Theorems · Theorem · commutative algebra
CharacterModule.dual_injective_iff_surjective
∀ {R : Type uR} [inst : CommRing R] {A : Type uA} [inst_1 : AddCommGroup A] {A' : Type u_1} [inst_2 : AddCommGroup A']
[inst_3 : Module R A] [inst_4 : Module R A'] {f : A →ₗ[R] A'},
Function.Injective ⇑(CharacterModule.dual f) ↔ Function.Surjective ⇑f- Defined in
- Mathlib.Algebra.Module.CharacterModule
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 99 from the axioms · uses propext, Classical.choice, 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 and proof · cited by 62,936
- Modulestatement and proof · cited by 20,661
- RingHom.idstatement and proof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- AddCommGroupstatement and proof · cited by 12,871
- LinearMapstatement and proof · cited by 10,215
- CharacterModulestatement · cited by 26
- CharacterModule.dualstatement and proof · cited by 13
- CharacterModule.dual_injective_of_surjectiveproof · cited by 1
- CharacterModule.surjective_of_dual_injectiveproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- CharacterModule.dual_bijective_iff_bijectiveproof · cited by 1