b = 1 ⊗ₜ[R] b
• Algebra.TensorProduct.intCast_def' 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} [CommSemiring R] [Ring A] [Algebra R A] {B : Type u_2} [Ring B] [Algebra R B] (z : ℤ) : ↑z = 1 ⊗ₜ[R] ↑z
• Algebra.TensorProduct.includeLeft_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] (a : A) : Algebra.TensorProduct.includeLeft a = a ⊗ₜ[R] 1
• Algebra.TensorProduct.includeLeftRingHom_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a✝ : A) : Algebra.TensorProduct.includeLeftRingHom a✝ = a✝ ⊗ₜ[R] 1
• Algebra.TensorProduct.tmul_pow 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) (b : B) (k : ℕ) : a ⊗ₜ[R] b ^ k = (a ^ k) ⊗ₜ[R] (b ^ k)
• Algebra.TensorProduct.algebraMap_apply' 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (r : R) : (algebraMap R (TensorProduct R A B)) r = 1 ⊗ₜ[R] (algebraMap R B) r
• Algebra.TensorProduct.right_algebraMap_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [CommSemiring B] [Algebra R B] (b : B) : (algebraMap B (TensorProduct R A B)) b = 1 ⊗ₜ[R] b
• Algebra.TensorProduct.algebraMap_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [CommSemiring S] [Algebra S A] [SMulCommClass R S A] (r : S) : (algebraMap S (TensorProduct R A B)) r = (algebraMap S A) r ⊗ₜ[R] 1
• Commute.tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] {a₁ a₂ : A} {b₁ b₂ : B} (ha : Commute a₁ a₂) (hb : Commute b₁ b₂) : Commute (a₁ ⊗ₜ[R] b₁) (a₂ ⊗ₜ[R] b₂)
• SemiconjBy.tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] {a₁ a₂ a₃ : A} {b₁ b₂ b₃ : B} (ha : SemiconjBy a₁ a₂ a₃) (hb : SemiconjBy b₁ b₂ b₃) : SemiconjBy (a₁ ⊗ₜ[R] b₁) (a₂ ⊗ₜ[R] b₂) (a₃ ⊗ₜ[R] b₃)
• LinearMap.liftBaseChange_one_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {M : Type u_2} {N : Type u_3} (A : Type u_4) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module A N] [IsScalarTower R A N] (l : M →ₗ[R] N) (y : M) : (LinearMap.liftBaseChange A l) (1 ⊗ₜ[R] y) = l y
• Algebra.TensorProduct.tmul_mul_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (a₁ a₂ : A) (b₁ b₂ : B) : a₁ ⊗ₜ[R] b₁ * a₂ ⊗ₜ[R] b₂ = (a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)
• LinearMap.liftBaseChange_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {M : Type u_2} {N : Type u_3} (A : Type u_4) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module A N] [IsScalarTower R A N] (l : M →ₗ[R] N) (x : A) (y : M) : (LinearMap.liftBaseChange A l) (x ⊗ₜ[R] y) = x • l y
• LinearMap.mulLeft_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [Module R A] [Module R B] [SMulCommClass R A A] [SMulCommClass R B B] [IsScalarTower R A A] [IsScalarTower R B B] (a : A) (b : B) : LinearMap.mulLeft R (a ⊗ₜ[R] b) = TensorProduct.map (LinearMap.mulLeft R a) (LinearMap.mulLeft R b)
• LinearMap.mulRight_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [NonUnitalNonAssocSemiring A] [NonUnitalNonAssocSemiring B] [Module R A] [Module R B] [SMulCommClass R A A] [SMulCommClass R B B] [IsScalarTower R A A] [IsScalarTower R B B] (a : A) (b : B) : LinearMap.mulRight R (a ⊗ₜ[R] b) = TensorProduct.map (LinearMap.mulRight R a) (LinearMap.mulRight R b)
• TensorProduct.Algebra.smul_def 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] [SMulCommClass A B M] (a : A) (b : B) (m : M) : a ⊗ₜ[R] b • m = a • b • m
• AlgHom.liftEquiv_tmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_4} {S : Type u_5} {A : Type u_6} {B : Type u_7} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] [Algebra R S] [Algebra R B] [IsScalarTower R S B] (f : A →ₐ[R] B) (s : S) (a : A) : ((AlgHom.liftEquiv R S A B) f) (s ⊗ₜ[R] a) = s • f a
• Algebra.TensorProduct.ext' 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} {C : Type uC} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [Semiring C] [Algebra R C] [CommSemiring S] [Algebra S A] [Algebra R S] [Algebra S C] [IsScalarTower R S A] [IsScalarTower R S C] {g h : TensorProduct R A B →ₐ[S] C} (H : ∀ (a : A) (b : B), g (a ⊗ₜ[R] b) = h (a ⊗ₜ[R] b)) : g = h
• TensorProduct.Algebra.moduleAux_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} [CommSemiring R] [AddCommMonoid M] [Module R M] [Semiring A] [Semiring B] [Module A M] [Module B M] [Algebra R A] [Algebra R B] [IsScalarTower R A M] [IsScalarTower R B M] (a : A) (b : B) (m : M) : (TensorProduct.Algebra.moduleAux (a ⊗ₜ[R] b)) m = a • b • m
• AlgHom.liftEquiv_symm_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_4} {S : Type u_5} {A : Type u_6} {B : Type u_7} [CommSemiring R] [CommSemiring S] [Semiring A] [Semiring B] [Algebra R A] [Algebra S B] [Algebra R S] [Algebra R B] [IsScalarTower R S B] (f : TensorProduct R S A →ₐ[S] B) (a : A) : ((AlgHom.liftEquiv R S A B).symm f) a = f (1 ⊗ₜ[R] a)
• Algebra.TensorProduct.adjoin_one_tmul_image_eq_top 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [CommSemiring A] [Semiring B] [Algebra R A] [Algebra R B] (s : Set B) (hs : Algebra.adjoin R s = ⊤) : Algebra.adjoin A ((fun x => 1 ⊗ₜ[R] x) '' s) = ⊤
• Algebra.TensorProduct.mul_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonUnitalNonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonUnitalNonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (a₁ a₂ : A) (b₁ b₂ : B) : (Algebra.TensorProduct.mul (a₁ ⊗ₜ[R] b₁)) (a₂ ⊗ₜ[R] b₂) = (a₁ * a₂) ⊗ₜ[R] (b₁ * b₂)
• Algebra.TensorProduct.mul_one 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (x : TensorProduct R A B) : (Algebra.TensorProduct.mul x) (1 ⊗ₜ[R] 1) = x
• Algebra.TensorProduct.one_mul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [NonAssocSemiring A] [Module R A] [SMulCommClass R A A] [IsScalarTower R A A] [NonAssocSemiring B] [Module R B] [SMulCommClass R B B] [IsScalarTower R B B] (x : TensorProduct R A B) : (Algebra.TensorProduct.mul (1 ⊗ₜ[R] 1)) x = x
• LinearMap.liftBaseChangeEquiv_symm_apply 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_4} {M : Type u_2} {N : Type u_3} (A : Type u_1) [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] [Module A N] [IsScalarTower R A N] (l : TensorProduct R A M →ₗ[A] N) (x : M) : ((LinearMap.liftBaseChangeEquiv A).symm l) x = l (1 ⊗ₜ[R] x)
• Algebra.baseChange_lmul 📋 Mathlib.RingTheory.TensorProduct.Basic
  {R : Type u_1} {B : Type u_2} [CommSemiring R] [Semiring B] [Algebra R B] {A : Type u_3} [CommSemiring A] [Algebra R A] (f : B) : LinearMap.baseChange A ((Algebra.lmul R B) f) = (Algebra.lmul A (TensorProduct R A B)) (1 ⊗ₜ[R] f)
• Algebra.TensorProduct.tmul_one_eq_one_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (r : R) : (algebraMap R A) r ⊗ₜ[R] 1 = 1 ⊗ₜ[R] (algebraMap R B) r
• Algebra.TensorProduct.lid_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {A : Type uA} [CommSemiring R] [Semiring A] [Algebra R A] (r : R) (a : A) : (Algebra.TensorProduct.lid R A) (r ⊗ₜ[R] a) = r • a
• Algebra.TensorProduct.lmul'_apply_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {S : Type uS} [CommSemiring R] [CommSemiring S] [Algebra R S] (a b : S) : (Algebra.TensorProduct.lmul' R) (a ⊗ₜ[R] b) = a * b
• Algebra.TensorProduct.lid_symm_apply 📋 Mathlib.RingTheory.TensorProduct.Maps
  (R : Type uR) {A : Type uA} [CommSemiring R] [Semiring A] [Algebra R A] (a : A) : (Algebra.TensorProduct.lid R A).symm a = 1 ⊗ₜ[R] a
• Algebra.TensorProduct.productMap_left_apply 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A) : (Algebra.TensorProduct.productMap f g) (a ⊗ₜ[R] 1) = f a
• Algebra.TensorProduct.productMap_right_apply 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (b : B) : (Algebra.TensorProduct.productMap f g) (1 ⊗ₜ[R] b) = g b
• Algebra.TensorProduct.productMap_apply_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} {S : Type uS} {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Semiring B] [CommSemiring S] [Algebra R A] [Algebra R B] [Algebra R S] (f : A →ₐ[R] S) (g : B →ₐ[R] S) (a : A) (b : B) : (Algebra.TensorProduct.productMap f g) (a ⊗ₜ[R] b) = f a * g b
• Algebra.TensorProduct.rid_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type uR} (S : Type uS) {A : Type uA} [CommSemiring R] [CommSemiring S] [Algebra R S] [Semiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] (r : R) (a : A) : (Algebra.TensorProduct.rid R S A) (a ⊗ₜ[R] r) = r • a
• Subalgebra.tmul_mem_baseChange 📋 Mathlib.RingTheory.TensorProduct.Maps
  {R : Type u_1} {A : Type u_2} {B : Type u_3} [CommSemiring R] [Semiring A] [CommSemiring B] [Algebra R A] [Algebra R B] {C : Subalgebra R A} {x : A} (hx : x ∈ C) (b : B) : b ⊗ₜ[R] x ∈ Subalgebra.baseChange B C
• Algebra.TensorProduct.comm_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  (R : Type uR) {A : Type uA} {B : Type uB} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] (a : A) (b : B) : (Algebra.TensorProduct.comm R A B) (a ⊗ₜ[R] b) = b ⊗ₜ[R] a
• Algebra.TensorProduct.lidOfCompatibleSMul_tmul 📋 Mathlib.RingTheory.TensorProduct.Maps
  (R : Type u_3) (S : Type u_4) (A : Type u_6) [CommSemiring R] [CommSemiring S] [Semiring A] [Algebra R A] [Algebra S A] [Algebra R S] [TensorProduct.CompatibleSMul R S S A] [TensorProduct.CompatibleSMul S R S A] (s : S) (a : A) : (Algebra.TensorProduct.lidOfCompatibleSMul R S A) (s ⊗ₜ[R] a) = s • 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:

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 69fae59