Theorems · Definition · linear algebra

Basis.piTensorProduct

{ι : Type u_1} →
  {R : Type u_2} →
    {M : ι → Type u_3} →
      {κ : ι → Type u_4} →
        [inst : CommSemiring R] →
          [inst_1 : (i : ι) → AddCommMonoid (M i)] →
            [inst_2 : (i : ι) → Module R (M i)] →
              [Finite ι] →
                ((i : ι) → Module.Basis (κ i) R (M i)) → Module.Basis ((i : ι) → κ i) R (PiTensorProduct R fun i => M i)

Let ι be a Finite type and M be a family of modules indexed by ι. If b i : κ i → M i is a basis for every i in ι, then fun (p : Π i, κ i) ↦ ⨂ₜ[R] i, b i (p i) is a basis of ⨂[R] i, M i.

Defined in

Mathlib.LinearAlgebra.PiTensorProduct.Basis

Cited by

5 results in Mathlib

Foundations

Depth 93 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidModuleFinite