Theorems · Theorem · group theory
Finset.prod_ne_zero_iff
∀ {ι : Type u_1} {M₀ : Type u_4} [inst : CommMonoidWithZero M₀] {f : ι → M₀} {s : Finset ι} [Nontrivial M₀]
[NoZeroDivisors M₀], ∏ x ∈ s, f x ≠ 0 ↔ ∀ a ∈ s, f a ≠ 0- Cited by
- 45 results in Mathlib
- Foundations
- Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites6
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Finsetstatement and proof · cited by 13,712
- Nontrivialstatement and proof · cited by 2,416
- Finset.prodstatement and proof · cited by 2,356
- CommMonoidWithZerostatement and proof · cited by 913
- NoZeroDivisorsstatement and proof · cited by 545
- Finset.prod_eq_zero_iffproof · cited by 9
Cited by45
Results whose statement or proof uses this declaration.
- Nat.mem_finMulAntidiagproof · cited by 7
- ENNReal.prod_inv_distribproof · cited by 3
- MeromorphicOn.extract_zeros_poles_logproof · cited by 3
- Nat.primeFactors_prodproof · cited by 3
- Field.nonempty_algHom_of_exists_rootproof · cited by 3
- Polynomial.natDegree_prodproof · cited by 3
- meromorphicNFAt_prodproof · cited by 3
- Finsupp.prod_ne_zero_iffproof · cited by 3
- Associates.finprod_ne_zeroproof · cited by 2
- IsDedekindDomain.exists_sup_span_eqproof · cited by 2
- RootPairing.finrank_range_polarization_eq_finrank_span_corootproof · cited by 2
- Algebra.IsAlgebraic.exists_integral_multiplesproof · cited by 2