Theorems · Theorem · group theory
Finset.mul_prod_erase
∀ {ι : Type u_1} {M : Type u_4} [inst : CommMonoid M] [inst_1 : DecidableEq ι] (s : Finset ι) (f : ι → M) {a : ι},
a ∈ s → f a * ∏ x ∈ s.erase a, f x = ∏ x ∈ s, f xTaking a product over s : Finset ι is the same as multiplying the value on a single element
f a by the product of s.erase a.
See Multiset.prod_map_erase for the Multiset version.
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommMonoidDecidableEq
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Finset.prodstatement and proof · cited by 2,356
- CommMonoidstatement and proof · cited by 2,264
- Finset.erasestatement · cited by 455
- Finset.prod_insertproof · cited by 109
- Finset.insert_eraseproof · cited by 65
- Finset.notMem_eraseproof · cited by 59
Cited by23
Results whose statement or proof uses this declaration.
- Finset.prod_erase_mulproof · cited by 20
- meromorphicNFAt_prodproof · cited by 3
- Function.FactorizedRational.extractFactorproof · cited by 2
- Lagrange.nodal_eq_mul_nodal_eraseproof · cited by 2
- Finsupp.mul_prod_eraseproof · cited by 1
- Matrix.isAddUnit_detp_mul_detpproof · cited by 1
- Lagrange.eval_basis_not_at_nodeproof · cited by 1
- NumberField.mixedEmbedding.adjust_fproof · cited by 1
- Finset.prod_filter_of_pairwise_eq_oneproof · cited by 1
- RingOfIntegers.dvd_normproof · cited by 1
- Polynomial.quo_add_sum_rem_mul_pow_inverse_uniqueproof · cited by 1
- Polynomial.quo_mul_prod_add_sum_rem_mul_prod_uniqueproof · cited by 1