Theorems · Theorem · group theory
map_eq_zero_iff
∀ {M : Type u_4} {N : Type u_5} {F : Type u_9} [inst : Zero M] [inst_1 : Zero N] [inst_2 : FunLike F M N]
[ZeroHomClass F M N] (f : F), Function.Injective ⇑f → ∀ {x : M}, f x = 0 ↔ x = 0- Defined in
- Mathlib.Algebra.Group.Hom.Defs
- Cited by
- 62 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- ZeroZeroFunLikeZeroHomClass
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_zeroproof · cited by 1,614
- ZeroHomClassstatement and proof · cited by 74
Cited by62
Results whose statement or proof uses this declaration.
- injective_iff_map_eq_zeroproof · cited by 62
- map_ne_zero_iffproof · cited by 24
- LinearMap.map_eq_zero_iffproof · cited by 10
- EmbeddingLike.map_eq_zero_iffproof · cited by 9
- IsNilpotent.map_iffproof · cited by 9
- FaithfulSMul.algebraMap_eq_zero_iffproof · cited by 8
- minpoly.algHom_eqproof · cited by 8
- IsAlgebraic.exists_integral_multipleproof · cited by 8
- Polynomial.map_eq_zero_iffproof · cited by 5
- Valuation.Integers.coe_span_singleton_eq_setOfPred_le_v_algebraMapproof · cited by 5
- charP_of_injective_ringHomproof · cited by 5
- map_le_nonZeroDivisors_of_injectiveproof · cited by 5