Theorems · Theorem · commutative algebra

TensorProduct.exists_finsupp_left

∀ {R : Type u_1} {M : Type u_2} {N : Type u_3} [inst : CommSemiring R] [inst_1 : AddCommMonoid M]
  [inst_2 : AddCommMonoid N] [inst_3 : Module R M] [inst_4 : Module R N] (x : TensorProduct R M N),
  ∃ S, x = S.sum fun m n => m ⊗ₜ[R] n

For any element x of M ⊗[R] N, there exists a finite subset { (m_i, n_i) } of M × N such that each m_i is distinct (we represent it as an element of M →₀ N), such that x is equal to the sum of m_i ⊗ₜ[R] n_i.

Defined in

Mathlib.LinearAlgebra.TensorProduct.Finiteness

Cited by

2 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringAddCommMonoidAddCommMonoidModuleModule