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Theorems · Theorem · ring theory

isGroupLikeElem_iff_of_mul_eq_one

∀ {R : Type u_1} {A : Type u_2} [inst : CommSemiring R] [inst_1 : Semiring A] [inst_2 : Bialgebra R A] {a b : A},
  a * b = 1 → b * a = 1 → (IsGroupLikeElem R a ↔ IsGroupLikeElem R b)

Group-like elements in a bialgebra are stable under inverses, when they exist.

Defined in
Mathlib.RingTheory.Bialgebra.GroupLike
Cited by
1 results in Mathlib
Foundations
Depth 81 from the axioms · uses propext, Quot.sound
Assumes
CommSemiringSemiringBialgebra
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Around this declaration

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

isGroupLikeElem_unitsInv · cited by 2isGroupLikeElem_unitsInvSemiring · cited by 13802SemiringCommSemiring · cited by 10911CommSemiringBialgebra · cited by 160BialgebraIsGroupLikeElem · cited by 39IsGroupLikeElemIsGroupLikeElem.of_mul_eq_one · cited by 2IsGroupLikeElem.of_mul_eq…isGroupLikeElem_iff_of_mul_eq…CITED BYCITES

Cites5

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Cited by1

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