Documentation

Mathlib.LinearAlgebra.PiTensorProduct

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 FreeAddMonoid (R × Π i, s i) quotiented by the appropriate equivalence relation. The treatment follows very closely that of the binary tensor product in LinearAlgebra/TensorProduct.lean.

Main definitions #

Notations #

Implementation notes #

TODO #

Tags #

multilinear, tensor, tensor product

inductive PiTensorProduct.Eqv {ι : Type u_1} (R : Type u_4) [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
FreeAddMonoid (R × ((i : ι) → s i))FreeAddMonoid (R × ((i : ι) → s i))Prop

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

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    noncomputable def PiTensorProduct {ι : Type u_1} (R : Type u_4) [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
    Type (max (max u_1 u_4) u_7)

    PiTensorProduct 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.

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      This enables the notation ⨂[R] i : ι, s i for the pi tensor product PiTensorProduct, given an indexed family of types s : ι → Type*.

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        Pretty printer defined by notation3 command.

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          noncomputable instance PiTensorProduct.instAddCommMonoid {ι : Type u_1} {R : Type u_4} [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
          AddCommMonoid (PiTensorProduct R fun (i : ι) => s i)
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          noncomputable instance PiTensorProduct.instInhabited {ι : Type u_1} {R : Type u_4} [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
          Inhabited (PiTensorProduct R fun (i : ι) => s i)
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          noncomputable def PiTensorProduct.tprodCoeff {ι : Type u_1} (R : Type u_4) [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (r : R) (f : (i : ι) → s i) :
          PiTensorProduct R fun (i : ι) => s i

          tprodCoeff 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.

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            theorem PiTensorProduct.zero_tprodCoeff {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (f : (i : ι) → s i) :
            theorem PiTensorProduct.zero_tprodCoeff' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (z : R) (f : (i : ι) → s i) (i : ι) (hf : f i = 0) :
            theorem PiTensorProduct.add_tprodCoeff {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [DecidableEq ι] (z : R) (f : (i : ι) → s i) (i : ι) (m₁ : s i) (m₂ : s i) :
            theorem PiTensorProduct.add_tprodCoeff' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (z₁ : R) (z₂ : R) (f : (i : ι) → s i) :
            theorem PiTensorProduct.smul_tprodCoeff_aux {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [DecidableEq ι] (z : R) (f : (i : ι) → s i) (i : ι) (r : R) :
            theorem PiTensorProduct.smul_tprodCoeff {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [DecidableEq ι] (z : R) (f : (i : ι) → s i) (i : ι) (r : R₁) [SMul R₁ R] [IsScalarTower R₁ R R] [SMul R₁ (s i)] [IsScalarTower R₁ R (s i)] :
            noncomputable def PiTensorProduct.liftAddHom {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {F : Type u_10} [AddCommMonoid F] (φ : R × ((i : ι) → s i)F) (C0 : ∀ (r : R) (f : (i : ι) → s i) (i : ι), f i = 0φ (r, f) = 0) (C0' : ∀ (f : (i : ι) → s i), φ (0, f) = 0) (C_add : ∀ [inst : DecidableEq ι] (r : R) (f : (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 : (i : ι) → s i), φ (r, f) + φ (r', f) = φ (r + r', f)) (C_smul : ∀ [inst : DecidableEq ι] (r : R) (f : (i : ι) → s i) (i : ι) (r' : R), φ (r, Function.update f i (r' f i)) = φ (r' * r, f)) :
            (PiTensorProduct R fun (i : ι) => s i) →+ F

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

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              theorem PiTensorProduct.induction_on' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {motive : (PiTensorProduct R fun (i : ι) => s i)Prop} (z : PiTensorProduct R fun (i : ι) => s i) (tprodCoeff : ∀ (r : R) (f : (i : ι) → s i), motive (PiTensorProduct.tprodCoeff R r f)) (add : ∀ (x y : PiTensorProduct R fun (i : ι) => s i), motive xmotive ymotive (x + y)) :
              motive z

              Induct using tprodCoeff

              noncomputable instance PiTensorProduct.hasSMul' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] :
              SMul R₁ (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable instance PiTensorProduct.instSMul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
              SMul R (PiTensorProduct R fun (i : ι) => s i)
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              • PiTensorProduct.instSMul = PiTensorProduct.hasSMul'
              theorem PiTensorProduct.smul_tprodCoeff' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] (r : R₁) (z : R) (f : (i : ι) → s i) :
              theorem PiTensorProduct.smul_add {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] (r : R₁) (x : PiTensorProduct R fun (i : ι) => s i) (y : PiTensorProduct R fun (i : ι) => s i) :
              r (x + y) = r x + r y
              noncomputable instance PiTensorProduct.distribMulAction' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] :
              DistribMulAction R₁ (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable instance PiTensorProduct.smulCommClass' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {R₂ : Type u_6} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] [Monoid R₂] [DistribMulAction R₂ R] [SMulCommClass R₂ R R] [SMulCommClass R₁ R₂ R] :
              SMulCommClass R₁ R₂ (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable instance PiTensorProduct.isScalarTower' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {R₂ : Type u_6} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Monoid R₁] [DistribMulAction R₁ R] [SMulCommClass R₁ R R] [Monoid R₂] [DistribMulAction R₂ R] [SMulCommClass R₂ R R] [SMul R₁ R₂] [IsScalarTower R₁ R₂ R] :
              IsScalarTower R₁ R₂ (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable instance PiTensorProduct.module' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {R₁ : Type u_5} {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [Semiring R₁] [Module R₁ R] [SMulCommClass R₁ R R] :
              Module R₁ (PiTensorProduct R fun (i : ι) => s i)
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              • PiTensorProduct.module' = let __src := PiTensorProduct.distribMulAction'; Module.mk
              noncomputable instance PiTensorProduct.instModule {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
              Module R (PiTensorProduct R fun (i : ι) => s i)
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              • PiTensorProduct.instModule = PiTensorProduct.module'
              noncomputable instance PiTensorProduct.instSMulCommClass {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
              SMulCommClass R R (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable instance PiTensorProduct.instIsScalarTower {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
              IsScalarTower R R (PiTensorProduct R fun (i : ι) => s i)
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              noncomputable def PiTensorProduct.tprod {ι : Type u_1} (R : Type u_4) [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
              MultilinearMap R s (PiTensorProduct R fun (i : ι) => s i)

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

              tprod R fun i => f i has notation ⨂ₜ[R] i, f i.

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

                tprod R fun i => f i has notation ⨂ₜ[R] i, f i.

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                  Pretty printer defined by notation3 command.

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                    theorem PiTensorProduct.tprod_eq_tprodCoeff_one {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                    @[simp]
                    theorem PiTensorProduct.tprodCoeff_eq_smul_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (z : R) (f : (i : ι) → s i) :
                    theorem FreeAddMonoid.toPiTensorProduct {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (p : FreeAddMonoid (R × ((i : ι) → s i))) :
                    p = (List.map (fun (x : R × ((i : ι) → s i)) => x.1 ⨂ₜ[R] (i : ι), x.2 i) p).sum

                    The image of an element p of FreeAddMonoid (R × Π i, s i) in the PiTensorProduct is equal to the sum of a • ⨂ₜ[R] i, m i over all the entries (a, m) of p.

                    noncomputable def PiTensorProduct.lifts {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (x : PiTensorProduct R fun (i : ι) => s i) :
                    Set (FreeAddMonoid (R × ((i : ι) → s i)))

                    The set of lifts of an element x of ⨂[R] i, s i in FreeAddMonoid (R × Π i, s i).

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                      theorem PiTensorProduct.mem_lifts_iff {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (x : PiTensorProduct R fun (i : ι) => s i) (p : FreeAddMonoid (R × ((i : ι) → s i))) :
                      p x.lifts (List.map (fun (x : R × ((i : ι) → s i)) => x.1 ⨂ₜ[R] (i : ι), x.2 i) p).sum = x

                      An element p of FreeAddMonoid (R × Π i, s i) lifts an element x of ⨂[R] i, s i if and only if x is equal to to the sum of a • ⨂ₜ[R] i, m i over all the entries (a, m) of p.

                      theorem PiTensorProduct.nonempty_lifts {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (x : PiTensorProduct R fun (i : ι) => s i) :
                      x.lifts.Nonempty

                      Every element of ⨂[R] i, s i has a lift in FreeAddMonoid (R × Π i, s i).

                      theorem PiTensorProduct.lifts_zero {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :

                      The empty list lifts the element 0 of ⨂[R] i, s i.

                      theorem PiTensorProduct.lifts_add {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {x : PiTensorProduct R fun (i : ι) => s i} {y : PiTensorProduct R fun (i : ι) => s i} {p : FreeAddMonoid (R × ((i : ι) → s i))} {q : FreeAddMonoid (R × ((i : ι) → s i))} (hp : p x.lifts) (hq : q y.lifts) :
                      p + q (x + y).lifts

                      If elements p,q of FreeAddMonoid (R × Π i, s i) lift elements x,y of ⨂[R] i, s i respectively, then p + q lifts x + y.

                      theorem PiTensorProduct.lifts_smul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {x : PiTensorProduct R fun (i : ι) => s i} {p : FreeAddMonoid (R × ((i : ι) → s i))} (h : p x.lifts) (a : R) :
                      List.map (fun (y : R × ((i : ι) → s i)) => (a * y.1, y.2)) p (a x).lifts

                      If an element p of FreeAddMonoid (R × Π i, s i) lifts an element x of ⨂[R] i, s i, and if a is an element of R, then the list obtained by multiplying the first entry of each element of p by a lifts a • x.

                      theorem PiTensorProduct.induction_on {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {motive : (PiTensorProduct R fun (i : ι) => s i)Prop} (z : PiTensorProduct R fun (i : ι) => s i) (smul_tprod : ∀ (r : R) (f : (i : ι) → s i), motive (r (PiTensorProduct.tprod R) f)) (add : ∀ (x y : PiTensorProduct R fun (i : ι) => s i), motive xmotive ymotive (x + y)) :
                      motive z

                      Induct using scaled versions of PiTensorProduct.tprod.

                      theorem PiTensorProduct.ext {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {φ₁ : (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E} {φ₂ : (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E} (H : φ₁.compMultilinearMap (PiTensorProduct.tprod R) = φ₂.compMultilinearMap (PiTensorProduct.tprod R)) :
                      φ₁ = φ₂
                      theorem PiTensorProduct.span_tprod_eq_top {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :

                      The pure tensors (i.e. the elements of the image of PiTensorProduct.tprod) span the tensor product.

                      noncomputable def PiTensorProduct.liftAux {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (φ : MultilinearMap R s E) :
                      (PiTensorProduct R fun (i : ι) => s i) →+ E

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

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                        theorem PiTensorProduct.liftAux_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (φ : MultilinearMap R s E) (f : (i : ι) → s i) :
                        theorem PiTensorProduct.liftAux_tprodCoeff {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (φ : MultilinearMap R s E) (z : R) (f : (i : ι) → s i) :
                        theorem PiTensorProduct.liftAux.smul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {φ : MultilinearMap R s E} (r : R) (x : PiTensorProduct R fun (i : ι) => s i) :
                        noncomputable def PiTensorProduct.lift {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] :
                        MultilinearMap R s E ≃ₗ[R] (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E

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

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                          @[simp]
                          theorem PiTensorProduct.lift.tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {φ : MultilinearMap R s E} (f : (i : ι) → s i) :
                          (PiTensorProduct.lift φ) ((PiTensorProduct.tprod R) f) = φ f
                          theorem PiTensorProduct.lift.unique' {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {φ : MultilinearMap R s E} {φ' : (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E} (H : φ'.compMultilinearMap (PiTensorProduct.tprod R) = φ) :
                          φ' = PiTensorProduct.lift φ
                          theorem PiTensorProduct.lift.unique {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {φ : MultilinearMap R s E} {φ' : (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E} (H : ∀ (f : (i : ι) → s i), φ' ((PiTensorProduct.tprod R) f) = φ f) :
                          φ' = PiTensorProduct.lift φ
                          @[simp]
                          theorem PiTensorProduct.lift_symm {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (φ' : (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] E) :
                          PiTensorProduct.lift.symm φ' = φ'.compMultilinearMap (PiTensorProduct.tprod R)
                          @[simp]
                          theorem PiTensorProduct.lift_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                          PiTensorProduct.lift (PiTensorProduct.tprod R) = LinearMap.id
                          noncomputable def PiTensorProduct.map {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) :
                          (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] PiTensorProduct R fun (i : ι) => t i

                          Let sᵢ and tᵢ be two families of R-modules. Let f be a family of R-linear maps between sᵢ and tᵢ, i.e. f : Πᵢ sᵢ → tᵢ, then there is an induced map ⨂ᵢ sᵢ → ⨂ᵢ tᵢ by ⨂ aᵢ ↦ ⨂ fᵢ aᵢ.

                          This is TensorProduct.map for an arbitrary family of modules.

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                            @[simp]
                            theorem PiTensorProduct.map_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (x : (i : ι) → s i) :
                            (PiTensorProduct.map f) ((PiTensorProduct.tprod R) x) = ⨂ₜ[R] (i : ι), (f i) (x i)
                            theorem PiTensorProduct.map_range_eq_span_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) :
                            LinearMap.range (PiTensorProduct.map f) = Submodule.span R {t_1 : PiTensorProduct R fun (i : ι) => t i | ∃ (m : (i : ι) → s i), (⨂ₜ[R] (i : ι), (f i) (m i)) = t_1}
                            noncomputable def PiTensorProduct.mapIncl {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (p : (i : ι) → Submodule R (s i)) :
                            (PiTensorProduct R fun (i : ι) => (p i)) →ₗ[R] PiTensorProduct R fun (i : ι) => s i

                            Given submodules p i ⊆ s i, this is the natural map: ⨂[R] i, p i → ⨂[R] i, s i. This is TensorProduct.mapIncl for an arbitrary family of modules.

                            Equations
                            Instances For
                              theorem PiTensorProduct.map_comp {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (g : (i : ι) → t i →ₗ[R] t' i) (f : (i : ι) → s i →ₗ[R] t i) :
                              (PiTensorProduct.map fun (i : ι) => g i ∘ₗ f i) = PiTensorProduct.map g ∘ₗ PiTensorProduct.map f
                              theorem PiTensorProduct.lift_comp_map {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (h : MultilinearMap R t E) :
                              PiTensorProduct.lift h ∘ₗ PiTensorProduct.map f = PiTensorProduct.lift (h.compLinearMap f)
                              @[simp]
                              theorem PiTensorProduct.map_id {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                              (PiTensorProduct.map fun (i : ι) => LinearMap.id) = LinearMap.id
                              @[simp]
                              theorem PiTensorProduct.map_one {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                              (PiTensorProduct.map fun (i : ι) => 1) = 1
                              theorem PiTensorProduct.map_mul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (f₁ : (i : ι) → s i →ₗ[R] s i) (f₂ : (i : ι) → s i →ₗ[R] s i) :
                              (PiTensorProduct.map fun (i : ι) => f₁ i * f₂ i) = PiTensorProduct.map f₁ * PiTensorProduct.map f₂
                              noncomputable def PiTensorProduct.mapMonoidHom {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                              ((i : ι) → s i →ₗ[R] s i) →* (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] PiTensorProduct R fun (i : ι) => s i

                              Upgrading PiTensorProduct.map to a MonoidHom when s = t.

                              Equations
                              • PiTensorProduct.mapMonoidHom = { toFun := PiTensorProduct.map, map_one' := , map_mul' := }
                              Instances For
                                @[simp]
                                theorem PiTensorProduct.mapMonoidHom_apply {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (f : (i : ι) → s i →ₗ[R] s i) :
                                PiTensorProduct.mapMonoidHom f = PiTensorProduct.map f
                                @[simp]
                                theorem PiTensorProduct.map_pow {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (f : (i : ι) → s i →ₗ[R] s i) (n : ) :
                                theorem PiTensorProduct.map_add {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) [DecidableEq ι] (i : ι) (u : s i →ₗ[R] t i) (v : s i →ₗ[R] t i) :
                                theorem PiTensorProduct.map_smul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) [DecidableEq ι] (i : ι) (c : R) (u : s i →ₗ[R] t i) :
                                noncomputable def PiTensorProduct.mapMultilinear {ι : Type u_1} (R : Type u_4) [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (t : ιType u_11) [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] :
                                MultilinearMap R (fun (i : ι) => s i →ₗ[R] t i) ((PiTensorProduct R fun (i : ι) => s i) →ₗ[R] PiTensorProduct R fun (i : ι) => t i)

                                The tensor of a family of linear maps from sᵢ to tᵢ, as a multilinear map of the family.

                                Equations
                                Instances For
                                  @[simp]
                                  theorem PiTensorProduct.mapMultilinear_apply {ι : Type u_1} (R : Type u_4) [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (t : ιType u_11) [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) :
                                  noncomputable def PiTensorProduct.piTensorHomMap {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] :
                                  (PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i) →ₗ[R] (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] PiTensorProduct R fun (i : ι) => t i

                                  Let sᵢ and tᵢ be families of R-modules. Then there is an R-linear map between ⨂ᵢ Hom(sᵢ, tᵢ) and Hom(⨂ᵢ sᵢ, ⨂ tᵢ) defined by ⨂ᵢ fᵢ ↦ ⨂ᵢ aᵢ ↦ ⨂ᵢ fᵢ aᵢ.

                                  This is TensorProduct.homTensorHomMap for an arbitrary family of modules.

                                  Note that PiTensorProduct.piTensorHomMap (tprod R f) is equal to PiTensorProduct.map f.

                                  Equations
                                  • PiTensorProduct.piTensorHomMap = PiTensorProduct.lift ∘ₗ PiTensorProduct.lift (MultilinearMap.piLinearMap (PiTensorProduct.tprod R))
                                  Instances For
                                    @[simp]
                                    theorem PiTensorProduct.piTensorHomMap_tprod_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (x : (i : ι) → s i) :
                                    (PiTensorProduct.piTensorHomMap ((PiTensorProduct.tprod R) f)) ((PiTensorProduct.tprod R) x) = ⨂ₜ[R] (i : ι), (f i) (x i)
                                    theorem PiTensorProduct.piTensorHomMap_tprod_eq_map {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) :
                                    PiTensorProduct.piTensorHomMap ((PiTensorProduct.tprod R) f) = PiTensorProduct.map f
                                    noncomputable def PiTensorProduct.congr {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i ≃ₗ[R] t i) :
                                    (PiTensorProduct R fun (i : ι) => s i) ≃ₗ[R] PiTensorProduct R fun (i : ι) => t i

                                    If s i and t i are linearly equivalent for every i in ι, then ⨂[R] i, s i and ⨂[R] i, t i are linearly equivalent.

                                    This is the n-ary version of TensorProduct.congr

                                    Equations
                                    Instances For
                                      @[simp]
                                      theorem PiTensorProduct.congr_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i ≃ₗ[R] t i) (m : (i : ι) → s i) :
                                      (PiTensorProduct.congr f) ((PiTensorProduct.tprod R) m) = ⨂ₜ[R] (i : ι), (f i) (m i)
                                      @[simp]
                                      theorem PiTensorProduct.congr_symm_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i ≃ₗ[R] t i) (p : (i : ι) → t i) :
                                      (PiTensorProduct.congr f).symm ((PiTensorProduct.tprod R) p) = ⨂ₜ[R] (i : ι), (f i).symm (p i)
                                      noncomputable def PiTensorProduct.map₂ {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (f : (i : ι) → s i →ₗ[R] t i →ₗ[R] t' i) :
                                      (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] (PiTensorProduct R fun (i : ι) => t i) →ₗ[R] PiTensorProduct R fun (i : ι) => t' i

                                      Let sᵢ, tᵢ and t'ᵢ be families of R-modules, then f : Πᵢ sᵢ → tᵢ → t'ᵢ induces an element of Hom(⨂ᵢ sᵢ, Hom(⨂ tᵢ, ⨂ᵢ t'ᵢ)) defined by ⨂ᵢ aᵢ ↦ ⨂ᵢ bᵢ ↦ ⨂ᵢ fᵢ aᵢ bᵢ.

                                      This is PiTensorProduct.map for two arbitrary families of modules. This is TensorProduct.map₂ for families of modules.

                                      Equations
                                      Instances For
                                        theorem PiTensorProduct.map₂_tprod_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (f : (i : ι) → s i →ₗ[R] t i →ₗ[R] t' i) (x : (i : ι) → s i) (y : (i : ι) → t i) :
                                        ((PiTensorProduct.map₂ f) ((PiTensorProduct.tprod R) x)) ((PiTensorProduct.tprod R) y) = ⨂ₜ[R] (i : ι), ((f i) (x i)) (y i)
                                        noncomputable def PiTensorProduct.piTensorHomMapFun₂ {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] :
                                        (PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i →ₗ[R] t' i)(PiTensorProduct R fun (i : ι) => s i) →ₗ[R] (PiTensorProduct R fun (i : ι) => t i) →ₗ[R] PiTensorProduct R fun (i : ι) => t' i

                                        Let sᵢ, tᵢ and t'ᵢ be families of R-modules. Then there is a function from ⨂ᵢ Hom(sᵢ, Hom(tᵢ, t'ᵢ)) to Hom(⨂ᵢ sᵢ, Hom(⨂ tᵢ, ⨂ᵢ t'ᵢ)) defined by ⨂ᵢ fᵢ ↦ ⨂ᵢ aᵢ ↦ ⨂ᵢ bᵢ ↦ ⨂ᵢ fᵢ aᵢ bᵢ.

                                        Equations
                                        • φ.piTensorHomMapFun₂ = PiTensorProduct.lift (PiTensorProduct.piTensorHomMap.compMultilinearMap ((PiTensorProduct.lift (MultilinearMap.piLinearMap (PiTensorProduct.tprod R))) φ))
                                        Instances For
                                          theorem PiTensorProduct.piTensorHomMapFun₂_add {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (φ : PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i →ₗ[R] t' i) (ψ : PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i →ₗ[R] t' i) :
                                          (φ + ψ).piTensorHomMapFun₂ = φ.piTensorHomMapFun₂ + ψ.piTensorHomMapFun₂
                                          theorem PiTensorProduct.piTensorHomMapFun₂_smul {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (r : R) (φ : PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i →ₗ[R] t' i) :
                                          (r φ).piTensorHomMapFun₂ = r φ.piTensorHomMapFun₂
                                          noncomputable def PiTensorProduct.piTensorHomMap₂ {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] :
                                          (PiTensorProduct R fun (i : ι) => s i →ₗ[R] t i →ₗ[R] t' i) →ₗ[R] (PiTensorProduct R fun (i : ι) => s i) →ₗ[R] (PiTensorProduct R fun (i : ι) => t i) →ₗ[R] PiTensorProduct R fun (i : ι) => t' i

                                          Let sᵢ, tᵢ and t'ᵢ be families of R-modules. Then there is an linear map from ⨂ᵢ Hom(sᵢ, Hom(tᵢ, t'ᵢ)) to Hom(⨂ᵢ sᵢ, Hom(⨂ tᵢ, ⨂ᵢ t'ᵢ)) defined by ⨂ᵢ fᵢ ↦ ⨂ᵢ aᵢ ↦ ⨂ᵢ bᵢ ↦ ⨂ᵢ fᵢ aᵢ bᵢ.

                                          This is TensorProduct.homTensorHomMap for two arbitrary families of modules.

                                          Equations
                                          • PiTensorProduct.piTensorHomMap₂ = { toFun := PiTensorProduct.piTensorHomMapFun₂, map_add' := , map_smul' := }
                                          Instances For
                                            @[simp]
                                            theorem PiTensorProduct.piTensorHomMap₂_tprod_tprod_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} {t' : ιType u_12} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] [(i : ι) → AddCommMonoid (t' i)] [(i : ι) → Module R (t' i)] (f : (i : ι) → s i →ₗ[R] t i →ₗ[R] t' i) (a : (i : ι) → s i) (b : (i : ι) → t i) :
                                            ((PiTensorProduct.piTensorHomMap₂ ((PiTensorProduct.tprod R) f)) ((PiTensorProduct.tprod R) a)) ((PiTensorProduct.tprod R) b) = ⨂ₜ[R] (i : ι), ((f i) (a i)) (b i)
                                            noncomputable def PiTensorProduct.reindex {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (s : ιType u_7) [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (e : ι ι₂) :
                                            (PiTensorProduct R fun (i : ι) => s i) ≃ₗ[R] PiTensorProduct R fun (i : ι₂) => s (e.symm i)

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

                                            Equations
                                            • One or more equations did not get rendered due to their size.
                                            Instances For
                                              @[simp]
                                              theorem PiTensorProduct.reindex_tprod {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (e : ι ι₂) (f : (i : ι) → s i) :
                                              (PiTensorProduct.reindex R s e) ((PiTensorProduct.tprod R) f) = (PiTensorProduct.tprod R) fun (i : ι₂) => f (e.symm i)
                                              @[simp]
                                              theorem PiTensorProduct.reindex_comp_tprod {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (e : ι ι₂) :
                                              ((PiTensorProduct.reindex R s e)).compMultilinearMap (PiTensorProduct.tprod R) = (MultilinearMap.domDomCongrLinearEquiv' R R s (PiTensorProduct R fun (i : ι₂) => s (e.symm i)) e).symm (PiTensorProduct.tprod R)
                                              theorem PiTensorProduct.lift_comp_reindex {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (e : ι ι₂) (φ : MultilinearMap R (fun (i : ι₂) => s (e.symm i)) E) :
                                              PiTensorProduct.lift φ ∘ₗ (PiTensorProduct.reindex R s e) = PiTensorProduct.lift ((MultilinearMap.domDomCongrLinearEquiv' R R s E e).symm φ)
                                              @[simp]
                                              theorem PiTensorProduct.lift_comp_reindex_symm {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (e : ι ι₂) (φ : MultilinearMap R s E) :
                                              PiTensorProduct.lift φ ∘ₗ (PiTensorProduct.reindex R s e).symm = PiTensorProduct.lift ((MultilinearMap.domDomCongrLinearEquiv' R R s E e) φ)
                                              theorem PiTensorProduct.lift_reindex {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (e : ι ι₂) (φ : MultilinearMap R (fun (i : ι₂) => s (e.symm i)) E) (x : PiTensorProduct R fun (i : ι) => s i) :
                                              (PiTensorProduct.lift φ) ((PiTensorProduct.reindex R s e) x) = (PiTensorProduct.lift ((MultilinearMap.domDomCongrLinearEquiv' R R s E e).symm φ)) x
                                              @[simp]
                                              theorem PiTensorProduct.lift_reindex_symm {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {E : Type u_9} [AddCommMonoid E] [Module R E] (e : ι ι₂) (φ : MultilinearMap R s E) (x : PiTensorProduct R fun (i : ι₂) => s (e.symm i)) :
                                              (PiTensorProduct.lift φ) ((PiTensorProduct.reindex R s e).symm x) = (PiTensorProduct.lift ((MultilinearMap.domDomCongrLinearEquiv' R R s E e) φ)) x
                                              @[simp]
                                              theorem PiTensorProduct.reindex_trans {ι : Type u_1} {ι₂ : Type u_2} {ι₃ : Type u_3} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (e : ι ι₂) (e' : ι₂ ι₃) :
                                              PiTensorProduct.reindex R s e ≪≫ₗ PiTensorProduct.reindex R (fun (i : ι₂) => s (e.symm i)) e' = PiTensorProduct.reindex R s (e.trans e')
                                              theorem PiTensorProduct.reindex_reindex {ι : Type u_1} {ι₂ : Type u_2} {ι₃ : Type u_3} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] (e : ι ι₂) (e' : ι₂ ι₃) (x : PiTensorProduct R fun (i : ι) => s i) :
                                              (PiTensorProduct.reindex R (fun (i : ι₂) => s (e.symm i)) e') ((PiTensorProduct.reindex R s e) x) = (PiTensorProduct.reindex R s (e.trans e')) x
                                              @[simp]
                                              theorem PiTensorProduct.reindex_symm {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {M : Type u_8} [AddCommMonoid M] [Module R M] (e : ι ι₂) :
                                              (PiTensorProduct.reindex R (fun (x : ι) => M) e).symm = PiTensorProduct.reindex R (fun (x : ι₂) => M) e.symm

                                              This lemma is impractical to state in the dependent case.

                                              @[simp]
                                              theorem PiTensorProduct.reindex_refl {ι : Type u_1} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] :
                                              theorem PiTensorProduct.map_comp_reindex_eq {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (e : ι ι₂) :
                                              (PiTensorProduct.map fun (i : ι₂) => f (e.symm i)) ∘ₗ (PiTensorProduct.reindex R s e) = (PiTensorProduct.reindex R t e) ∘ₗ PiTensorProduct.map f

                                              Re-indexing the components of the tensor product by an equivalence e is compatible with PiTensorProduct.map.

                                              theorem PiTensorProduct.map_reindex {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (e : ι ι₂) (x : PiTensorProduct R fun (i : ι) => s i) :
                                              (PiTensorProduct.map fun (i : ι₂) => f (e.symm i)) ((PiTensorProduct.reindex R s e) x) = (PiTensorProduct.reindex R t e) ((PiTensorProduct.map f) x)
                                              theorem PiTensorProduct.map_comp_reindex_symm {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (e : ι ι₂) :
                                              PiTensorProduct.map f ∘ₗ (PiTensorProduct.reindex R s e).symm = (PiTensorProduct.reindex R t e).symm ∘ₗ PiTensorProduct.map fun (i : ι₂) => f (e.symm i)
                                              theorem PiTensorProduct.map_reindex_symm {ι : Type u_1} {ι₂ : Type u_2} {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] {t : ιType u_11} [(i : ι) → AddCommMonoid (t i)] [(i : ι) → Module R (t i)] (f : (i : ι) → s i →ₗ[R] t i) (e : ι ι₂) (x : PiTensorProduct R fun (i : ι₂) => s (e.symm i)) :
                                              (PiTensorProduct.map f) ((PiTensorProduct.reindex R s e).symm x) = (PiTensorProduct.reindex R t e).symm ((PiTensorProduct.map fun (i : ι₂) => f (e.symm i)) x)
                                              noncomputable def PiTensorProduct.isEmptyEquiv (ι : Type u_1) {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [IsEmpty ι] :
                                              (PiTensorProduct R fun (i : ι) => s i) ≃ₗ[R] R

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

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                                              • One or more equations did not get rendered due to their size.
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                                                @[simp]
                                                theorem PiTensorProduct.isEmptyEquiv_symm_apply (ι : Type u_1) {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [IsEmpty ι] (r : R) :
                                                @[simp]
                                                theorem PiTensorProduct.isEmptyEquiv_apply_tprod (ι : Type u_1) {R : Type u_4} [CommSemiring R] {s : ιType u_7} [(i : ι) → AddCommMonoid (s i)] [(i : ι) → Module R (s i)] [IsEmpty ι] (f : (i : ι) → s i) :
                                                noncomputable def PiTensorProduct.subsingletonEquiv {ι : Type u_1} {R : Type u_4} [CommSemiring R] {M : Type u_8} [AddCommMonoid M] [Module R M] [Subsingleton ι] (i₀ : ι) :
                                                (PiTensorProduct R fun (x : ι) => M) ≃ₗ[R] M

                                                Tensor product of M over a singleton set is equivalent to M

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                                                • One or more equations did not get rendered due to their size.
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                                                  @[simp]
                                                  theorem PiTensorProduct.subsingletonEquiv_symm_apply {ι : Type u_1} {R : Type u_4} [CommSemiring R] {M : Type u_8} [AddCommMonoid M] [Module R M] [Subsingleton ι] (i₀ : ι) (m : M) :
                                                  (PiTensorProduct.subsingletonEquiv i₀).symm m = (PiTensorProduct.tprod R) fun (x : ι) => m
                                                  @[simp]
                                                  theorem PiTensorProduct.subsingletonEquiv_apply_tprod {ι : Type u_1} {R : Type u_4} [CommSemiring R] {M : Type u_8} [AddCommMonoid M] [Module R M] [Subsingleton ι] (i : ι) (f : ιM) :
                                                  noncomputable def PiTensorProduct.tmulEquiv {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] :
                                                  TensorProduct R (PiTensorProduct R fun (x : ι) => M) (PiTensorProduct R fun (x : ι₂) => M) ≃ₗ[R] PiTensorProduct R fun (x : ι ι₂) => M

                                                  Equivalence between a TensorProduct of PiTensorProducts and a single PiTensorProduct indexed by a Sum type.

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

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                                                    @[simp]
                                                    theorem PiTensorProduct.tmulEquiv_apply {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] (a : ιM) (b : ι₂M) :
                                                    (PiTensorProduct.tmulEquiv R M) (((PiTensorProduct.tprod R) fun (i : ι) => a i) ⊗ₜ[R] (PiTensorProduct.tprod R) fun (i : ι₂) => b i) = (PiTensorProduct.tprod R) fun (i : ι ι₂) => Sum.elim a b i
                                                    @[simp]
                                                    theorem PiTensorProduct.tmulEquiv_symm_apply {ι : Type u_1} {ι₂ : Type u_2} (R : Type u_4) [CommSemiring R] (M : Type u_8) [AddCommMonoid M] [Module R M] (a : ι ι₂M) :
                                                    (PiTensorProduct.tmulEquiv R M).symm ((PiTensorProduct.tprod R) fun (i : ι ι₂) => a i) = ((PiTensorProduct.tprod R) fun (i : ι) => a (Sum.inl i)) ⊗ₜ[R] (PiTensorProduct.tprod R) fun (i : ι₂) => a (Sum.inr i)
                                                    noncomputable instance PiTensorProduct.instAddCommGroup {ι : Type u_1} {R : Type u_2} [CommRing R] {s : ιType u_3} [(i : ι) → AddCommGroup (s i)] [(i : ι) → Module R (s i)] :
                                                    AddCommGroup (PiTensorProduct R fun (i : ι) => s i)
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