Theorems · Theorem · commutative algebra
Module.End.isUnit_iff
∀ {R : Type u_1} {M : Type u_5} [inst : Semiring R] [inst_1 : AddCommMonoid M] [inst_2 : Module R M]
(f : Module.End R M), IsUnit f ↔ Function.Bijective ⇑f- Defined in
- Mathlib.Algebra.Module.Equiv.Basic
- Cited by
- 21 results in Mathlib
- Foundations
- Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- SemiringAddCommMonoidModule
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
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
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- Semiringstatement and proof · cited by 13,802
- AddCommMonoidstatement and proof · cited by 12,281
- Equivproof · cited by 8,337
- LinearEquivproof · cited by 3,317
- IsUnitstatement and proof · cited by 1,602
- LinearEquiv.symmproof · cited by 1,461
- LinearEquiv.toLinearMapproof · cited by 1,171
- Function.Bijectivestatement and proof · cited by 863
- LinearMap.extproof · cited by 844
Cited by21
Results whose statement or proof uses this declaration.
- Module.End.algebraMap_isUnit_inv_apply_eq_iffproof · cited by 8
- IsLocalizedModule.eq_iff_existsproof · cited by 8
- Module.mem_freeLocus_of_isLocalizationproof · cited by 3
- exists_bijective_map_powersproof · cited by 3
- AlgebraicGeometry.StructureSheaf.Localizations.comapFun_mkproof · cited by 3
- Module.End.algebraMap_isUnit_inv_apply_eq_iff'proof · cited by 2
- AlgebraicGeometry.Scheme.Modules.isUnit_algebraMap_end_of_le_basicOpenproof · cited by 2
- Module.injective_of_isLocalizedModuleproof · cited by 2
- IsLocalizedModule.smul_injectiveproof · cited by 2
- Module.FinitePresentation.exists_lift_equiv_of_isLocalizedModuleproof · cited by 1
- Module.exists_localizedMap_surjective_of_surjectiveproof · cited by 1
- ContinuousLinearMap.isUnit_iff_isUnit_toLinearMapproof · cited by 1