Skip to content

Commit

Permalink
feat(GroupTheory/GroupAction/Group): invOf smul lemmas (#9972)
Browse files Browse the repository at this point in the history
Give `smul` versions of some existing `mul` lemmas:

- `invOf_smul_smul`
- smul_invOf_smul (c.f. mul_invOf_self_assoc)
- `invOf_smul_eq_iff` (c.f. `invOf_mul_eq_iff_eq_mul_left`) (required for #7569)
- `smul_eq_iff_eq_invOf_smul` (c.f `mul_left_eq_iff_eq_invOf_mul`)



Co-authored-by: Christopher Hoskin <[email protected]>
Co-authored-by: Christopher Hoskin <[email protected]>
  • Loading branch information
3 people committed Jan 27, 2024
1 parent e4e10f9 commit 09b44c1
Showing 1 changed file with 23 additions and 0 deletions.
23 changes: 23 additions & 0 deletions Mathlib/GroupTheory/GroupAction/Group.lean
Original file line number Diff line number Diff line change
Expand Up @@ -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"
Expand Down Expand Up @@ -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 :=
Expand Down

0 comments on commit 09b44c1

Please sign in to comment.