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Commute.lean
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Commute.lean
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/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
Ported by: Frédéric Dupuis
! This file was ported from Lean 3 source module algebra.hom.commute
! leanprover-community/mathlib commit 6eb334bd8f3433d5b08ba156b8ec3e6af47e1904
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
import Mathlib.Algebra.Hom.Group
import Mathlib.Algebra.Group.Commute
/-!
# Multiplicative homomorphisms respect semiconjugation and commutation.
-/
section Commute
variable {F M N : Type _} [Mul M] [Mul N] {a x y : M}
@[to_additive (attr := simp)]
protected theorem SemiconjBy.map [MulHomClass F M N] (h : SemiconjBy a x y) (f : F) :
SemiconjBy (f a) (f x) (f y) := by simpa only [SemiconjBy, map_mul] using congr_arg f h
#align semiconj_by.map SemiconjBy.map
#align add_semiconj_by.map AddSemiconjBy.map
@[to_additive (attr := simp)]
protected theorem Commute.map [MulHomClass F M N] (h : Commute x y) (f : F) : Commute (f x) (f y) :=
SemiconjBy.map h f
#align commute.map Commute.map
#align add_commute.map AddCommute.map
end Commute