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Theorems · Definition · group theory

Submonoid.equivMapOfInjective

{N : Type u_2} →
  [inst : MulOneClass N] →
    {M : Type u_5} →
      [inst_1 : MulOneClass M] → (S : Submonoid M) → (f : M →* N) → Function.Injective ⇑f → ↥S ≃* ↥(Submonoid.map f S)

A Subgroup is isomorphic to its image under an injective function. If you have an isomorphism, use MulEquiv.submonoidMap for better definitional equalities.

Defined in
Mathlib.Algebra.Group.Submonoid.Operations
Cited by
2 results in Mathlib
Foundations
Depth 16 from the axioms · uses Classical.choice
Assumes
MulOneClassMulOneClass
Mathlib docsSourceLoogleLeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Submonoid.coe_equivMapOfInjective_apply · cited by 0Submonoid.coe_equivMapOfI…Submonoid.equivMapOfInjective_coe_mulEquiv · cited by 0Submonoid.equivMapOfInjec…DFunLike.coe · cited by 62936DFunLike.coeEquiv · cited by 8337EquivSetLike.coe · cited by 8199SetLike.coeSet.Elem · cited by 7166Set.ElemSet.image · cited by 5609Set.imageMonoidHom · cited by 3629MonoidHomSubmonoid · cited by 3086SubmonoidMulEquiv · cited by 1142MulEquivMulOneClass · cited by 1018MulOneClassSubmonoid.map · cited by 190Submonoid.mapEquiv.Set.image · cited by 15Set.imageSubmonoid.equivMapOfInjectiveCITED BYCITES

Cites11

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Cited by2

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