one_le_finprod' — Mathlib

English

If every f i is at least 1, then finprod f is at least 1.

Русский

Если каждый f i не меньше 1, то финпроизводство не меньше 1.

LaTeX

$$$\\forall i, 1 \\le f(i) \\Rightarrow 1 \\le \\mathrm{finprod}(f)$$$

Lean4

@[to_additive finsum_nonneg]
theorem one_le_finprod' {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedMonoid M] {f : α → M}
    (hf : ∀ i, 1 ≤ f i) : 1 ≤ ∏ᶠ i, f i :=
  finprod_induction _ le_rfl (fun _ _ => one_le_mul) hf

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