Theorems · Theorem · linear algebra

Module.Basis.mem_submodule_iff

∀ {ι : Type u_1} {R : Type u_3} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
  {P : Submodule R M} (b : Module.Basis ι R ↥P) {x : M}, x ∈ P ↔ ∃ c, x = c.sum fun i x => x • ↑(b i)

If the submodule P has a basis, x ∈ P iff it is a linear combination of basis vectors.

Defined in

Mathlib.LinearAlgebra.Basis.Submodule

Cited by

4 results in Mathlib

Foundations

Depth 87 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringAddCommMonoidModule