Theorems · Theorem · group theory
GroupWithZero.mul_inv_cancel
∀ {G₀ : Type u} [self : GroupWithZero G₀] (a : G₀), a ≠ 0 → a * a⁻¹ = 1Every nonzero element of a group with zero is invertible.
- Defined in
- Mathlib.Algebra.GroupWithZero.Defs
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 4 from the axioms · uses no axioms
- Assumes
- GroupWithZero
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.
- GroupWithZerostatement and proof · cited by 691
- Monoid.toOnestatement · cited by 34
- MonoidWithZero.toZerostatement · cited by 8
- GroupWithZero.toInvstatement · cited by 4
Cited by8
Results whose statement or proof uses this declaration.
- mul_inv_cancel₀proof · cited by 210
- Fintype.divisionRingOfIsDomainproof · cited by 1
- commGroupWithZeroOfIsUnitOrEqZeroproof · cited by 0
- Function.Injective.divisionRingproof · cited by 0
- Function.Injective.commGroupWithZeroproof · cited by 0
- Function.Injective.divisionSemiringproof · cited by 0
- DivisionRing.ofIsUnitOrEqZeroproof · cited by 0
- Function.Surjective.commGroupWithZeroproof · cited by 0