Theorems · Theorem · linear algebra
LinearMap.lsmul_injective
∀ {R : Type u_1} {M : Type u_2} [inst : CommRing R] [IsDomain R] [inst_2 : AddCommGroup M] [inst_3 : Module R M]
[Module.IsTorsionFree R M] {x : R}, x ≠ 0 → Function.Injective ⇑((LinearMap.lsmul R M) x)- Defined in
- Mathlib.LinearAlgebra.BilinearMap
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 37 from the axioms · uses propext, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- RingHom.idstatement · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement · cited by 10,215
- IsDomainstatement and proof · cited by 2,196
- Module.IsTorsionFreestatement and proof · cited by 600
- LinearMap.lsmulstatement · cited by 50
- smul_right_injectiveproof · cited by 24
Cited by3
Results whose statement or proof uses this declaration.
- Module.Flat.flat_iff_torsion_eq_bot_of_isBezoutproof · cited by 1
- Submodule.finiteQuotient_iffproof · cited by 1
- LinearMap.ker_lsmulproof · cited by 0