Theorems · Theorem · group theory
map_eq_one_iff
∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : One M] [inst_1 : One N] [inst_2 : FunLike F M N]
[OneHomClass F M N] (f : F), Function.Injective ⇑f → ∀ {x : M}, f x = 1 ↔ x = 1- Defined in
- Mathlib.Algebra.Group.Hom.Defs
- Cited by
- 15 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- OneOneFunLikeOneHomClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites4
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
- FunLikestatement and proof · cited by 2,560
- map_oneproof · cited by 861
- OneHomClassstatement and proof · cited by 38
Cited by15
Results whose statement or proof uses this declaration.
- injective_iff_map_eq_oneproof · cited by 9
- EmbeddingLike.map_eq_one_iffproof · cited by 4
- Set.powersetCard.fixedPoints_ne_univ_of_faithfulSMulproof · cited by 3
- MonoidHom.injective_of_surjective_of_injective_of_injectiveproof · cited by 2
- MonoidHom.surjective_of_surjective_of_surjective_of_injectiveproof · cited by 2
- DirichletCharacter.changeLevel_eq_one_iffproof · cited by 1
- Monoid.PushoutI.NormalWord.eq_one_of_smul_normalizedproof · cited by 1
- IsDedekindDomain.selmerGroup.fromUnit_kerproof · cited by 1
- FaithfulSMul.algebraMap_eq_one_iffproof · cited by 1
- map_ne_one_iffproof · cited by 1
- alternatingGroup.mem_map_kleinFour_ofSubtypeproof · cited by 1
- TensorAlgebra.algebraMap_eq_one_iffproof · cited by 0