Theorems · Theorem · linear algebra

Submodule.projectionOnto_comp_surjective_of_exact

∀ {R : Type u_1} {M : Type u_2} {K : Type u_3} {P : Type u_4} [inst : Ring R] [inst_1 : AddCommGroup M]
  [inst_2 : AddCommGroup K] [inst_3 : AddCommGroup P] [inst_4 : Module R M] [inst_5 : Module R K] [inst_6 : Module R P]
  {f : K →ₗ[R] M} {g : M →ₗ[R] P},
  Function.Exact ⇑f ⇑g →
    ∀ {p q : Submodule R M} (hpq : IsCompl p q),
      Submodule.map g q = ⊤ → Function.Surjective ⇑(p.projectionOnto q hpq ∘ₗ f)

Defined in

Mathlib.LinearAlgebra.Basis.Exact

Cited by

2 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupAddCommGroupAddCommGroupModuleModuleModule