Theorems · Theorem · commutative algebra
eq_zero_of_ne_zero_of_mul_left_eq_zero
∀ {M₀ : Type u_2} [inst : MonoidWithZero M₀] {x y : M₀} [NoZeroDivisors M₀], x ≠ 0 → x * y = 0 → y = 0- Cited by
- 7 results in Mathlib
- Foundations
- Depth 7 from the axioms · uses no axioms
- Assumes
- MonoidWithZeroNoZeroDivisors
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.
- NoZeroDivisorsstatement and proof · cited by 545
- MonoidWithZerostatement and proof · cited by 456
- NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zeroproof · cited by 14
Cited by7
Results whose statement or proof uses this declaration.
- mem_nonZeroDivisors_of_ne_zeroproof · cited by 34
- Real.geom_mean_eq_arith_mean_weighted_iff_of_pos'proof · cited by 4
- IsPrimitiveRoot.geom_sum_eq_zeroproof · cited by 2
- map_mem_nonZeroDivisorsproof · cited by 1
- minpoly.prime_of_isIntegrallyClosedproof · cited by 1
- Polynomial.eval₂_primPart_eq_zeroproof · cited by 0
- Polynomial.aeval_primPart_eq_zeroproof · cited by 0