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[Merged by Bors] - feat(GroupTheory/GroupAction/Group): invOf smul lemmas #9972

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[Merged by Bors] - feat(GroupTheory/GroupAction/Group): invOf smul lemmas #9972

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invOf smul lemmas
mans0954 Jan 24, 2024
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More invOf smul lemmas
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Merge branch 'master' into mans0954/smul-invOf
mans0954 Jan 24, 2024
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Remove duplicate instance
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Remove lemmas
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Reinstate lemma
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Explicit var
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Merge branch 'master' into mans0954/smul-invOf
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Update Mathlib/GroupTheory/GroupAction/Group.lean
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Only import Basic
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22 changes: 22 additions & 0 deletions Mathlib/GroupTheory/GroupAction/Group.lean
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Expand Up @@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Aut
import Mathlib.Algebra.Invertible.Defs
import Mathlib.Algebra.Invertible.Basic
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import Mathlib.GroupTheory.GroupAction.Units

#align_import group_theory.group_action.group from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
Expand Down Expand Up @@ -178,6 +180,26 @@ 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) _
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c and x should be explicit for these lemmas.

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Okay, done. Please can you explain why though so I know for next time.

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@eric-wieser eric-wieser Jan 26, 2024 •
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The simple answer here is that this lemma is almost identical to smul_inv_smul, and so you should try to match the argument explicitness too.

The slightly more general answer is that lemmas with = in their goal should ensure that every variable in their goal is captured in an an explicit variable (either as (x : X) or {x : X} (hx : P x)).


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 :=
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