English
If R has no zero divisors, then multiplication commutes with finsum over a finite index set: (r * ∑ᶠ a ∈ s, f a) = ∑ᶠ a ∈ s, r * f a.
Русский
Если в кольце без нулевых делителей, то умножение молчит через finsum по конечному индексу: (r · ∑ᶠ a ∈ s, f a) = ∑ᶠ a ∈ s, r · f a.
LaTeX
$$$ (NoZeroDivisors\\, R) \\\\Rightarrow (r \\cdot \\\\sum^\\\\mathrm{f}_{a \\in s} f(a)) = \\\\sum^\\\\mathrm{f}_{a \\in s} (r \\cdot f(a)) $$$
Lean4
/-- If `R` has no zero divisors, then multiplication commutes with `finsum_mem`. See `mul_finsum_mem'`
for a statement assuming finiteness of support.
-/
theorem mul_finsum_mem {R : Type*} [NonUnitalNonAssocSemiring R] [NoZeroDivisors R] {s : Set α} (f : α → R) (r : R) :
(r * ∑ᶠ a ∈ s, f a) = ∑ᶠ a ∈ s, r * f a := by
rw [mul_finsum]
congr
ext a
by_cases h : a ∈ s <;> simp_all
