mathlib documentation

linear_algebra.pi_tensor_product

Tensor product of an indexed family of modules over commutative semirings #

We define the tensor product of an indexed family s : ι → Type* of modules over commutative semirings. We denote this space by ⨂[R] i, s i and define it as free_add_monoid (R × Π i, s i) quotiented by the appropriate equivalence relation. The treatment follows very closely that of the binary tensor product in linear_algebra/tensor_product.lean.

Main definitions #

Notations #

Implementation notes #

TODO #

Tags #

multilinear, tensor, tensor product

inductive pi_tensor_product.eqv {ι : Type u_1} [decidable_eq ι] (R : Type u_4) [comm_semiring R] (s : ι → Type u_7) [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
free_add_monoid (R × Π (i : ι), s i)free_add_monoid (R × Π (i : ι), s i) → Prop

The relation on free_add_monoid (R × Π i, s i) that generates a congruence whose quotient is the tensor product.

def pi_tensor_product {ι : Type u_1} [decidable_eq ι] (R : Type u_4) [comm_semiring R] (s : ι → Type u_7) [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
Type (max u_4 u_1 u_7)

pi_tensor_product R s with R a commutative semiring and s : ι → Type* is the tensor product of all the s i's. This is denoted by ⨂[R] i, s i.

Equations
@[protected, instance]
def pi_tensor_product.add_comm_monoid {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] (s : ι → Type u_7) [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
add_comm_monoid (⨂[R] (i : ι), s i)
Equations
@[protected, instance]
def pi_tensor_product.inhabited {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] (s : ι → Type u_7) [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
inhabited (⨂[R] (i : ι), s i)
Equations
def pi_tensor_product.tprod_coeff {ι : Type u_1} [decidable_eq ι] (R : Type u_4) [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (r : R) (f : Π (i : ι), s i) :
⨂[R] (i : ι), s i

tprod_coeff R r f with r : R and f : Π i, s i is the tensor product of the vectors f i over all i : ι, multiplied by the coefficient r. Note that this is meant as an auxiliary definition for this file alone, and that one should use tprod defined below for most purposes.

Equations
theorem pi_tensor_product.zero_tprod_coeff {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (f : Π (i : ι), s i) :
theorem pi_tensor_product.zero_tprod_coeff' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z : R) (f : Π (i : ι), s i) (i : ι) (hf : f i = 0) :
theorem pi_tensor_product.add_tprod_coeff {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z : R) (f : Π (i : ι), s i) (i : ι) (m₁ m₂ : s i) :
theorem pi_tensor_product.add_tprod_coeff' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z₁ z₂ : R) (f : Π (i : ι), s i) :
theorem pi_tensor_product.smul_tprod_coeff_aux {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z : R) (f : Π (i : ι), s i) (i : ι) (r : R) :
theorem pi_tensor_product.smul_tprod_coeff {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z : R) (f : Π (i : ι), s i) (i : ι) (r : R₁) [has_scalar R₁ R] [is_scalar_tower R₁ R R] [has_scalar R₁ (s i)] [is_scalar_tower R₁ R (s i)] :
def pi_tensor_product.lift_add_hom {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {F : Type u_10} [add_comm_monoid F] (φ : (R × Π (i : ι), s i) → F) (C0 : ∀ (r : R) (f : Π (i : ι), s i) (i : ι), f i = 0φ (r, f) = 0) (C0' : ∀ (f : Π (i : ι), s i), φ (0, f) = 0) (C_add : ∀ (r : R) (f : Π (i : ι), s i) (i : ι) (m₁ m₂ : s i), φ (r, function.update f i m₁) + φ (r, function.update f i m₂) = φ (r, function.update f i (m₁ + m₂))) (C_add_scalar : ∀ (r r' : R) (f : Π (i : ι), s i), φ (r, f) + φ (r', f) = φ (r + r', f)) (C_smul : ∀ (r : R) (f : Π (i : ι), s i) (i : ι) (r' : R), φ (r, function.update f i (r' f i)) = φ (r' * r, f)) :
⨂[R] (i : ι), s i →+ F

Construct an add_monoid_hom from (⨂[R] i, s i) to some space F from a function φ : (R × Π i, s i) → F with the appropriate properties.

Equations
@[protected]
theorem pi_tensor_product.induction_on' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {C : ⨂[R] (i : ι), s i → Prop} (z : ⨂[R] (i : ι), s i) (C1 : ∀ {r : R} {f : Π (i : ι), s i}, C (pi_tensor_product.tprod_coeff R r f)) (Cp : ∀ {x y : ⨂[R] (i : ι), s i}, C xC yC (x + y)) :
C z
@[protected, instance]
def pi_tensor_product.has_scalar' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] :
has_scalar R₁ (⨂[R] (i : ι), s i)
Equations
@[protected, instance]
def pi_tensor_product.has_scalar {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
has_scalar R (⨂[R] (i : ι), s i)
Equations
theorem pi_tensor_product.smul_tprod_coeff' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] (r : R₁) (z : R) (f : Π (i : ι), s i) :
@[protected]
theorem pi_tensor_product.smul_add {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] (r : R₁) (x y : ⨂[R] (i : ι), s i) :
r (x + y) = r x + r y
@[protected, instance]
def pi_tensor_product.distrib_mul_action' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] :
distrib_mul_action R₁ (⨂[R] (i : ι), s i)
Equations
@[protected, instance]
def pi_tensor_product.smul_comm_class' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {R₂ : Type u_6} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] [monoid R₂] [distrib_mul_action R₂ R] [smul_comm_class R₂ R R] [smul_comm_class R₁ R₂ R] :
smul_comm_class R₁ R₂ (⨂[R] (i : ι), s i)
@[protected, instance]
def pi_tensor_product.is_scalar_tower' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {R₂ : Type u_6} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [monoid R₁] [distrib_mul_action R₁ R] [smul_comm_class R₁ R R] [monoid R₂] [distrib_mul_action R₂ R] [smul_comm_class R₂ R R] [has_scalar R₁ R₂] [is_scalar_tower R₁ R₂ R] :
is_scalar_tower R₁ R₂ (⨂[R] (i : ι), s i)
@[protected, instance]
def pi_tensor_product.module' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {R₁ : Type u_5} {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] [semiring R₁] [module R₁ R] [smul_comm_class R₁ R R] :
module R₁ (⨂[R] (i : ι), s i)
Equations
@[protected, instance]
def pi_tensor_product.module {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
module R (⨂[R] (i : ι), s i)
Equations
@[protected, instance]
def pi_tensor_product.smul_comm_class {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
smul_comm_class R R (⨂[R] (i : ι), s i)
@[protected, instance]
def pi_tensor_product.is_scalar_tower {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
is_scalar_tower R R (⨂[R] (i : ι), s i)
def pi_tensor_product.tprod {ι : Type u_1} [decidable_eq ι] (R : Type u_4) [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
multilinear_map R s (⨂[R] (i : ι), s i)

The canonical multilinear_map R s (⨂[R] i, s i).

Equations
@[simp]
theorem pi_tensor_product.tprod_coeff_eq_smul_tprod {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] (z : R) (f : Π (i : ι), s i) :
@[protected]
theorem pi_tensor_product.induction_on {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {C : ⨂[R] (i : ι), s i → Prop} (z : ⨂[R] (i : ι), s i) (C1 : ∀ {r : R} {f : Π (i : ι), s i}, C (r (pi_tensor_product.tprod R) f)) (Cp : ∀ {x y : ⨂[R] (i : ι), s i}, C xC yC (x + y)) :
C z
@[ext]
theorem pi_tensor_product.ext {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] {φ₁ φ₂ : ⨂[R] (i : ι), s i →ₗ[R] E} (H : φ₁.comp_multilinear_map (pi_tensor_product.tprod R) = φ₂.comp_multilinear_map (pi_tensor_product.tprod R)) :
φ₁ = φ₂
def pi_tensor_product.lift_aux {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] (φ : multilinear_map R s E) :
⨂[R] (i : ι), s i →+ E

Auxiliary function to constructing a linear map (⨂[R] i, s i) → E given a multilinear map R s E with the property that its composition with the canonical multilinear_map R s (⨂[R] i, s i) is the given multilinear map.

Equations
theorem pi_tensor_product.lift_aux_tprod {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] (φ : multilinear_map R s E) (f : Π (i : ι), s i) :
theorem pi_tensor_product.lift_aux_tprod_coeff {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] (φ : multilinear_map R s E) (z : R) (f : Π (i : ι), s i) :
theorem pi_tensor_product.lift_aux.smul {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] {φ : multilinear_map R s E} (r : R) (x : ⨂[R] (i : ι), s i) :
def pi_tensor_product.lift {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] :
multilinear_map R s E ≃ₗ[R] ⨂[R] (i : ι), s i →ₗ[R] E

Constructing a linear map (⨂[R] i, s i) → E given a multilinear_map R s E with the property that its composition with the canonical multilinear_map R s E is the given multilinear map φ.

Equations
@[simp]
theorem pi_tensor_product.lift.tprod {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] {φ : multilinear_map R s E} (f : Π (i : ι), s i) :
theorem pi_tensor_product.lift.unique' {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] {φ : multilinear_map R s E} {φ' : ⨂[R] (i : ι), s i →ₗ[R] E} (H : φ'.comp_multilinear_map (pi_tensor_product.tprod R) = φ) :
theorem pi_tensor_product.lift.unique {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] {φ : multilinear_map R s E} {φ' : ⨂[R] (i : ι), s i →ₗ[R] E} (H : ∀ (f : Π (i : ι), s i), φ' ((pi_tensor_product.tprod R) f) = φ f) :
@[simp]
theorem pi_tensor_product.lift_symm {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] {E : Type u_9} [add_comm_monoid E] [module R E] (φ' : ⨂[R] (i : ι), s i →ₗ[R] E) :
@[simp]
theorem pi_tensor_product.lift_tprod {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {s : ι → Type u_7} [Π (i : ι), add_comm_monoid (s i)] [Π (i : ι), module R (s i)] :
def pi_tensor_product.reindex {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] (R : Type u_4) [comm_semiring R] (M : Type u_8) [add_comm_monoid M] [module R M] (e : ι ι₂) :
⨂[R] (i : ι), M ≃ₗ[R] ⨂[R] (i : ι₂), M

Re-index the components of the tensor power by e.

For simplicity, this is defined only for homogeneously- (rather than dependently-) typed components.

Equations
@[simp]
theorem pi_tensor_product.reindex_tprod {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] (e : ι ι₂) (f : ι → M) :
@[simp]
theorem pi_tensor_product.reindex_comp_tprod {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] (e : ι ι₂) :
@[simp]
theorem pi_tensor_product.lift_comp_reindex {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] {E : Type u_9} [add_comm_monoid E] [module R E] (e : ι ι₂) (φ : multilinear_map R (λ (_x : ι₂), M) E) :
@[simp]
theorem pi_tensor_product.lift_reindex {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] {E : Type u_9} [add_comm_monoid E] [module R E] (e : ι ι₂) (φ : multilinear_map R (λ (_x : ι₂), M) E) (x : ⨂[R] (i : ι), M) :
@[simp]
theorem pi_tensor_product.reindex_trans {ι : Type u_1} {ι₂ : Type u_2} {ι₃ : Type u_3} [decidable_eq ι] [decidable_eq ι₂] [decidable_eq ι₃] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] (e : ι ι₂) (e' : ι₂ ι₃) :
@[simp]
theorem pi_tensor_product.reindex_reindex {ι : Type u_1} {ι₂ : Type u_2} {ι₃ : Type u_3} [decidable_eq ι] [decidable_eq ι₂] [decidable_eq ι₃] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] (e : ι ι₂) (e' : ι₂ ι₃) (x : ⨂[R] (i : ι), M) :
@[simp]
theorem pi_tensor_product.reindex_symm {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] (e : ι ι₂) :
@[simp]
theorem pi_tensor_product.reindex_refl {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] :
def pi_tensor_product.is_empty_equiv (ι : Type u_1) [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] [is_empty ι] :
⨂[R] (i : ι), M ≃ₗ[R] R

The tensor product over an empty index type ι is isomorphic to the base ring.

Equations
@[simp]
@[simp]
theorem pi_tensor_product.is_empty_equiv_apply_tprod (ι : Type u_1) [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] [is_empty ι] (f : ι → M) :
@[simp]
theorem pi_tensor_product.subsingleton_equiv_symm_apply {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] [subsingleton ι] (i₀ : ι) (m : M) :
def pi_tensor_product.subsingleton_equiv {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] [subsingleton ι] (i₀ : ι) :
⨂[R] (i : ι), M ≃ₗ[R] M

The tensor product over an single index is isomorphic to the module

Equations
@[simp]
theorem pi_tensor_product.subsingleton_equiv_apply_tprod {ι : Type u_1} [decidable_eq ι] {R : Type u_4} [comm_semiring R] {M : Type u_8} [add_comm_monoid M] [module R M] [subsingleton ι] (i : ι) (f : ι → M) :
def pi_tensor_product.tmul_equiv {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] (R : Type u_4) [comm_semiring R] (M : Type u_8) [add_comm_monoid M] [module R M] :
⨂[R] (i : ι), M ⨂[R] (i : ι₂), M ≃ₗ[R] ⨂[R] (i : ι ι₂), M

Equivalence between a tensor_product of pi_tensor_products and a single pi_tensor_product indexed by a sum type.

For simplicity, this is defined only for homogeneously- (rather than dependently-) typed components.

Equations
@[simp]
theorem pi_tensor_product.tmul_equiv_apply {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] (R : Type u_4) [comm_semiring R] (M : Type u_8) [add_comm_monoid M] [module R M] (a : ι → M) (b : ι₂ → M) :
(pi_tensor_product.tmul_equiv R M) ((pi_tensor_product.tprod R) (λ (i : ι), a i) ⊗ₜ[R] (pi_tensor_product.tprod R) (λ (i : ι₂), b i)) = (pi_tensor_product.tprod R) (λ (i : ι ι₂), sum.elim a b i)
@[simp]
theorem pi_tensor_product.tmul_equiv_symm_apply {ι : Type u_1} {ι₂ : Type u_2} [decidable_eq ι] [decidable_eq ι₂] (R : Type u_4) [comm_semiring R] (M : Type u_8) [add_comm_monoid M] [module R M] (a : ι ι₂ → M) :
((pi_tensor_product.tmul_equiv R M).symm) ((pi_tensor_product.tprod R) (λ (i : ι ι₂), a i)) = (pi_tensor_product.tprod R) (λ (i : ι), a (sum.inl i)) ⊗ₜ[R] (pi_tensor_product.tprod R) (λ (i : ι₂), a (sum.inr i))
@[protected, instance]
def pi_tensor_product.add_comm_group {ι : Type u_1} [decidable_eq ι] {R : Type u_2} [comm_ring R] {s : ι → Type u_3} [Π (i : ι), add_comm_group (s i)] [Π (i : ι), module R (s i)] :
add_comm_group (⨂[R] (i : ι), s i)
Equations