Theorems · Theorem · linear algebra
LinearMap.ker_eq_top
∀ {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}
{f : M →ₛₗ[τ₁₂] M₂}, f.ker = ⊤ ↔ f = 0- Defined in
- Mathlib.Algebra.Module.Submodule.Ker
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 29 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Modulestatement and proof · cited by 20,661
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- LinearMapstatement and proof · cited by 10,215
- RingHomstatement and proof · cited by 10,189
- Top.topstatement and proof · cited by 9,680
- Submodulestatement · cited by 7,192
- LinearMap.kerstatement and proof · cited by 848
- LinearMap.extproof · cited by 844
- LinearMap.mem_kerproof · cited by 66
- LinearMap.ker_zeroproof · cited by 12
Cited by12
Results whose statement or proof uses this declaration.
- LinearMap.range_le_ker_iffproof · cited by 9
- polarCoord_source_ae_eq_univproof · cited by 3
- LinearMap.injective_or_eq_zeroproof · cited by 3
- CommRingCat.KaehlerDifferential.extproof · cited by 1
- LinearMap.range_le_bot_iffproof · cited by 1
- Module.Finite.exists_smul_of_comp_eq_of_isLocalizedModuleproof · cited by 1
- LinearIsometryEquiv.reflections_generate_dim_auxproof · cited by 1
- Module.End.IsSemisimple.eq_zero_iff_forall_eigenvalueproof · cited by 0
- PiTensorProduct.submodule_span_eq_topproof · cited by 0
- LinearMap.surjective_iff_eq_zero_of_exactproof · cited by 0
- Module.exists_surjective_quotient_of_finiteproof · cited by 0