Theorems · Theorem · linear algebra
LinearMap.ker_eq_bot
∀ {R : Type u_1} {R₂ : Type u_2} {M : Type u_5} {M₂ : Type u_7} [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}, f.ker = ⊥ ↔ Function.Injective ⇑f- Defined in
- Mathlib.Algebra.Module.Submodule.Ker
- Cited by
- 92 results in Mathlib
- Foundations
- Depth 68 from the axioms · uses propext, Classical.choice, 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.
- DFunLike.coestatement · cited by 62,936
- 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
- Ringstatement and proof · cited by 7,463
- Submodulestatement · cited by 7,192
- Bot.botstatement and proof · cited by 4,720
- LinearMap.kerstatement · cited by 848
- inf_of_le_rightproof · cited by 128
- LinearMap.disjoint_ker_iff_injOnproof · cited by 2
Cited by92
Results whose statement or proof uses this declaration.
- ModuleCat.mono_iff_injectiveproof · cited by 21
- lTensor_exactproof · cited by 11
- groupHomology.map_chainsFunctor_shortExactproof · cited by 8
- LinearMap.normDet_ne_zero_tfaeproof · cited by 6
- LinearMap.exists_leftInverse_of_injectiveproof · cited by 6
- Algebra.FormallySmooth.comp_surjectiveproof · cited by 6
- LinearMap.normDet_eq_zero_iff_ker_ne_botproof · cited by 5
- Module.FaithfullyFlat.zero_iff_lTensor_zeroproof · cited by 5
- iSupIndep.dfinsupp_lsum_injectiveproof · cited by 5
- Module.End.HasUnifEigenvalue.ltproof · cited by 4
- Algebra.Extension.subsingleton_h1Cotangentproof · cited by 3
- LieHom.ker_eq_botproof · cited by 3