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feat: missing lemma on finprod, modelled on the one on Finset.prod (#…
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sgouezel committed Jul 24, 2023
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8 changes: 8 additions & 0 deletions Mathlib/Algebra/BigOperators/Finprod.lean
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Expand Up @@ -590,6 +590,14 @@ theorem finprod_eq_one_of_forall_eq_one {f : α → M} (h : ∀ x, f x = 1) :
#align finprod_eq_one_of_forall_eq_one finprod_eq_one_of_forall_eq_one
#align finsum_eq_zero_of_forall_eq_zero finsum_eq_zero_of_forall_eq_zero

@[to_additive finsum_pos']
theorem one_lt_finprod' {M : Type _} [OrderedCancelCommMonoid M] {f : ι → M}
(h : ∀ i, 1 ≤ f i) (h' : ∃ i, 1 < f i) (hf : (mulSupport f).Finite) : 1 < ∏ᶠ i, f i := by
rcases h' with ⟨i, hi⟩
rw [finprod_eq_prod _ hf]
refine Finset.one_lt_prod' (fun i _ ↦ h i) ⟨i, ?_, hi⟩
simpa only [Finite.mem_toFinset, mem_mulSupport] using ne_of_gt hi

/-!
### Distributivity w.r.t. addition, subtraction, and (scalar) multiplication
-/
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