Theorems · Theorem · group theory
eq_zero_of_zero_eq_one
∀ {M₀ : Type u_1} [inst : MulZeroOneClass M₀], 0 = 1 → ∀ (a : M₀), a = 0In a monoid with zero, if zero equals one, then zero is the only element.
- Defined in
- Mathlib.Algebra.GroupWithZero.Basic
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 6 from the axioms · uses no axioms
- Assumes
- MulZeroOneClass
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites3
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- mul_oneproof · cited by 3,885
- MulZeroClass.mul_zeroproof · cited by 2,091
- MulZeroOneClassstatement and proof · cited by 184
Cited by3
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_of_isNilpotentproof · cited by 1
- zero_ne_one_or_forall_eq_0proof · cited by 0
- uniqueOfZeroEqOneproof · cited by 0