English
For a finite index set, the product over the complement of s times the product over s equals the product over all indices.
Русский
Для конечного множества индексов произведение по дополнению s на множества и произведение по s给 равны произведению по всем индексам.
LaTeX
$$$\\left( \\prod_{i \\in s^{c}} f(i) \\right) \\left( \\prod_{i \\in s} f(i) \\right) = \\prod_{i} f(i)$$$
Lean4
/-- Multiplying the products of a function over `s` and over `sᶜ` gives the whole product.
For a version expressed with subtypes, see `Fintype.prod_subtype_mul_prod_subtype`. -/
@[to_additive /-- Adding the sums of a function over `s` and over `sᶜ` gives the whole sum.
For a version expressed with subtypes, see `Fintype.sum_subtype_add_sum_subtype`. -/
]
theorem prod_mul_prod_compl [Fintype ι] [DecidableEq ι] (s : Finset ι) (f : ι → M) :
(∏ i ∈ s, f i) * ∏ i ∈ sᶜ, f i = ∏ i, f i :=
IsCompl.prod_mul_prod isCompl_compl f
