Theorems · Theorem · group theory
isUnit_iff_exists
∀ {M : Type u_1} [inst : Monoid M] {x : M}, IsUnit x ↔ ∃ b, x * b = 1 ∧ b * x = 1See isUnit_iff_exists_and_exists for a similar lemma with two existentials.
- Defined in
- Mathlib.Algebra.Group.Units.Defs
- Cited by
- 12 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- Monoid
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Monoidstatement and proof · cited by 3,887
- Unitsproof · cited by 2,804
- Units.valproof · cited by 1,966
- IsUnitstatement and proof · cited by 1,602
- Units.invproof · cited by 35
- Units.inv_valproof · cited by 8
- Units.val_invproof · cited by 8
Cited by12
Results whose statement or proof uses this declaration.
- Polynomial.Sequence.span_degreeLTproof · cited by 3
- MonoidHom.isUnit_eqLocusM_mk_iffproof · cited by 2
- IsStrictlyPositive.smulproof · cited by 1
- exists_integral_inj_algHom_of_quotientproof · cited by 1
- isUnit_iff_exists_and_existsproof · cited by 1
- RestrictedProduct.isUnit_of_eventually_isUnitproof · cited by 1
- IsNilpotent.isUnit_expproof · cited by 0
- factorPowSucc.isUnit_of_isUnit_imageproof · cited by 0
- IsLocalRing.eq_of_eval_eq_zero_of_not_isUnit_subproof · cited by 0
- StarAlgEquiv.eq_linearIsometryEquivConjStarAlgEquivproof · cited by 0
- ZMod.isUnit_invproof · cited by 0
- MvPolynomial.X_primeproof · cited by 0