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[Merged by Bors] - feat(Topology/Algebra/InfiniteSum/NatInt): hasProd versions of ℕ+/ℕ transfer lemmas #41502

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[Merged by Bors] - feat(Topology/Algebra/InfiniteSum/NatInt): hasProd versions of ℕ+/ℕ transfer lemmas #41502

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18 changes: 18 additions & 0 deletions Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -541,10 +541,28 @@ lemma multipliable_pnat_iff_multipliable_nat [TopologicalSpace G] [IsTopological
{f : ℕ → G} : Multipliable (fun n : ℕ+ ↦ f n) ↔ Multipliable f := by
rw [multipliable_pnat_iff_multipliable_succ, multipliable_nat_add_iff]

@[to_additive]
theorem hasProd_pnat_iff_hasProd_succ {f : ℕ → M} :
HasProd (fun x : ℕ+ ↦ f x) m ↔ HasProd (fun x : ℕ ↦ f (x + 1)) m :=
Equiv.pnatEquivNat.symm.hasProd_iff.symm

@[to_additive]
theorem hasProd_pnat_iff [TopologicalSpace G] [IsTopologicalGroup G] {f : ℕ → G} {a : G} :
HasProd (fun x : ℕ+ ↦ f x) a ↔ HasProd f (a * f 0) := by
simp [hasProd_pnat_iff_hasProd_succ, hasProd_nat_add_iff]

@[to_additive]
theorem tprod_pnat_eq_tprod_succ {f : ℕ → M} : ∏' n : ℕ+, f n = ∏' n, f (n + 1) :=
(Equiv.pnatEquivNat.symm.tprod_eq _).symm

@[to_additive]
theorem tprod_pnat_eq_tprod_of_eq_one {f : ℕ → M} (hf : f 0 = 1) :
∏' n : ℕ+, f n = ∏' n : ℕ, f n :=
PNat.coe_injective.tprod_eq fun n hn ↦ by
rcases Nat.eq_zero_or_pos n with rfl | h
· exact absurd hf hn
· exact ⟨⟨n, h⟩, rfl⟩

@[to_additive]
lemma tprod_zero_pnat_eq_tprod_nat [TopologicalSpace G] [IsTopologicalGroup G] [T2Space G]
{f : ℕ → G} (hf : Multipliable f) :
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