Theorems · Theorem · commutative algebra
Submonoid.isUnit_iff_and
∀ {α : Type u_2} {C : Type u_4} [inst : SetLike C α] [inst_1 : GroupWithZero α] [inst_2 : SubmonoidClass C α] {S : C}
{a : ↥S}, IsUnit a ↔ ↑a ≠ 0 ∧ (↑a)⁻¹ ∈ S- Defined in
- Mathlib.RingTheory.Ideal.Nonunits
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 26 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- IsUnitstatement and proof · cited by 1,602
- SetLikestatement and proof · cited by 1,084
- GroupWithZerostatement and proof · cited by 691
- inv_mul_cancel₀proof · cited by 267
- mul_inv_cancel₀proof · cited by 210
- IsUnit.mapproof · cited by 104
- SubmonoidClassstatement and proof · cited by 60
- IsUnit.ne_zeroproof · cited by 36
- Submonoid.subtypeproof · cited by 26
- Submonoid.ofClassproof · cited by 6
- Submonoid.inv_mem_of_isUnitproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- Submonoid.mem_nonunits_iff_orproof · cited by 1
- Submonoid.isUnit_iff_of_ne_zeroproof · cited by 0