English
Let s be a finite multiset of a nonunital nonassoc semiring. If every element b ∈ s commutes with a, then a commutes with s.sum: (s.sum) a = a (s.sum).
Русский
Пусть s — конечное мультимножество в полусемиринге. Если каждый элемент b ∈ s commuting с a, то произведение суммы s по переставлению также равно: a commutes with sum(s).
LaTeX
$$$$ (\\forall b \\in s,\\ Commut e(b,a)) \\Rightarrow \\left( \\left( \\sum_{b \in s} b \\right) a = a \\left( \\sum_{b \in s} b \\right) \\right). $$$$
Lean4
theorem even_sum_iff_even_card_odd {s : Finset ι} (f : ι → ℕ) : Even (∑ i ∈ s, f i) ↔ Even #({x ∈ s | Odd (f x)}) :=
by
rw [← Finset.sum_filter_add_sum_filter_not _ (fun x ↦ Even (f x)), Nat.even_add]
simp only [Finset.mem_filter, and_imp, imp_self, implies_true, Finset.even_sum, true_iff]
rw [Nat.even_iff, Finset.sum_nat_mod, Finset.sum_filter]
simp +contextual only [Nat.not_even_iff_odd, Nat.odd_iff.mp]
simp_rw [← Finset.sum_filter, ← Nat.even_iff, Finset.card_eq_sum_ones]
