/- Copyright 2025 The Formal Conjectures Authors. Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at https://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. -/ module public import Mathlib.Topology.Algebra.InfiniteSum.Group@[expose] public sectionopen Filteropen scoped Topologyvariable {ι α β : Type*} [CommGroup α] [TopologicalSpace α] [IsTopologicalGroup α] {f : β → α} {l : Filter ι} {s : ι → Set β} @[to_additive] lemma Multipliable.tendsto_tprod_one (hf : Multipliable f) (hs : ∀ b, ∀ᶠ i in l, b ∉ s i) : Tendsto (fun i ↦ ∏' b : s i, f b) l (𝓝 1) := byι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s i⊢ Tendsto (fun i ↦ ∏' (b : ↑(s i)), f ↑b) l (𝓝 1) rw [tendsto_iff_eventuallyι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s i⊢ ∀ ⦃p : α → Prop⦄, (∀ᶠ (y : α) in 𝓝 1, p y) → ∀ᶠ (x : ι) in l, p (∏' (b : ↑(s x)), f ↑b)ι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s i⊢ ∀ ⦃p : α → Prop⦄, (∀ᶠ (y : α) in 𝓝 1, p y) → ∀ᶠ (x : ι) in l, p (∏' (b : ↑(s x)), f ↑b)]ι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s i⊢ ∀ ⦃p : α → Prop⦄, (∀ᶠ (y : α) in 𝓝 1, p y) → ∀ᶠ (x : ι) in l, p (∏' (b : ↑(s x)), f ↑b) rintro p hpι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s ip:α → Prophp:∀ᶠ (y : α) in 𝓝 1, p y⊢ ∀ᶠ (x : ι) in l, p (∏' (b : ↑(s x)), f ↑b) obtain ⟨t, ht⟩ := hf.tprod_vanishing hpι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s ip:α → Prophp:∀ᶠ (y : α) in 𝓝 1, p yt:Finset βht:∀ (t_1 : Set β), Disjoint t_1 ↑t → ∏' (b : ↑t_1), f ↑b ∈ {x | (fun y ↦ p y) x}⊢ ∀ᶠ (x : ι) in l, p (∏' (b : ↑(s x)), f ↑b) filter_upwards [t.eventually_all.2 fun b _ ↦ hs b] with i hiι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s ip:α → Prophp:∀ᶠ (y : α) in 𝓝 1, p yt:Finset βht:∀ (t_1 : Set β), Disjoint t_1 ↑t → ∏' (b : ↑t_1), f ↑b ∈ {x | (fun y ↦ p y) x}i:ιhi:∀ i_1 ∈ t, i_1 ∉ s i⊢ p (∏' (b : ↑(s i)), f ↑b) exact ht _ <| byι:Type u_1α:Type u_2β:Type u_3inst✝²:CommGroup αinst✝¹:TopologicalSpace αinst✝:IsTopologicalGroup αf:β → αl:Filter ιs:ι → Set βhf:Multipliable fhs:∀ (b : β), ∀ᶠ (i : ι) in l, b ∉ s ip:α → Prophp:∀ᶠ (y : α) in 𝓝 1, p yt:Finset βht:∀ (t_1 : Set β), Disjoint t_1 ↑t → ∏' (b : ↑t_1), f ↑b ∈ {x | (fun y ↦ p y) x}i:ιhi:∀ i_1 ∈ t, i_1 ∉ s i⊢ Disjoint (s i) ↑t simpa [Set.disjoint_right]All goals completed! 🐙