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Co-authored-by: Kim Morrison <477956+kim-em@users.noreply.github.com>
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Mathlib/Topology/Algebra/Group/Units.lean

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/-
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Copyright (c) 2025 Imperial College London. All rights reserved.
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Copyright (c) 2025 Ruben Van de Velde and David Ledvinka. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Imperial College London FLT Project Contributors
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Authors: Ruben Van de Velde and David Ledvinka
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-/
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import Mathlib.Algebra.Group.Pi.Units
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import Mathlib.Algebra.Group.Submonoid.Units
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import Mathlib.Topology.Algebra.Constructions
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import Mathlib.Topology.Algebra.ContinuousMonoidHom
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import Mathlib.Topology.Algebra.Group.Basic
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/-!
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# Topological properties of units
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/-- If a submonoid is open in a topological monoid, then its units form an open subset
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of the units of the monoid. -/
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lemma Submonoid.units_isOpen {M : Type*} [TopologicalSpace M] [Monoid M]
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@[to_additive]
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lemma Submonoid.isOpen_units {M : Type*} [TopologicalSpace M] [Monoid M]
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{U : Submonoid M} (hU : IsOpen (U : Set M)) : IsOpen (U.units : Set Mˣ) :=
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(hU.preimage Units.continuous_val).inter (hU.preimage Units.continuous_coe_inv)
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/-- The monoid homeomorphism between the units of a product of topological monoids
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and the product of the units of the monoids. -/
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@[to_additive]
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def ContinuousMulEquiv.piUnits {ι : Type*}
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{M : ι → Type*} [(i : ι) → Monoid (M i)] [(i : ι) → TopologicalSpace (M i)] :
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(Π i, M i)ˣ ≃ₜ* Π i, (M i)ˣ where

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