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[Merged by Bors] - feat(Algebra/Module/Submodule/Pointwise): generalize singleton_set_smul #14723

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[Merged by Bors] - feat(Algebra/Module/Submodule/Pointwise): generalize singleton_set_smul #14723

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12 changes: 4 additions & 8 deletions Mathlib/Algebra/Module/Submodule/Pointwise.lean
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Expand Up @@ -3,11 +3,9 @@ Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Span
import Mathlib.RingTheory.Ideal.Basic

#align_import algebra.module.submodule.pointwise from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
Expand Down Expand Up @@ -494,14 +492,12 @@ lemma mem_set_smul (x : M) [SMulCommClass R R N] :
@[simp] lemma set_smul_bot : s • (⊥ : Submodule R M) = ⊥ :=
eq_bot_iff.mpr fun x hx ↦ by induction x, hx using set_smul_inductionOn <;> aesop

-- TODO: `r • N` should be generalized to allow `r` to be an element of `S`.
lemma singleton_set_smul [SMulCommClass R R M] (r : R) :
({r} : Set R) • N = r • N := by
lemma singleton_set_smul [SMulCommClass S R M] (r : S) : ({r} : Set S) • N = r • N := by
apply set_smul_eq_of_le
· rintro r m rfl hm; exact ⟨m, hm, rfl⟩
· rintro _ m rfl hm; exact ⟨m, hm, rfl⟩
· rintro _ ⟨m, hm, rfl⟩
rw [mem_set_smul_def, Submodule.mem_sInf]
intro p hp; exact hp rfl hm
intro _ hp; exact hp rfl hm

lemma mem_singleton_set_smul [SMulCommClass R S M] (r : S) (x : M) :
x ∈ ({r} : Set S) • N ↔ ∃ (m : M), m ∈ N ∧ x = r • m := by
Expand Down
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