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Equiv.lean
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Equiv.lean
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/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupWithZero.Units.Basic
/-!
# Multiplication by a nonzero element in a `GroupWithZero` is a permutation.
-/
assert_not_exists DenselyOrdered
variable {G : Type*}
namespace Equiv
section GroupWithZero
variable [GroupWithZero G]
/-- Left multiplication by a nonzero element in a `GroupWithZero` is a permutation of the
underlying type. -/
@[simps! (config := .asFn)]
protected def mulLeft₀ (a : G) (ha : a ≠ 0) : Perm G :=
(Units.mk0 a ha).mulLeft
theorem _root_.mulLeft_bijective₀ (a : G) (ha : a ≠ 0) : Function.Bijective (a * · : G → G) :=
(Equiv.mulLeft₀ a ha).bijective
/-- Right multiplication by a nonzero element in a `GroupWithZero` is a permutation of the
underlying type. -/
@[simps! (config := .asFn)]
protected def mulRight₀ (a : G) (ha : a ≠ 0) : Perm G :=
(Units.mk0 a ha).mulRight
theorem _root_.mulRight_bijective₀ (a : G) (ha : a ≠ 0) : Function.Bijective ((· * a) : G → G) :=
(Equiv.mulRight₀ a ha).bijective
end GroupWithZero
end Equiv