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Theorems · Theorem · group theory

subsingleton_iff_zero_eq_one

∀ {M₀ : Type u_1} [inst : MulZeroOneClass M₀], 0 = 1 ↔ Subsingleton M₀

In a monoid with zero, zero equals one if and only if all elements of that semiring are equal.

Defined in
Mathlib.Algebra.GroupWithZero.Basic
Cited by
6 results in Mathlib
Foundations
Depth 8 from the axioms · uses no axioms
Assumes
MulZeroOneClass
Mathlib docsSourceLoogleLeanSearch

Around this declaration

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

Polynomial.monic_zero_iff_subsingleton · cited by 4Polynomial.monic_zero_iff…IsSimpleRing.injective_ringHom_or_subsingleton_codomain · cited by 2IsSimpleRing.injective_ri…OreLocalization.subsingleton_iff · cited by 1OreLocalization.subsingle…subsingleton_of_zero_eq_one · cited by 1subsingleton_of_zero_eq_o…Polynomial.not_monic_zero_iff · cited by 1Polynomial.not_monic_zero…ringChar.ringChar_eq_one · cited by 0ringChar.ringChar_eq_oneMulZeroOneClass · cited by 184MulZeroOneClasssubsingleton_iff_zero_eq_oneCITED BYCITES

Cites1

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Cited by6

Results whose statement or proof uses this declaration.