/- 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.Order@[expose] public sectionvariable {ι α : Type*} [PartialOrder α] [CommGroup α] [IsOrderedMonoid α] [UniformSpace α] [IsUniformGroup α] [CompleteSpace α] [OrderClosedTopology α] {f : ι → α} {s : Finset ι} {t : Set ι} @[to_additive] protected lemma Multipliable.prod_le_tprod_set (hst : ↑s ⊆ t) (hfst : ∀ i ∉ s, i ∈ t → 1 ≤ f i) (hf : Multipliable f) : ∏ i ∈ s, f i ≤ ∏' i : t, f i := byι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∏ i ∈ s, f i ≤ ∏' (i : ↑t), f ↑i rw [tprod_subtype,ι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∏ i ∈ s, f i ≤ ∏' (x : ι), t.mulIndicator f xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ s.prod ?m.53 ≤ ∏' (x : ι), t.mulIndicator f xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∀ x ∈ s, f x = ?m.53 xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ι → α Finset.prod_congr rflι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ s.prod ?m.53 ≤ ∏' (x : ι), t.mulIndicator f xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∀ x ∈ s, f x = ?m.53 xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ι → αι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ s.prod ?m.53 ≤ ∏' (x : ι), t.mulIndicator f xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∀ x ∈ s, f x = ?m.53 xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ι → α]ι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ s.prod ?m.53 ≤ ∏' (x : ι), t.mulIndicator f xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∀ x ∈ s, f x = ?m.53 xι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ι → α refine (hf.mulIndicator t).prod_le_tprod _ (byι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable f⊢ ∀ i ∉ s, 1 ≤ t.mulIndicator f i simpa [Set.mulIndicator, apply_ite]All goals completed! 🐙) rintro i hiι:Type u_1α:Type u_2inst✝⁶:PartialOrder αinst✝⁵:CommGroup αinst✝⁴:IsOrderedMonoid αinst✝³:UniformSpace αinst✝²:IsUniformGroup αinst✝¹:CompleteSpace αinst✝:OrderClosedTopology αf:ι → αs:Finset ιt:Set ιhst:↑s ⊆ thfst:∀ i ∉ s, i ∈ t → 1 ≤ f ihf:Multipliable fi:ιhi:i ∈ s⊢ f i = t.mulIndicator f i simp [Set.mulIndicator, hst hi]All goals completed! 🐙