Theorems · Theorem · linear algebra

Submodule.span_preimage_eq

∀ {R : Type u_1} {M : Type u_2} [inst : Ring R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] {R₂ : Type u_3}
  {M₂ : Type u_4} [inst_3 : Ring R₂] [inst_4 : AddCommGroup M₂] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
  [inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {s : Set M₂},
  s.Nonempty → s ⊆ ↑f.range → Submodule.span R (⇑f ⁻¹' s) = Submodule.comap f (Submodule.span R₂ s)

Defined in

Mathlib.LinearAlgebra.Quotient.Basic

Cited by

1 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RingAddCommGroupModuleRingAddCommGroupModuleRingHomSurjective