[Merged by Bors] - feat(GroupTheory/GroupAction/Group): invOf smul lemmas

Mathlib/GroupTheory/GroupAction/Group.lean

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes
 -/
 import Mathlib.Algebra.Group.Aut
+import Mathlib.Algebra.Invertible.Basic
 import Mathlib.GroupTheory.GroupAction.Units
 
 #align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
@@ -178,6 +179,28 @@ theorem smul_eq_iff_eq_inv_smul (g : α) {x y : β} : g • x = y ↔ x = g⁻¹
 
 end Group
 
+section Monoid
+
+variable [Monoid α] [MulAction α β]
+
+variable (c : α) (x y : β) [Invertible c]
+
+@[simp]
+theorem invOf_smul_smul : ⅟c • c • x = x := inv_smul_smul (unitOfInvertible c) _
+
+@[simp]
+theorem smul_invOf_smul : c • (⅟ c • x) = x := smul_inv_smul (unitOfInvertible c) _
+
+variable {c x y}
+
+theorem invOf_smul_eq_iff : ⅟c • x = y ↔ x = c • y :=
+  inv_smul_eq_iff (a := unitOfInvertible c)
+
+theorem smul_eq_iff_eq_invOf_smul : c • x = y ↔ x = ⅟c • y :=
+  smul_eq_iff_eq_inv_smul (g := unitOfInvertible c)
+
+end Monoid
+
 /-- `Monoid.toMulAction` is faithful on nontrivial cancellative monoids with zero. -/
 instance CancelMonoidWithZero.faithfulSMul [CancelMonoidWithZero α] [Nontrivial α] :
     FaithfulSMul α α :=