Theorems · Theorem · linear algebra

Submodule.biSup_comap_eq_top_of_surjective

∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_4} {M₂ : Type u_5} [inst : Semiring R] [inst_1 : Semiring R₂]
  [inst_2 : AddCommGroup M] [inst_3 : Module R M] [inst_4 : AddCommGroup M₂] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
  [RingHomSurjective τ₁₂] {ι : Type u_8} (s : Set ι),
  s.Nonempty →
    ∀ (p : ι → Submodule R₂ M₂),
      ⨆ i ∈ s, p i = ⊤ → ∀ (f : M →ₛₗ[τ₁₂] M₂), Function.Surjective ⇑f → ⨆ i ∈ s, Submodule.comap f (p i) = ⊤

Defined in

Mathlib.LinearAlgebra.Span.Basic

Cited by

1 results in Mathlib

Foundations

Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringSemiringAddCommGroupModuleAddCommGroupModuleRingHomSurjective