Theorems · Theorem · group theory
AddMonoidHom.ker_eq_bot_iff
∀ {G : Type u_1} [inst : AddGroup G] {M : Type u_7} [inst_1 : AddZeroClass M] (f : G →+ M),
f.ker = ⊥ ↔ Function.Injective ⇑f- Defined in
- Mathlib.Algebra.Group.Subgroup.Ker
- Cited by
- 14 results in Mathlib
- Foundations
- Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AddGroupAddZeroClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
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
- Bot.botstatement and proof · cited by 4,720
- AddGroupstatement and proof · cited by 4,410
- AddSubgroupstatement and proof · cited by 3,232
- AddMonoidHomstatement and proof · cited by 3,230
- AddZeroClassstatement and proof · cited by 1,237
- AddMonoidHom.kerstatement and proof · cited by 158
- bot_uniqueproof · cited by 57
- AddMonoidHom.map_zeroproof · cited by 47
- AddSubgroup.mem_botproof · cited by 18
- neg_add_eq_zeroproof · cited by 14
- AddMonoidHom.eq_iffproof · cited by 2
Cited by14
Results whose statement or proof uses this declaration.
- AddMonoidHom.ker_eq_botproof · cited by 7
- AddCommGrpCat.mono_iff_injectiveproof · cited by 6
- AddChar.doubleDualEmb_injectiveproof · cited by 2
- AddGrpCat.mono_iff_injectiveproof · cited by 2
- AddCommGrpCat.mono_iff_ker_eq_botproof · cited by 1
- AddMonoidHom.injective_noncommPiCoprod_of_iSupIndepproof · cited by 1
- AddGroup.isAddCyclic_of_coprime_card_range_card_kerproof · cited by 1
- CategoryTheory.Abelian.Ext.postcomp_mk₀_injective_of_monoproof · cited by 1
- CategoryTheory.Abelian.Ext.precomp_mk₀_injective_of_epiproof · cited by 1
- AddGrpCat.mono_iff_ker_eq_botproof · cited by 1
- QuotientAddGroup.injective_lift_iffproof · cited by 0
- AddSubgroup.nsmulAddMonoidHom_injective_of_isTorsionFreeproof · cited by 0