Theorems · Theorem · linear algebra

LinearMap.le_ker_iff_map

∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_7} [inst : Semiring R] [inst_1 : Semiring R₂]
  [inst_2 : AddCommMonoid M] [inst_3 : AddCommMonoid M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
  [inst_6 : RingHomSurjective τ₁₂] {f : M →ₛₗ[τ₁₂] M₂} {p : Submodule R M}, p ≤ f.ker ↔ Submodule.map f p = ⊥

Defined in

Mathlib.Algebra.Module.Submodule.Ker

Cited by

3 results in Mathlib

Foundations

Depth 27 from the axioms · uses propext, Quot.sound

Assumes

SemiringSemiringAddCommMonoidAddCommMonoidModuleModuleRingHomSurjective