Theorems · Theorem · commutative algebra
LinearMap.ker_le_iff
∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_6} [inst : Ring R] [inst_1 : Ring R₂]
[inst_2 : AddCommGroup M] [inst_3 : AddCommGroup M₂] [inst_4 : Module R M] [inst_5 : Module R₂ M₂] {τ₁₂ : R →+* R₂}
{f : M →ₛₗ[τ₁₂] M₂} [inst_6 : RingHomSurjective τ₁₂] {p : Submodule R M}, f.ker ≤ p ↔ ∃ y ∈ f.range, ⇑f ⁻¹' {y} ⊆ ↑p- Defined in
- Mathlib.Algebra.Module.Submodule.Range
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- Setstatement and proof · cited by 53,352
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- SetLike.coestatement and proof · cited by 8,199
- Ringstatement and proof · cited by 7,463
- Submodulestatement and proof · cited by 7,192
- Set.preimagestatement and proof · cited by 4,946
- zero_addproof · cited by 2,366
- map_zeroproof · cited by 1,614
Cited by1
Results whose statement or proof uses this declaration.
- Submodule.span_preimage_eqproof · cited by 1